Controllable Few State Quantum Systems for Information Processing

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Controllable few state quantum systems for information processing

Jared Heath Cole BAppSc(AppPhys) BEng(Comm) (Hons) RMIT

Submitted in total fulﬁlment of the requirements for the degree of

Doctor of Philosophy

School of Physics The University of Melbourne Australia

October, 2006

The University of Melbourne Australia

ABSTRACT

Controllable few state quantum systems for information processing

by Jared Heath Cole

Chairperson of Supervisory Committee: Prof. G. N. Taylor School of Physics

This thesis investigates several different aspects of the physics of few state quantum systems and their use in information processing applications. The main focus is performing high precision computations or experiments using imperfect quantum systems. Specifically looking at methods to calibrate a quantum system once it has been manufactured or performing useful tasks, using a quantum system with only limited spatial or temporal coherence. A novel method for characterising an unknown two-state Hamiltonian is presented which is based on the measurement of coherent oscillations. The method is subsequently extended to include the effects of decoherence and enable the estimation of uncertainties. Using the uncertainty estimates, the achievable precision for a given number of measurements is computed. This method is tested experimentally using the nitrogen-vacancy defect in diamond as an example of a two-state quantum system of interest for quantum information processing. The method of characterisation is extended to higher dimensional systems and this is illustrated using the Heisenberg interaction between spins as an example. The use of buried donors in silicon is investigated as an architecture for realising quantum-dot cellular automata as an example of quantum systems used for classical information processing. The interaction strengths and time scales are calculated and both coherent and incoherent evolution are assessed as possible switching mechanisms. The effects of decoherence on the operation of a single cell and the scaling behaviour of a line of cells is investigated. The use of type-II quantum computers for simulating classical systems is studied as an application of small scale quantum computing. An algorithm is developed for simulating the classical Ising model using Metropolis Monte-Carlo where random number generation is incorporated using quantum superposition. This suggests that several new algorithms could be developed for a type-II quantum computer based on probabilistic cellular automata. This is to certify that

(i) the thesis comprises only my original work,

(ii) due acknowledgement has been made in the text to all other material used,

(iii) the thesis is less than 100,000 words in length, exclusive of table, maps, bibli- ographies, appendices and footnotes.

I authorize the Head of the School of Physics to make or have made a copy of this thesis to any person judged to have an acceptable reason for access to the information, i.e., for research, study or instruction.

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ACKNOWLEDGMENTS

There are a number of people, without whom this thesis would never have seen the light of day. First of all, I must thank both my official and unofficial supervisors. Steven Prawer for getting me excited about quantum computing in the first place, and for many useful discussions. Lloyd Hollenberg for his untiring support and as- sistance, both technical and professional, and for teaching me how to be a theorist. Cameron Wellard and Andrew Greentree for answering my many questions (especially the stupid ones) and for always being available to argue over technicalities. I also thank Salvy Russo who first introduced me to the world of theoretical physics and who has provided many a coffee and a chat.

During my candidature, I was able to visit several other institutions and received support and warm hospitality from all of them. Thanks must go to Frank Wilhelm and Jan von Delft at LMU München and Jason Ralph at the University of Liver- pool. I’m especially indebted to Torsten Gäbel, Fedor Jelezko and Jörg Wrachtrup at the University of Stuttgart for the beautiful experimental data which constitutes chapter 6. I must also thank Sonia Schirmer and Daniel Oi at the University of Cam- bridge for encouragement and technical support for all of the work on Hamiltonian characterisation, both in Melbourne and during my visit to Cambridge.

Closer to home, special thanks go to Simon Devitt for useful discussions, both technical and nontechnical, in many locations all over the world. Also, the inhab- itants of room 607, Vince, Damien and Joo Chew and the other members of the DMP group and the 6th ﬂoor. They made the School of Physics a fantastic place to work, where I always felt welcome, challenged and stimulated. It was also these people, ably assisted by the School of Physics pool table, who kept me sane ...... or at least limited the damage.

Finally I’d like to thank my family, without them I would not have the drive

v or conﬁdence to get through so many years of university. Sharna for the use of her hideaway for the dark days during writeup and Tahlia for proof reading all 200+ pages. My parents, without whom I would not be here (ﬁguratively and literally), Dad for encouraging my never ending questions about the world and Mum for never ending support. Last but certainly not least, I must thank Danielle for everything, for being both a source of strength and inspiration and for always being there. LIST OF PUBLICATIONS

During the course of this project, a number of public presentations have been made, which are based on the work presented in this thesis. They are listed here for reference.

REFEREED PUBLICATIONS J. H. Cole, L. C. L. Hollenberg and S. Prawer. An algorithm for simulating • the Ising model on a type-II quantum computer. Computer Physics Commu- nication, vol. 161, no. 1-2, pg. 18-26, 2004.

A. D. Greentree, J. H. Cole, A. R. Hamilton and L. C. L. Hollenberg. Coherent • electronic transfer in quantum dot systems using adiabatic passage. Physical Review B, vol. 70, no. 235317, 2004.

J. H. Cole, A. D. Greentree, C. J. Wellard, L. C. L. Hollenberg and S. Prawer. • Quantum-dot cellular automata using buried dopants. Physical Review B, vol. 71, no. 115302, 2005.

J. H. Cole, S. G. Schirmer, A. D. Greentree, C. J. Wellard, D. K. L. Oi and • L. C. L. Hollenberg. Identifying an experimental two-state Hamiltonian to arbitrary accuracy. Physical Review A, vol. 71, no. 062312, 2005.

J. H. Cole, A. D. Greentree, D. K. L. Oi, S. G. Schirmer. C. J. Wellard and • L. C. L. Hollenberg. Identifying a two-state Hamiltonian in the presence of decoherence. Physical Review A, vol. 73, no. 062333, 2006.

S. J. Devitt, J. H. Cole and L. C. L. Hollenberg. Scheme for direct measurement • of a general two-qubit Hamiltonian. Physical Review A, vol. 73, no. 052317, 2006.

J. H. Cole, S. J. Devitt and L. C. L. Hollenberg. Precision characterisation • of two-qubit Hamiltonians via entanglement mapping. Journal of Physics A, vol. 39, no. 47, pg. 14649-14658, 2006.

A. D. Greentree, C. Tahan, J. H. Cole and L. C. L. Hollenberg. Quantum • phase transitions of light. Nature Physics, vol. 2, no. 12, pg. 856-861, 2006.

vii REFEREED CONFERENCE PROCEEDINGS S. G. Schirmer, A. Kolli, D. K. L. Oi and J. H. Cole. Experimental Hamiltonian • identiﬁcation for qubits subject to multiple independent control mechanisms. Quantum Communication, Measurement and Computing, vol. 734, pg. 79-82, 2004.

J. H. Cole, A. D. Greentree, C. J. Wellard, L. C. L. Hollenberg and S. Prawer. • Measuring decoherence properties of charge qubits using buried donor cellular automata. SPIE Conference Proceedings, vol. 5650, pg. 96, 2005.

A. D. Greentree, J. H. Cole, A. R. Hamilton and L. C. L. Hollenberg. Scaling of • coherent tunneling adiabatic passage in solid-state coherent quantum systems. SPIE Conference Proceedings, vol. 5650, pg. 15, 2005.

L. C. L. Hollenberg, A. D. Greentree, C. J. Wellard, A. G. Fowler, S. J. Devitt • and J. H. Cole. Qubit transport and fault-tolerant architectures. Proceed- ings of the 2006 International conference on nanoscience and nanotechnology, Brisbane, Australia , 2006.

CONFERENCE ABSTRACTS J. H. Cole, A. D. Greentree, C. J. Wellard and L. C. L. Hollenberg. Quantum- • dot cellular automata using buried dopants. 5th Quantum Information Pro- cessing and Communication Workshop, Rome, Italy, 2004.

J. H. Cole, A. D. Greentree, C. J. Wellard and L. C. L. Hollenberg. Systematic • Hamiltonian identiﬁcation of two-level systems. Australian Institute of Physics - National Congress, Canberra, Australia, 2005.

J. H. Cole, A. D. Greentree, C. J. Wellard, S. G. Schirmer, D. K. L. Oi and • L. C. L. Hollenberg. Systematic Hamiltonian identiﬁcation of two-level systems with decoherence. CMI Summer School, Belfast, Northern Ireland, 2005.

J. H. Cole, A. D. Greentree, C. J. Wellard, S. G. Schirmer, D. K. L. Oi • and L. C. L. Hollenberg. Systematic Hamiltonian identiﬁcation of ﬁnite state systems. Sir Marcus Oliphant conference on the frontiers of quantum nanoscience, Noosa, Australia, 2006. LIST OF ABBREVIATIONS

The following acronyms are used throughout this thesis.

BDCA Buried Donor Cellular Automata CA Cellular Automata CMOS Complementary Metal-Oxide-Semiconductor DFT Discrete Fourier Transform EM Electromagnetic FFT Fast Fourier Transform FWHM Full Width Half Maximum MC Monte Carlo MLE Maximum Likelihood Estimation MPP Minimum Phase Point NMR Nuclear Magnetic Resonance NV Nitrogen Vacancy ODMR Optically Detected Magnetic Resonance QCA Quantum Cellular Automata QDCA Quantum-Dot Cellular Automata QED Quantum ElectroDynamics QFT Quantum Fourier Transform QIP Quantum Information Processing QPT Quantum Process Tomography QST Quantum State Tomography SEP Someone Else’s Problem T2QC Type-II Quantum Computer

CONTENTS

1 Introduction 1 1.1 Layout and publication of material in this thesis ...... 3

2 Background 5 2.1 (Quantum)informationprocessing ...... 5 2.2 Systemidentiﬁcation ...... 6 2.2.1 Spectroscopy ...... 7 2.2.2 Quantum state and process tomography ...... 9 2.2.3 Parameter estimation ...... 13 2.3 CA,QCAandQDCA ...... 13 2.4 Quantum-dotcellularautomata ...... 16 2.4.1 Theoretical work on QDCA ...... 17 2.4.2 Experimental QDCA demonstrations ...... 18 2.5 Type-IIquantumcomputers ...... 18 2.5.1 Algorithms for type-II quantum computers ...... 20 2.5.2 Eﬃciency of a type-II quantum computer ...... 21 2.5.3 Simulating a cellular automata system on a T2QC ...... 22

3 Characterising a two-state Hamiltonian 25 3.1 Characterising a two-state Hamiltonian ...... 25 3.2 The eﬀects of imperfect measurement ...... 29 3.3 EstimatingHusingFouriercomponents...... 30 3.4 Determiningtherelativephaseangle ...... 32 3.5 Chaptersummary...... 35

4 Including decoherence 37 4.1 Modellingtheeﬀectofdecoherence ...... 38 4.2 Puredephasing ...... 39 4.3 Including a more general decoherence model ...... 43 4.4 Imperfect measurement with decoherence ...... 47 4.5 Chaptersummary...... 49

5 Uncertainty estimation 51 5.1 Estimatingtheuncertainties ...... 52 5.2 Determining the precession frequency accurately ...... 54 5.3 Estimating the uncertainty in φˆ ...... 57 5.4 Estimating the uncertainty in the Hamiltonian parameters ...... 58 5.5 Uncertainties with decoherence ...... 59 5.6 Examplesimulations ...... 60 5.6.1 Accuracy of the uncertainty estimate ...... 61 5.6.2 Estimating the uncertainty with decoherence ...... 64

xi 5.7 Scaling behaviour of the uncertainty ...... 66 5.7.1 Scaling with no decoherence ...... 66 5.7.2 Scaling with decoherence ...... 68 5.8 Implications for single-qubit rotations in QIP ...... 69 5.9 Chaptersummary...... 71

6 Experimental Demonstration 73 6.1 TheNVcentre ...... 74 6.1.1 Eﬀective Hamiltonian ...... 76 6.1.2 Pumpingandreadout ...... 77 6.2 Experimentalsetup...... 78 6.3 Observation of Rabi oscillations ...... 79 6.4 Includingthehyperﬁneinteraction ...... 79 6.5 Hamiltonian characterisation ...... 81 6.6 Chaptersummary...... 86

7 Two-qubit characterisation 87 7.1 Characterisation of higher dimensional systems ...... 88 7.2 TheHeisenbergHamiltonian...... 90 7.3 Mappingtheentanglement ...... 93 7.4 Uncertainty estimation and gate errors ...... 97 7.5 Eﬀect of single qubit terms in the Hamiltonian ...... 101 7.6 Eﬀect of imperfectly prepared input states ...... 102 7.7 Chaptersummary...... 103

8 Buried Donor Cellular Automata 105 8.1 Quantum-dotcellularautomata ...... 106 8.2 Buried donors and the hydrogenic approximation ...... 107 8.3 EﬀectiveHamiltonian...... 108 8.4 BDCAswitching ...... 112 8.5 Incoherentswitching ...... 115 8.6 Chaptersummary...... 119

9 Coherent Buried Donor Cellular Automata 121 9.1 Coherentswitching ...... 121 9.2 Timedependantbehaviour...... 124 9.3 Theeﬀectofdephasing...... 125 9.4 Scalability of the buried donor scheme ...... 126 9.5 Chaptersummary...... 131

10 Simulating the Ising Model on a T2QC 133 10.1 Type-II QCs for simulating quantum cellular automata ...... 133 10.2 Metropolis simulation of the Ising model ...... 135 10.31DIsingmodel ...... 137 10.42DIsingmodel ...... 140 10.5 Streaming and parallelisation ...... 141 10.6 Possibleimplementations...... 142 10.7 Ensemblestreaming...... 143 10.8 Scalability of a type-II quantum computer ...... 146 10.9 Chaptersummary...... 147

11 Conclusions 149

A Quantum circuit formalism 155

LIST OF FIGURES

2.1 Illustration of a type-II quantum computer using NMR technology. The computer is comprised of a volume of solution containing many copies of an organic molecule, the spins of which can be addressed via magnetic resonance. The volume is partitioned using gradient magnetic ﬁelds and each region corresponds to a single node in the computer with each node consisting of many identical copies of the molecule...... 20

3.1 Bloch sphere representation of the state of a qubit (s) and its trajectory given an arbitrary Hamiltonian d. If the system is not in an eigenstate of the Hamiltonian, the state (given by the unit vector s) precesses around an axis deﬁned by d. The components of d are given by the Hamiltonian using Eq. (3.1) where the d gives the angular | |T precession frequency around the vector (dx,dy,dz) ...... 26 3.2 To map z(t), the system must be repeatedly initialised, allowed to evolve under the system Hamiltonian and then measured. To map the time evolution of the system, the Hamiltonian step is applied for progressively longer time intervals (i∆t for i = 1, 2,...,n) where ∆t is the minimum controllable time interval and tob = n∆t is the maximum time over which the system is observed...... 28 3.3 The left hand plot shows a numerical example of a sampled time signal z(t) = [cos(2πt) + 1]/2 with the number of ensemble measurements Ne set to (a) Ne = 1, (b) 2, (c) 8 and (d) 500 at each time point, where each measurement is a projection onto the (1,-1) axis. The corresponding DFT for each signal is shown on the right for ν 0, illustrating the signal-to-noise improvement as more measurem≥ents aretakenateachtimepoint...... 32

4.1 The z-projection of the time evolution of a two state system after preparation in the z = 1 state, for several different dephasing rates. The magnitude of the Hamiltonian (d = 1) is kept constant while the angle between the components is varied. (a) For θ = π/2, the oscillations decay exponentially. (b) With θ = π/4, the system undergoes a different evolution, but still decays to the mixed state z( ) = 0. Note the difference in scales on the vertical axes in (a) and (b).∞ . . . 40 4.2 The Fourier transform of the system evolution given in Fig. 4.1, in the presence of dephasing, plotted for ω 0. The magnitude of the ≥ Hamiltonian, d, is kept constant and results displayed for (a) θ = π/2, and (b) θ = π/4. The peak is reduced when θ = π/2, as expected, and the zero-frequency component increases. As the6 decoherence rate increases, the peaks broaden and the frequency shifts slightly. . . . . 41

xv 4.3 The evolution of a two-state system under the inﬂuence of both spontaneous absorption and emission, in the limit of large detuning (θ 0). For this example, the emission rate is ﬁve times that of spon-→ taneous absorption (Γ /Γ+ = 5). The path taken by the system − depends on which initial state is used, though the asymptotic behaviour is the same. The two paths are labelled z 1 and z1 depending − on whether the system is initialised in the ground, z(0) = 1, or excited state, z(0)=1,respectively...... − . . . . . 46

5.1 Example Fourier transform of a simulated measurement record, showing the Fourier component peaks clearly above the noise ﬂoor. The measurement record consists of Ns = 500 time points with Ne = 2 ensemble measurements, giving NT = 1000 total measurements. . . . 53 5.2 The left hand plot shows time signals which are truncated at various time points to produce a net phase diﬀerence of (a) ∆ϕ = π, (b) ∆ϕ = π/2 and (c) ∆ϕ = 0 between the start and end of the signal. The corresponding DFT for each signal is shown on the right, where the peak approaches a δ-function only for ∆ϕ 0...... 56 ≈

5.3 The test function P (tp) used to locate the point at which there is zero phase diﬀerence between the ﬁrst and last sample point. The amount of the time signal to use in the DFT is given by tp and the uncertainty is given by the FWHM of P (tp)...... 57 5.4 An example of the systematic reduction in the uncertainty of the Hamiltonian parameters as the number of measurements is increased. The error bars are given by three times the uncertainty estimate for each point and the solid line gives the ‘true’ value (Hr,x = 0.1, Hr,z = 0.05). The estimates are seen to converge to the true value as the number of measurements are increased...... 61

5.5 The distribution of for the estimated (a) Hr and (b) Hk over 5000 simulated runs. ForD these simulations, (a) 98.4% and (b) 98.7% of the estimates are found to lie within the average uncertainty interval (3δ ). The absolute uncertainty in Hk is greater than for Hr as more stepsD are required, giving a larger accumulated error...... 63

5.6 The distribution of the estimated measurement error (ˆη) for 5000 simulated runs. For this simulation, 99.5% of the estimates lie within the uncertainty 3δη...... 64 ± 5.7 Simulated data for evolution due to the example Hamiltonian, H = 0.1σx + 0.05σz. (a) The time series data for Ne = 100 (points) which approximates the ensemble average of the evolution (solid line). (b) The Fourier transform of the time domain data (points) plotted with the fitted function (solid line) using the estimated parameters in Ta- ble5.1...... 65 5.8 Error ellipses for the fit used to obtain the parameters in Table 5.1. The three dimensional ellipsoid has been projected down on to the three pairwise combinations of the parameters. The blue ellipse corresponds to the 3-sigma confidence interval whereas the red box corresponds to the outer boundaries used to compute the uncertainties in Table 5.1. The black circle indicates the parameter estimate while theblackcrossshowsthetruevalue...... 66

5.9 The average uncertainty δ of the estimate for the Hamiltonian H = Dr 0.1σx + 0.05σz as a function of total number of measurements. Each data point is the average of 10 simulation runs. The solid line shows 1/√NT where NT is the total number of measurements. As the total number of measurements increases, the overall precision to which the Hamiltonian is known increases. For a random measurement error, the achievable precision is reduced but still asymptotes to a scaling of one over the square-root of the number of measurements...... 67

5.10 The uncertainty estimate as a function of total number measurements for an example Hamiltonian undergoing pure dephasing in the bare qubit basis. Each data point is the average of 10 simulation runs. The fractional uncertainty is plotted for each of components of the Hamiltonian and the dephasing rate Γz, where Γz/d = 0.1 for this example. The average uncertainty measure, δ r, is also plotted for comparison with Fig. 5.9. The scaling is approximatelyD proportional to 1/√NT and the absolute fractional uncertainty is approximately independentofthedecoherencerate...... 69

6.1 The crystal structure of the NV centre showing the substitutional nitrogen adjacent to the vacancy defect in the unit cell of a diamond lattice...... 75

6.2 The energy levels of the NV centre illustrating the ground (3A) and excited (3E) states. The transition between these levels is optically active with a wavelength of λ 637nm. Alternatively, driving the system with a shorter wavelength≈ (λ< 637) results in a transition to higher levels, followed by a non-radiative transition down to the 3E state. The ground spin state of the centre is split due to zero ﬁeld splitting, while under nonzero magnetic ﬁeld the ms = 1 states are split. For these experiments, we will use the magnetic| ± spini levels m = 0 and m = 1 as our two level system...... 76 | s i | s − i 6.3 Confocal microscope image of diamond surface under laser illumination. The colour centre circled in red was used for all of the following measurements...... 79

6.4 The pulse sequence applied to obtain the Rabi oscillation data. . . . . 80 6.5 Example Rabi oscillation data for the on-resonance case, fMW = fres. The solid curve is a theoretical model with the system parameters determined using the method detailed in section 6.5. There is no beating due to the hyperfine interaction as the system is tuned to the central line, so the effects of the other two hyperfine lines cancel at each point in time. Note that the fluorescence level cannot be used to determine the state absolutely as the readout is not ‘single-shot’. . 80

6.6 Splitting due to the hyperfine interaction at finite magnetic field, 0 < B < 102.8mT, with the spin allowed transitions labelled in blue. For our purposes, we will treat transition c as our target transition to use as a qubit. Note that the spacing between levels is not to scale. . . . 81

6.7 ODMR spectra for an example NV centre showing the hyperﬁne splitting due to the 14N nucleus. The transitions a, b and c correspond to the three spin allowed transitions illustrated in Fig. 6.6...... 82

6.8 Example Rabi oscillation data for a system detuned from resonance, fMW = fres 40MHz. The beating of the oscillations is due to the hyperﬁne interaction− with the nitrogen nucleus. The decay envelope of the oscillations is enitrely due to this beating as the dephasing time of the system is 3.2µs, considerably longer than the period of observation. The solid line corresponds to a theoretical curve using the system parameters determined by ﬁtting in the Fourier domain. . 83

6.9 Example Rabi oscillation data at resonance, fMW = fres (top). The Fourier transform of this data (bottom) with the ﬁt obtained using the Levenberg-Marquardt algorithm and Eq. 6.2...... 84

6.10 Effective detuning as measured from Rabi oscillation data, as a function of detuning as measured from the microwave generator. Error bars indicate the 1-sigma confidence level obtained from the fitting process. Notice that the errors become larger for large detuning as the visibility of the oscillations is reduced. The solid line shows the expected linear behaviour while the dashed lines indicate the detuning from the other two hyperfine levels. The insert shows the residuals of the data when compared to the expected behaviour which show a systematic trend above 20MHz. Also, note that the effective detuning should not reach zero as the system is always detuned from at least twoofthethreehyperfinelevels...... 85

6.11 Effective interaction strength (Ω) as measured from Rabi oscillation data, as a function of detuning as measured from the microwave generator. Error bars indicate the 1-sigma confidence level obtained from the fitting process. Notice that the errors become larger for large detuning as the visibility of the oscillations is reduced. The dashed line shows a weighted linear fit to the data, giving a value of Ω = 27.43MHz while the insert shows the residuals of the fit...... 86 7.1 Plot of the sampled entanglement, C2(t), as a function of time (in inverse units of the Hamiltonian energy) for the input states given in table 7.3, for an example Hamiltonian H = 1.2XX +0.6YY +1.4ZZ. Each time point is the average of Ne = 10 measurements and there are Ns = 200 time points. In each case the observation time has been chosen to obtain approximately the same number of sample points per oscillation period, independent of input state...... 97 7.2 Discrete Fourier transform of the data shown in Fig. 7.1 for different input states. From the position of the peaks, the values of the Hamiltonianparameterscanbedetermined...... 98 7.3 The uncertainty in the Hamiltonian parameters as a function of the total number of measurements NT = 2Ns +2Ne, obtained from Eq. 7.16. The curves are plotted for initial values of Ns = 10 and Ns = 100, for increasing Ne. The right hand axis shows the effective probability of a discrete gate error (peff ) for the Ising case (the Heisenberg case differs by a factor of 3/4)...... 101

8.1 Two possible states for a basic QDCA cell where the 0 and 1 states constitute the ground or ‘computational’ states and ‘e’ labels the po- sitionoftheelectrons...... 106 8.2 Layout for a QDCA wire (a) and inverter (b) which demonstrate information transfer and binary inversion respectively after Tougaw andLent[TL94]...... 107 8.3 (a) The ground states of the buried donor BDCA cell where the positions of the electrons ‘e’ are designated by top (T) and bottom (B). These computational states are referred to as T B and BT , and are assigned the logical values of 0 and 1 respectively.| i (b) The| excii ted or ‘non-computational’ states are labelled T T and BB respectively and correspond to the ﬁrst excited state| of thei system.| i ...... 109

8.4 Energy difference between the ground (Egs) and first excited (Eex) states of a square BDCA cell made up of phosphorous donors in silicon. The energy difference is computed for various donor separations (R) by numerically integrating the Schrödinger equation. The points are full quantum mechanical calculations using the LCAO approach and the solid line is the energy difference determined analytically from simple electrostatic arguments, Eq. (8.9)...... 112 8.5 Simplified layout of a BDCA chain, where the cell size is labelled R and the cell spacing (S) is the distance between the centre of neighbouring cells. The circles represent the position of the donors. The position of the electrons ‘e’ is shown for the ground state configuration which corresponds to some non-zero bias on the control gates with the labelled polarity. The position of the electrons are measured withsingle-electrontransistors(SET)...... 113 8.6 Illustration of the BDCA model used in this and the subsequent chapter with the electron densities plotted for the ground state, given the gate polarities shown. The symmetry and barrier surface gates are used to perform switching and control the tunnelling rate respectively. The charge density of the electrons is computed from the wavefunction corresponding to one of the computation states of the cells. . . . 113

8.7 Eigenspectrum for a four donor cell as the symmetry potential (ES) is swept from -2 to 2 (meV) with no applied barrier potential (EB = 0) and a cell size R =15nm...... 115 8.8 Energy level diagram for a single BDCA cell in the presence of a well localised neighbouring cell, where the cell size R = 15nm and there is a cell spacing S = 30nm. The direct transition ( BT T B ) is suppressed as the interaction is phonon mediated and| musti → theref | iore proceed via single-electron transitions. The two (ﬁrst order) decay paths from one computational state to the other are illustrated with their associated transition rates for absorption (A) and emission (E). 117 8.9 Incoherent switching time calculated for a range of cell sizes (R) with the spacing between the neighbouring cells given by the cell centre- to-centre distance S = 2R. Higher operating temperatures result in faster switching times but also result in higher excited state populations,reducingtheoverallﬁdelity...... 118

9.1 Eigenspectrum for a four donor cell as the symmetry potential (ES) is swept from -2 to 2 (meV) with an applied barrier potential (EB) of 1.2meV ...... 123

9.2 The symmetry (ES) and splitting (EB) potentials which are applied to the system to achieve adiabatic evolution, where tp is the time over which the pulse is applied and the standard deviation of the pulses (ς) is set to tp/6. Bmax and Smax are the maximum barrier and symmetry potentials and are used to control the amount of tunnelling and localisation respectively...... 124 9.3 Eigenspectrum for a four donor cell as the pulse scheme given in Eq. (9.3) and (9.4) is applied to the barrier and symmetry gates respectively, where Smax = 2meV and Bmax = 1.2meV...... 125 9.4 Population of states as a function of time showing complete population transfer from the state T B to state BT while only transiently populating the non-computational| i states (| BBi and T T ). The time | i | i over which the pulse sequence is applied is tp = 100ps and the eﬀects ofdecoherenceareignored...... 126

9.5 Final population of states as a function of total pulse time (tp), ignoring the eﬀects of decoherence. High ﬁdelity transfer ( 99.95%) ≥ between computational states is observed for pulse times greater than 20ps. For pulse times of less than 0.1ps the system does not have time to evolve from its initial state. Between these times, the non- computational states are partially occupied...... 127 9.6 Probability of successful transfer as a function of both total pulse time and dephasing time. The region of 99% successful transfer is enclosed by the dotted contour line in the≥ bottom right corner. . . . . 128

9.7 Scaling behaviour for the maximum barrier coupling (Bmax) which can be applied while still maintaining the minimum energy gap at the centre of the eigenspectrum (the degeneracy point). Bmax is calculated for increasing numbers of donors pairs and then ﬁtted to an exponentialfunction...... 129

9.8 Scaling behaviour of tadiab as a function of the number of donor pairs in a BDCA chain, for various cell sizes (R). This gives an estimate for the scaling of the minimum allowable evolution time for high ﬁdelity transfer...... 130

10.1 Design of a Type-II Quantum Computer, composed of many small conventional quantum computers ...... 134 10.2 Mapping between sites on an Ising spin lattice (a) and nodes of a T2QC (b), showing the nearest neighbour assignments for a one di- mensionallattice...... 137 10.3 Quantum circuit for the evolution of the on-site spin S for the 1D Isingmodel ...... 138| i 10.4 Simpliﬁed circuit for the 1D Ising model, due to Miakisz [MPS04] . . 139 10.5 Quantumcircuitfor2DIsingmodel...... 140 10.6 Mapping between sites on an Ising spin lattice (a) and nodes of a T2QC (b), showing the nearest neighbour assignments for a one di- mensionallattice...... 140

10.7 Each node is initialised with the on-site spin Si and copies of the neighbouring on-site spins Si 1 and Si+1 ...... 141 − 10.8 The checkerboard update scheme with the spin value Si,j and its four nearest neighbours A = Si,j+1, B = Si 1,j, C = Si,j 1 and D = Si+1,j − − illustrated ...... 142 10.9 Maximum allowable error rate required for a temperature resolution of δT = 0.1, as a function of Ising lattice temperature. The critical temperature for the 2D lattice is plotted to illustrate the region of interest. As the temperature approaches zero, the required accuracy increases exponentially and therefore the allowable gate error decreases exponentially. In this regime, the lattice displays trivial behaviour which is independent of temperature and therefore insensitive togateaccuracyfluctuations...... 144 10.10Magnetisation as a function of temperature for the 2D Ising model. The ‘Exact’ solution is the analytic solution for an infinite lattice, ‘Mean field’ is the mean field approximation for an infinite lattice, ‘MC’ is a classical Metropolis monte-carlo run using a 1000 1000 lattice and ‘Ensemble T2QC’ is a simulated run of the Ising algor× ithm for a T2QC using ensemble streaming on 4 nodes ...... 146 A.1 Example of a quantum circuit showing example single qubit unitaries and multiqubit gates acting on the initial state q1 q2 q3 q4 . The gate ordering in time is indicated by horizontal| i⊗| gatei⊗| posii⊗|tion,i with computation proceeding from left to right...... 156 A.2 Notation for a CNOT gate and the truth-table showing classical operation...... 157 A.3 Notation for a Toffoli gate and the truth-table showing classical operation...... 158 A.4 (a) Alternative notation for a NOT gate. (b) Notation for a swap gateand(c)adestructivemeasurement...... 159 A.5 Notation for a CNOT gate where the target bit is inverted if the controlbitisequalto0...... 159 LIST OF TABLES

5.1 Example values from a simulated run of the ﬁtting procedure discussed above, using the data shown in Fig. 5.7. The true value (x), its estimate (ˆx), the uncertainty (δx) and the fractional uncertainty (δx/xˆ) are given for the three system parameters d, θ and Γz with Ns = 500 and Ne =100...... 66 6.1 Experimentally measured parameters for the Hamiltonian given in Eq. 6.1 for the magnetic spin levels of the NV centre, as given in reference[HMF93a]...... 77

7.1 The analytic form of the various measurement probabilities for a system evolving due to a general Heisenberg Hamiltonian, given different initial states. After evolution, the system is measured in the ZZ basis. 92 7.2 The analytic form of the time evolution for the various measurement probabilities (in the XZ basis) for particular input states. The other input states obtained by applying bit flips to each of the input states resultinswappingthesineandcosineterms...... 93 7.3 The analytic form of the entanglement generated by Eq. (7.10) for fourdifferentinputstates...... 95

10.1 Truth-table for the quantum circuit in 10.3, showing the equivalence to the classical Metropolis algorithm once the superposition state S0 ismeasured ...... 139| i

xxiii

Chapter 1

INTRODUCTION

One of the great technological triumphs of the 20th century has been the ability to process binary information quickly and accurately. This development would not have been possible without the advent of modern electronics and ultimately computers. Inextricably linked to this technological leap, was the conceptual leap provided by the advent of quantum mechanics, the overarching laws of nature at the micro-scale. As we enter the next century, one of the great goals of both physics and engineering is to find ways of combining these two great advances. Exploiting the strange and counter-intuitive realm of quantum mechanics to perform computing faster and more efficiently than ever before and even to tackle problems which are not tractable on conventional computers. While there has been much theoretical work on controllable quantum systems, it is often assumed that high precision, repeatable fabrication or incredibly large numbers of identical quantum systems are available. Instead, this thesis investigates what happens if the world is not perfect. How can we calibrate imperfectly manufactured systems? What can we do if we only have small numbers of qubits or even fleeting amounts of coherence? The central theme of this thesis is therefore, ‘what does quantum mechanics allow us to do in an imperfect world and how can we make devices to take advantage of this?’ Initially we investigate methods of characterising few state quantum systems for use as qubits in quantum computing. Without accurate information about the devices we are constructing, it will be impossible to use these devices for high precision computing tasks. We present a method called Hamiltonian characterisation which allows the measurement of a quantum system given minimal assumptions about controllability. This provides an alternative characterisation technique to the traditional methods such as spectroscopy and quantum state and process tomography.

1 2 Chapter 1. INTRODUCTION

The construction of large scale quantum computing devices requires techniques to allow assembly line manufacture of qubits, including all system identiﬁcation and calibration steps necessary to operate the computer. If we are to construct these machines using existing semiconductor fabrication techniques, we must be able to account for fabrication variation within and between diﬀerent devices. It is with this in mind that we investigate methods such as Hamiltonian characterisation which require a minimum number of measurements and almost no human intervention.

Chapters 3-7 introduce the concept of Hamiltonian characterisation and explores the application of this technique to parameter estimation for few-state systems. A detailed procedure is developed for identifying an arbitrary two-state Hamiltonian from minimal initial information. This technique is then extended to include the eﬀects of decoherence and provide uncertainty estimates on all the parameters of interest. Experimental results are then presented in chapter 6 which demonstrate the use of this technique on a two-level system comprised of electronic spin states of a defect centre in diamond. In chapter 7 an extension of the Hamiltonian characterisation technique to higher dimensions is considered, using the Heisenberg interaction between two qubits as an example.

In the second part of this thesis we investigate alternative methods of computation, where some amount of quantum coherence provides an advantage when performing classical computing tasks. This is in contrast to conventional quantum computing, which is the use of quantum systems to perform computing tasks involving information which is inherently quantum in nature. This provides both an interesting study of the controllability of quantum systems and provides an interim goal within the continued eﬀort to build a fully coherent quantum computer.

In chapters 8 and 9, quantum-dot cellular automata are investigated, specifically the implementation of these devices using single phosphorous donors in silicon. These devices constitute an experimental realisation of a classical cellular automata rule, where the physics of quantum-dots and the electrostatic interaction both define the states and the evolution rules of the system. The possibility of both incoherent and coherent operation is investigated and we estimate both switching times and scaling parameters. Chapter 10 then considers a type-II quantum computer 1.1. LAYOUT AND PUBLICATION OF MATERIAL IN THIS THESIS 3 as an example of a hybrid quantum-classical computer. An algorithm is presented which allows the simulation of the classical Ising model on a few qubit type-II quantum computer and we discuss the implementation of this algorithm on an existing hybrid machine. The algorithm presented is a specific case of a probabilistic cellular automata rule, the probabilistic aspect of which is incorporated using quantum superposition and the measurement hypothesis.

1.1 Layout and publication of material in this thesis

The material in this thesis is organised by topic, rather than chronologically, to improve the clarity of the presentation. All the work in this thesis is original, except for the background material in chapter 2 and at the start of each chapter. All the material presented, except for chapter 6, has previously appeared in refereed journal articles. In chapter 2 we provide a brief overview of the central themes of the thesis and review the relevant literature. The concept of Hamiltonian characterisation is presented in chapter 3 and this process is applied to the problem of identifying an unknown two-state Hamiltonian. A model for decoherence is introduced in chapter 4 and the characterisation method is extended to allow the identification of phenomenological decoherence parameters. Chapter 5 investigates the computing of uncertainties in the characterisation process. The material in chapters 3-5 has been published in references [CSG+05] and [CGO+06]. In chapter 6, experimental results are presented using the nitrogen-vacancy colour centre in diamond as an example two-level system. These results are in excellent aggreement with the theoretical analysis of the earlier chapters and represent the first experimental demonstration of Hamiltonian characterisation. Chapter 7 extends the analysis of chapters 3-5 to the case of higher dimensional systems and looks at the characterisation of the Heisenberg Hamiltonian as an example of a two-qubit Hamiltonian. The material in this chapter has been published 4 Chapter 1. INTRODUCTION in reference [CDH06] and [DCH06]. The concept of a quantum-dot cellular automata using phosphorous donors in silicon is presented in chapter 8 and the relevant time and energy scales calculated. The use of coherent evolution to switch a chain of QDCA cells is then presented in chapter 9 and detailed simulations are performed using the energies appropriate for phosphorous donors in silicon. The material of chapters 8 and 9 has appeared in reference [CGW+05a] and [CGW+05b]. Finally, in chapter 10 a novel algorithm for a type-II quantum computer is presented. This algorithm uses quantum superposition and projective measurement to simulate the Ising model, using the Metropolis Monte-Carlo method. This algorithm uses fewer quantum bits to simulate the Ising model than is possible using classical bits. This algorithm and most of the material in this chapter has appeared in reference [CHP04]. Since publication of that work, a refined version of the one dimensional algorithm circuit has been presented in reference [MPS04] which is also presented here, with further refinements, for completeness. Chapter 2

BACKGROUND

In this chapter we present a broad overview of the major topics covered in this thesis, both to provide background on these topics and to highlight the links between them. A brief review of the current literature is also included for each of the main topics. At the beginning of each chapter, this review is then extended and used to present the material in each chapter in context.

2.1 (Quantum) information processing

This thesis is primarily concerned with the use of quantum systems for both classical and quantum information processing. In the former case, we are interested in new ways of performing traditional computation using nano-scale devices whose operation must be described using the laws of coherent quantum mechanics. The investigation of alternative forms of classical computation promises advantages such as increased miniaturisation, higher bit densities and reductions in power consumption and heat generation [Gra98]. In the later case of quantum information processing, the information processed is quantum as well, utilising the concepts of superposition and entanglement to perform computing tasks which are diﬃcult using conventional information processing [Sho97]. It is this use of quantum coherent systems to perform quantum information processing which has attracted great interest and lead to a global eﬀort to build a quantum computer [NC00]. The basic components of a quantum computer, like a classical computer, are (qu)bits and gates (unitary operations). Using the circuit model for quantum computation, we can employ the action of gates upon a set of qubits (combined with appropriate measurement) to realise the computation. While there are a number of other models for quantum computation, including cluster state [RB01a], topologi-

5 6 Chapter 2. BACKGROUND cal [Kit03], adiabatic [FGGS00] and globally addressable [Ben00] quantum computing, in this thesis we will be primarily concerned with the circuit model. Much of the motivation for the early chapters of this thesis is the identification and characterisation of finite state Hamiltonians for quantum information proces- sion (QIP) applications, so the language is often that of quantum computing, though the topics covered are also applicable to the general study of controllable quantum systems. The remainder of this thesis focuses on the use of quantum systems for classical information processing, though concepts and techniques from quantum information theory, such as qubits, are often employed. In its simplest form, a qubit is a two-level quantum system, the state of which can be manipulated by controlling the system Hamiltonian. Single qubit gates are realised by manipulating the Hamiltonian of a given qubit while two qubit gates are often realised using a controllable interaction between the qubits. In this way, the qubits can be prepared in any given state and then two qubit gates are used to entangle qubits together, providing the necessary interference effects for non-trivial quantum computation. A brief review of single and two qubit gates and the quantum circuit formalism is given in appendix A.

2.2 System identiﬁcation

Methods for writing down the Hamiltonian description of a physical system in classical mechanics have been known for over 170 years [Ham34, Ham35]. The concept of a Hamiltonian came into its own when first Heisenberg [Hei25, BJ25, vdW67] and then Schrödinger [Sch26a, Sch26b, Sch78] used it to formulate quantum mechanics as we know it. Central to the solution of almost any non-relativistic quantum mechanical problem is the form of the Hamiltonian. This is particularly important with the advent of quantum computing, where the manipulation of the state of a quantum system via controllable Hamiltonians has given birth to the field of quantum information processing [NC00]. When performing precision quantum control experiments using ‘designer Hamil- tonians’ for QIP applications, an accurate knowledge of the system is crucial. This, 2.2. SYSTEM IDENTIFICATION 7 in turn, requires accurate and reliable methods for determining the system Hamil- tonian. The traditional mode of operation for experimental quantum mechanics has been to construct a Hamiltonian model, predicting various parameters of interest from this model and then comparing these to experimental results. As we move to higher and higher precision experiments, it is instructive to look for more efficient and systematic methods to measure the system parameters directly, rather than via this retrodictive process. In quantum computing, the system parameters are regularly measured using a range of old and new techniques which have their own terminology as well as strengths and weaknesses. In this chapter we will briefly discuss some of the key techniques, namely spectroscopy, quantum state tomography, quantum process tomography and parameter estimation. In subsequent chapters we present a novel technique we term Hamiltonian characterisation, which allows for the direct measurement of Hamiltonian and decoherence parameters for a finite state, quantum coherent system. This allows us to invert the traditional approach to experimental quantum physics experiments and measure the system directly. This knowledge can be then used to conduct more sophisticated experiments, such as those required for quantum information processing.

2.2.1 Spectroscopy

The oldest and most common method for studying a quantum mechanical system experimentally is via spectroscopy. This ﬁeld is large and varied (see for instance [Cha70, SW76]) and so will not be covered in any great depth, though a brief discussion is appropriate as it is the simplest and often most accurate way to determine the structure of a quantum system. While there are a variety of methods and techniques which come under the heading of spectroscopy, they all include the interaction of electromagnetic radiation with matter in some form. This electromagnetic radiation can come from any region of the spectrum, from far-infrared or radio-frequency through to X-ray and gamma ray spectroscopy. The quantum mechanical system can consist of electronic, nuclear, rotational, vibrational, orbital or spin degrees of freedom. The common element is the determination of the en- 8 Chapter 2. BACKGROUND ergy separation between eigenstates of the quantum system via the absorption or emission of photons. The energy of these photons,

hf =∆E = E E , (2.1) b − a

determines the diﬀerence between the energy levels Ea and Eb and thus provides a direct probe of the energy level structure.

Spectroscopy has a number of advantages for studying few state quantum systems. It can measure many transitions at once, is very accurate and is often the easiest experiment to conduct. A disadvantage of spectroscopy for few state systems is that, in its simplest form, it is only sensitive to the spacing between energy levels and therefore phase effects can be hidden and therefore not all components of the Hamiltonian can be reconstructed. More sophisticated spectroscopic experiments involving multiple energy levels or driving frequencies are often required to fully identify a given system Hamiltonian. Spectrocopy also requires the generation and detection of a electromagnetic (EM) field and the system must couple to the field to be detectable, which is especially important when working at the level of a single quantum system. These issues are particularly crucial when constructing qubits in the solid-state, for these systems often do not use an applied EM field to mediate the transitions. This means to perform a spectroscopy experiment requires addition apparatus to generate and detect the appropriate fields.

In order to study solid-state systems spectroscopically before using them for quantum computing, a method of coupling an EM ﬁeld must be included in the architecture and even then, it may not provide the necessary phase information. If the system is then to be scaled up to 100’s or 1000’s of qubits and each one must be coupled to the EM ﬁeld separately for calibration purposes, this negates the advantages of miniaturisation and mass manufacture which are inherent to solid- state architectures.

There are an almost uncountable number of papers on the topic of spectroscopy so we will just highlight several of particular interest to quantum computing in the solid state. It is common practice to analyse a two-state system comprised of 2.2. SYSTEM IDENTIFICATION 9 a cooper-pair box using microwave spectroscopy [NCT97, vdWtHW+00, WSB+04, BXR+03, XSD+05]. Spectroscopic methods have also been applied to transient voltage/current measurements in quantum dots [OFvdW+98, FAT+02] and proposed as a characterisation method using oscillating control signals for superconducting qubits [GRGC06]. In addition, the nitrogen-vacancy colour centre in diamond (which will be discussed as an example two-state system in chapter 6) has been studied extensively, both in bulk and single centre samples, using a combination of electron-spin, optical and magnetic-resonance spectroscopy. Spectroscopy has so far been the dominant characterisation technique for quantum computing research, even in the solid-state but as we move to assembly line construction of qubits, we need to look for other methods of characterisation, especially for systems which don’t have a native electromagnetic interaction.

2.2.2 Quantum state and process tomography

The use of tomographic methods to reconstruct the quantum state of a system was ﬁrst proposed by Fano in 1957 [Fan57]. The basic concept is analogous to medical tomography where one or two dimensional slices of data, taken from a range of diﬀerent directions, are used to reconstruct a two or three dimensional image. In quantum state tomography, measurements of the system in various bases are used to reconstruct the entire state of the system. Of course, due to the restrictions of the Heisenberg uncertainty principle, these measurements cannot be achieved simultaneously. The no cloning theorem further restricts our ability to copy the state [WZ82] so we generally speak of performing tomography on multiple identically prepared states. In principle, given an unlimited number of such identically prepared states, the state of the system could be reconstructed with arbitrary precision, given a suitable set of measurement bases. This is termed quantum state tomography or QST. To reconstruct a process or operator (rather than a single state), state tomography must be repeated on the output state resulting from the application of the operator in question upon a set of input states. This process of reconstructing the operator which acts on an arbitrary state is called quantum process tomography or QPT. 10 Chapter 2. BACKGROUND

The ﬁrst experimental demonstrations of quantum state tomography involved the reconstruction of the quantum state of photons using homodyne or intensity detection [VR89, SBRF93, RBM94]. A good review of both the historical development and the current state of the art is given in [PR04]. As we are primarily concerned with the use of tomographic methods for QIP applications, we will concentrate on more recent developments in the tomography of qubits.

When performing quantum state tomography on a single qubit, identically prepared copies of the state are measured in three linearly independent (though not necessarily orthogonal) bases. Using the averages of these measurements, the complete density matrix of the state can be reconstructed. In principle, if we assume an inﬁnite number of measurements and that the measurement process is error free, then the reconstructed density matrix is a faithful representation of the quantum state being measured. In practice this is not possible, even with error free measurement, due to the discrete nature of a projective measurement in quantum mechanics. The accuracy of the reconstructed state is therefore dependent on the accuracy of the averaged measurement results, which in turn depends on the number of measurements. A good discussion of this procedure as applied to qubits is given in chapter 4 of [PR04] and in [NC00].

The naive tomographic approach discussed so far is quite susceptible to errors, both in the measurement process and in the reconstruction (due to ﬁnite counting statistics). If the reconstruction process is applied blindly to experimental data, often density matrices are constructed which are unphysical and therefore more sophisticated methods are required, such as Monte-Carlo, Bayesian or maximum entropy methods [PR04]. An approach which has been used extensively in recent experiments on qubit tomography is the method of maximum likelihood estimation (MLE) [Hra97, PR04]. This method requires a parameterised density matrix which is physically legitimate by construction. A likelihood function is then constructed which can be maximised with respect to the free parameters. This results in a physically allowable density matrix which represents the closest match to the experimental data.

Quantum process tomography was ﬁrst suggested as a means of ‘black-box’ char- 2.2. SYSTEM IDENTIFICATION 11 acterisation for quantum computing by Poyatos et al. and independently by Chuang and Nielsen [PCZ97, CN97] and has subsequently been reﬁned and improved by several authors, see for example [JKMW01, DLP01, PR04]. The focus in this case was the reconstruction of some unknown process or black-box which acts on an arbitrary state. This was immediately applied to the problem of reconstructing gates for quantum computation and is presently the most common method used for this pur- pose. The concept of QPT follows directly from state estimation in that a series of carefully chosen input states [PCZ97] are prepared and the unknown gate is applied to each in turn. The resulting density matrix of the output state is reconstructed for each input state and from these results, the underlying operator can be reconstructed. As for QST, measurement errors can lead to unphysical reconstructions but these can also be eliminated by using estimation techniques such as maximum likelihood.

To date, quantum state and process tomography has been demonstrated on a range of experimental systems. It is especially suited to photon based schemes where quantum gates are performed using macroscopic components such as phase plates and beam splitters. This allows the preparation of arbitrary inputs states and more importantly measurement in arbitrary bases, making single qubit tomography of photonic qubits a straightforward and reliable procedure. The reconstruction of two-qubit gates is more difficult but also benefits from the discrete nature of photonic gates and the ease of state preparation and measurement in a photonic system. For this reason, the most accurate and repeatable QPT of a CNOT gate demonstrated to date is using linear optics [OPG+04]. Other systems in which QST or QPT have been demonstrated include qubits based on ions confined in electromagnetic traps [LMK+96, RRH+04], nuclear magnetic resonance [NKL98, CCL01], superconducting devices [SAM+06], colour centres in diamond [HTW+06] and even vibrational modes of atoms in an optical lattice [MFMS05].

There are both advantages and disadvantages to using tomographic methods to characterise and fine-tune systems for QIP. In the early developmental stages, a good estimate of the density matrix is vital as not only does this tell you about the precision of the gate but it allows the effect of decoherence to be estimated. In fact 12 Chapter 2. BACKGROUND by performing QST on a system undergoing free evolution, an effective decoherence model can be constructed directly [BHPC03, HTW+06]. For these reasons, the experimental reconstruction of an operator via QPT has become the default proof that you have in fact realised a quantum gate. Even in recent experiments on the generation of multi-ion entangled states in ion traps, it is the QST reconstruction which is used to demonstrate the existence of large scale entanglement [HHR+05, LKS+05].

Despite its prevalence in the literature, QPT has some disadvantages. The more obvious ones include the enormous amount of experimental data and computing resources required to completely reconstruct a multi-qubit operator. As the number of qubits increases, the number of measurement bases and input states required increases exponentially.

For many quantum computing architectures, the gates are not discrete but depend strongly on the pulse shape applied to control ﬁelds or a laser. This means QPT must be performed many times at diﬀerent pulse shapes or times until an optimal setting is found. During this process, the reconstructed operator does not necessarily indicate how these pulses must be changed to improve the required gate and so a search of the parameter space must be performed.

Another, more subtle restriction of QPT is that various input states and measurement bases are required. In many architectures, each qubit only has a single readout/initialisation channel and therefore single qubit operations are required to achieve the necessary set of basis measurements. This is difficult to achieve when it is the single qubit operations which you wish to characterise first. This difficulty does not arise when using ions in traps as the ions can be studied extensively using spectroscopy. Once one ion is characterised, all other ions of the same species be- have the same. In solid state systems, not only is spectroscopy not always available, but the characteristics of each qubit can vary wildly due to fabrication variation. This means that some form of calibration is essential for each qubit, especially when considering scaling up fabrication to 10’s or 1000’s of qubits. 2.3. CA, QCA AND QDCA 13

2.2.3 Parameter estimation

Closely linked to the previously discussed methods is the general approach of quantum parameter estimation [Hel76, Hol82]. The basic concept of parameter estimation is to take a known state and look at the action upon this state of an unknown process. The process or system in question is a function of some unknown parameters, the value of which can be estimated by measurement of the state after it is acted upon. Using this definition, it is obvious that, in fact, quantum state and process tomography can be thought of as specific application of parameter estimation. This is also true of some of the work presented in this thesis (chapters 3-7), though the approach and application here are novel. A more recent application of parameter estimation is in the identification of the parameters of a cavity QED experiment using single ions and continuous measurement, investigated by Gambetta and Wiseman [GW01]. In their system, they assumed some parameters of the system Hamiltonian were unknown and then analysed various measurement modes which would allow the estimation of these parameters. Reference [KWR04] contains a detailed discussion on the concepts of quantum state and process tomography, state estimation and Hamiltonian parameter estimation, and the links between them. This work was particularly focused on the reconstruction of photonic states and the use of MLE for these techniques. There is a vast amount of literature on both classical and quantum system identification, though the accuracy and efficiency of these techniques is strongly dependent on the particular application at hand.

2.3 Cellular automata, quantum cellular automata and quantum-dot cellular automata

The concept of a cellular or finite state automata (CA) was first discussed by von Neuman in 1948 as a result of his investigations into methods of modelling the function of the human brain [vNB66, vNAB87]. In its simplest form, a cellular automaton is a series of cells or sites where each cell can take on a finite set of values. 14 Chapter 2. BACKGROUND

As the calculation progresses, the value of a particular cell at each time-step is a function of its previous value and the value of its neighbouring cells. In general the rule used to determine the state of the cells at each time step is the same for every cell and is therefore a global rule. The programmability of the system then stems from the choice of this rule and from the initial state of the cells before computation is initiated. For a one dimensional system, this deﬁnition of a CA rule can be expressed as [Wol86]

ai0 = φ(ai 1,ai,ai+1), (2.2) − where ai is the state of the ith cell, ai0 is the new state of the cell and φ is the boolean function used to describe evolution of the system. In general, CA rules have the following deﬁning characteristics, as stated by Wolfram [Wol86],

1. Cell distribution is discrete in space.

2. Evolution is discrete in time.

3. Each cell consists of a discrete number of states.

4. Cell type and placement is homogeneous.

5. Cells are updated in a synchronised fashion.

6. The update rule is deterministic.

7. The update rule is spatially local.

8. The update rule is temporally local.

However, not all authors restrict themselves to these definitions. For instance, the global update rule can be probabilistic and in fact, this constitutes a separate subfield of CA research [Paz71]. There has been extensive research into the properties of cellular automata, both as mathematical models of computation and as models for physical systems. CA models have been proposed to model a plethora of problems including spin systems, fluid dynamics, crystal growth, percolation and even cardiac arrhythmias [Man89, 2.3. CA, QCA AND QDCA 15

Gut91, DM99]. A good review of the state of the art up until 1986 as well as repro- ductions of many of the key papers can be found in [Wol86] while [Cha97] provides a more modern overview including examples of interest to modern information processing such as error correction, hashing and constructing ciphers using CA rules. An important subfield of CA studies is that of lattice-gas hydrodynamics, where a CA rule on a hexagonal two or three dimensional lattice is used to simulate different types of fluid flow [WG00, RB01b, Man89, RZ97]. The lattice-gas algorithms are particularly important as it can be shown analytically that particular lattice-gas rules are equivalent asymptotically to the incompressible Navier-Stokes equation for modelling non-linear fluid flow [FHP86]. This is therefore a concrete example of CA rules which can be used to model physical systems.

Most work on CA systems considers ﬁnite state classical computation but if the state of the cell is now a quantum state we can generalise the concept of a CA to include quantum information theoretic ideas. This has variously been studied as a means to simulate quantum systems [Mey96, Mey97, Mey98, BT98, Bog99, LB05] and simply as a interesting mathematical extension of the classical cellular automaton [GZ88a, GZ88b, GZ91, FGSS93, Wat95, AA04, PDC05]. In most work on quantum CA’s, the quantum state of the system is preserved across the entire lattice of states. In our discussion in chapter 10 we also consider a quantum CA whose evolution rule can maintain quantum coherence but the state of each cell is measured at the end of each time step.

There is some confusion in the literature with the acronym QCA being used to describe both the quantum extension of the classical celluar automata (which we refer to as a QCA) and the physical realisation of a classical cellular automata using quantum-dot systems (which we term QDCA). While this ambiguity doesn’t cause problems in general, due to the relative isolation of the relevant ﬁelds, this thesis deals with both areas and so it is crucial to distinguish between them.

To add to this confusion, there has also been work on a quantum computing model using a ‘always-on-interaction’ between sites which is often described as a QCA architecture [Ben00, Ben01, BB03, AAB+03]. It has been proposed that this QC model could be realised using endohedral fullerenes [Twa03, FT04b, FT04a], 16 Chapter 2. BACKGROUND which can be considered a physical realisation of the original mathematical concept of a quantum cellular automata, ie. the state of the entire lattice is coherent. A detailed study of this architecture is beyond the scope of this thesis but we mention it to illustrate another use of the acronym QCA.

2.4 Quantum-dot cellular automata

In its simplest form, a quantum-dot cellular automata is an experimental analogue of cellular automata at the micro or even nanometer scale [Mac06]. In this sense a QDCA is a particular realisation of a classical cellular automaton where the evolution rules and state definitions are those allowed by the laws of physics. In this case, the confinement properties of low occupation number quantum-dots define the internal states of the system and the Coulomb interaction provides the neighbour coupling. This could provide an alternative architecture with which to build standard logic gates and information channels, which have low dissipation and fewer control gates compared to conventional transistor based logic [LTPB93, TL03]. As the Coulombic interaction depends directly on the geometry of the charge configuration, it is possible to make the cell switching dependent on the orientation of the neighbouring cells. In this way, logic gates and other components are possible given the correct arrangement of the QDCA cells. A detailed review of QDCA’s, including both theoretical analysis and experimental progress, is given in [Mac06]. A more technical overview of the key principles of operation are given at the start of chapter 8. While a QDCA is composed of quantum objects, the information they contain is actually classical (0 or 1) and therefore a QDCA is a physical realisation of a classical CA using quantum dots. Considering Wolfram’s CA characteristics from section 2.3, we find that a QDCA is deterministic, has a spatially and temporally local rule (the electrostatic interaction) and consists of a finite number of states per cell (2). While synchronous and discrete evolution is not guaranteed, in practice both these requirements are satisfied as the input states of a QDCA system are switched in a discrete fashion and the system is always found in its ground state. 2.4. QUANTUM-DOT CELLULAR AUTOMATA 17

2.4.1 Theoretical work on QDCA

The original concept of a QDCA was presented by Tougaw and Lent [LTPB93, Mac06] and subsequently developed by Lent et al. [TL94, LT94, LT97]. These include the design of logic gates [LTPB93, TL94], analysis of sensitivity to fabrication variation [YZWB99, Mac06] and introduction of the concept of ‘memory in motion’ [Fro05]. A number of authors have improved on these ideas with novel designs for memory [BF99, WVJD03, OPV+06], a full adder circuit [LT94, TL94] and even a simple microprocessor [WMSJ05]. The QDCA topology has also been investigated as a possible architecture for quantum computing using QDCA qubits [TL01, JFTS02] and more recently as a candidate for a decoherence free subspace [OSGS05].

All of the initial designs for QDCA used solid-state quantum-dots for conﬁnement and therefore required cryogenic temperatures for operation, with operating temperatures of 1-20K predicted for a range of diﬀerent semiconductor materials [BIM01]. For these systems, cell switching times of 1 1000 ps have been predicted, sug- − gesting the potential for clocking speeds of 10 105 MHz [LT97, BIM01]. There − has been some work on molecular QDCA as a pathway to high temperature operation [LIL03, HL01], though this work is still in its very early stages.

Walus and coworkers have developed a simulation package, ‘QCA Designer’, for simulating large scale QDCA circuits. This package allows both layout and testing of high level QDCA circuits and has been used to design and simulate memory cells, complex circuits and even a design for a ﬁeld programmable gate array [WDJB04].

In chapters 8 and 9, we investigate the possibility of using buried donors in silicon as an alternative QDCA architecture. It is not clear whether the conventional theoretical tools used to describe quantum-dots and tunnelling are applicable at the single donor level. For this reason we apply techniques used in the study of single donors for quantum computing to this problem. In this way we calculate energy levels and timescales which are appropriate for QDCA cells comprised of single atom sites. 18 Chapter 2. BACKGROUND

2.4.2 Experimental QDCA demonstrations

Signiﬁcant progress has been made in demonstrating the operation of functional QDCA components. Aluminium quantum-dots with aluminium-oxide tunnel junctions operating at 50-70 mK have been used to demonstrate the switching of a QDCA cell [OAB+97, SOA+99a], a binary wire [SOA+99b] as well as the operation of a QDCA logic gate [SOA+99b, AOT+99, OKR+03]. Based on the measured junction capacitances, these devices have a switching speed of 1ns, giving a theoretical clock speed of 100MHz [SOA+99b, Mac06]. Cowburn et al. have demonstrated a QDCA like binary wire at room temperature using a magnetic alloy which they term a magnetic quantum cellular automata [CW00]. They estimate this system is capable of clocking speeds of up to 100MHz based on a switching time of 1 ns, at much higher temperatures. Proof of principle experiments have also been conducted using silicon on insu- lator [SPK01], GaAs/AlGaAs heterostructures [GSC+03] and more recently buried phosphorous clusters in silicon [MCP+06]. All of these experimental demonstrations use the electronic ground state of mesoscopic quantum-dots (with sizes of 100-1000nm) to deﬁne the states of the QDCA cell.

2.5 Type-II quantum computers

Type-II quantum computers (T2QC) represent the natural extension of ordinary quantum computers to include the concept of distributed computing, via classical communication channels [Yep01c]. We assume that a classical ‘control’ processor is available to perform the routing between nodes and any required classical processing. Through the use of many linked quantum processors consisting of a few qubits each, computing tasks can be performed which take advantage of the small amounts of coherence available while still allowing the computation of problems of significance. This may be particularly important for solving fluid dynamics, finite element or field theoretic problems where the lattice quantisation of the problem is obvious. The advantage of this form of quantum computer is that it takes advantage of available technology, rather than requiring 1000’s of qubits to demonstrate any speed up over 2.5. TYPE-II QUANTUM COMPUTERS 19 a conventional computer. The disadvantage is that for an algorithm to be useful, it must display a quantum speed up of the type expected for quantum algorithms but at the same time must be able to be parallelised in the classical information sense. In theory, if this were achieved, a type-II quantum computer would exhibit linear scaling with the number of nodes and exponential scaling with the number of qubits per node. A more detailed discussion of the scaling properties is given in section 2.5.2.

Any architecture which has been proposed for conventional quantum computing can, in principle, be also used to construct a T2QC. The key requirement being that the basic building block or ‘node’ can be replicated many times with minimal overheads in terms of cost and operation. An obvious example of this is solid-state architectures where mass fabrication is central to construction and therefore a node could be replicated thousands or millions of times. A more timely example is an architecture based on Nuclear Magnetic Response (NMR) as few qubit quantum information processing as been demonstrated for up to 7 qubits successfully [VSB+01] using this technology. While NMR systems have fallen out of favour with main- stream quantum computing, due to a lack of scalability, this restriction does not seriously aﬀect the existing type-II algorithms as all the known algorithms require between 2 and 11 qubits per node.

A T2QC based on a NMR architecture is illustrated in Fig. 2.1, where the computer is comprised of a volume of liquid containing an organic molecule which can be used as a coherent array of qubits. The volume is partitioned in sections using gradient magnetic ﬁelds and each section is designated a node in the computer. Each node is initialised with a particular input state, a global set of pulse sequences are applied to the entire volume and then the state of each node is read. The nodes are then reinitialised with the result of its neighbouring nodes and the process is repeated. A more detailed discussion of the operation of a T2QC is given in chapter 10 and a more detailed discussion on an NMR based T2QC is given in reference [Yep01c].

One important thing to note with an NMR implementation of T2QC is that the readout is an ensemble readout, as each node contains a macroscopic number of copies of the organic molecule which constitutes the qubits of the computer. As the 20 Chapter 2. BACKGROUND

Qubits q1 | i NMR Liquid q2 | i q3 | i Node

Figure 2.1: Illustration of a type-II quantum computer using NMR technology. The computer is comprised of a volume of solution containing many copies of an organic molecule, the spins of which can be addressed via magnetic resonance. The volume is partitioned using gradient magnetic fields and each region corresponds to a single node in the computer with each node consisting of many identical copies of the molecule. measurement of spins in a single organic molecule is yet to be demonstrated experimentally (due to its technical difficulty) a large number of molecules are required to achieve a detectable signal strength. This limits which QIP operations can be realised (due to the lack of projective measurement) and ensures the communication step is fundamentally classical in nature. This is a result of the fact that performing a measurement on an ensemble of states returns the statistical average of those states rather than a single binary result. On the other hand, ensemble readout allows the state of each node to be effectively stored as a real number (as opposed to a binary result) whose precision is limited by the signal sensitivity of the equipment. Throughout our discussion, we will refer to both forms of T2QC, with and without ensemble readout.

2.5.1 Algorithms for type-II quantum computers

The original motivation for type-II quantum computers was to use the massive par- allelism of a NMR type-II quantum computer to simulate fluid dynamics problems. As a result, the existing algorithms are all based on simulating a quantum lattice-gas representation of differential equations [Yep98, LB06]. These include implementing a quantum lattice-gas algorithm for simulating the diffusion equation [Yep01b], the Burgers equation [Yep02], classical solitons [VYV03] and magnetohydrodynamic tur- 2.5. TYPE-II QUANTUM COMPUTERS 21 bulence [VVY03]. When these algorithms are analysed in details it turns out that the central gate operation of all these algorithms acts on the value of two qubits and it is unclear how to scale up the algorithm to take advantage of more than two qubits per node. To date, type-II quantum computing has been demonstrated experimentally using a 16 node, NMR based T2QC. Both the diffusion [PCCY03] and Burgers [CYC04] equations have been successfully simulated using this machine. It should be kept in mind that, at two qubits per node across 16 nodes, the complexity of these problems are still well within that achievable with a conventional computer or cluster of computers but it is an important step nonetheless.

2.5.2 Eﬃciency of a type-II quantum computer

Given that the lattice-gas algorithms suggested to date do not demonstrate scalable improvement in computational efficiency with increasing qubit number, it is then interesting to ask can we demonstrate an improvement at all? When type-II quantum computing was first suggested, it was assumed that if a quantum algorithm displays exponential scaling with the number of qubits, then a T2QC would display this same scaling with the number of qubits per node, while giving linear scaling with the number of nodes. This suggests the computation power would scale as (V 22nq ), for a T2QC consisting of V nodes, each comprised of n qubits [Yep01c]. O q Vianna et al. attempted to confirm this scaling behaviour by investigated the scaling laws for a type-II (or semi-quantum) implementation of Grover’s search algorithm and Shor’s quantum Fourier transform (QFT) algorithm [VRM03]. For a T2QC, Grover’s algorithm scales trivially, as the search space can be distributed across the nodes of the machine during the initialisation of the computer. This means an unsorted database consisting of N elements can be searched in (2nq/2) steps with a T2QC consisting of V nodes, each with n = log (N/V ) O q 2 qubits per node. While this is an example of an algorithm which both displays a quantum speed and can be classically parallelised, this does not take into account the overhead required to input the original database to be searched. The implementation of Shor’s QFT algorithm on a T2QC is more interesting as 22 Chapter 2. BACKGROUND this algorithm displays an exponential (rather than polynomial) speedup and it is known to require large scale entanglement across the entire computer [PP02, SSB05, KM06]. As yet, there is no implementation of Shor’s algorithm for a T2QC using conventional ‘single-shot’ measurement. Vianna et al. assumed that ensemble readout is available and used this to reconstruct the phase component of each node, which was then propagated across to the other nodes. Using this process, they demonstrated that the number of operations required to perform a QFT on N numbers scales as (n2V )+ (N log V ) (where n and V are defined as for Grover’s algorithm), O q O 2 q provided the cost of reconstructing the phase at each step was negligible. In the limit of no qubits per node (V = N) we regain the classical result (N log N), O 2 where all of the processing is performed by the control processor. When V = 1 we obtain the expected (n2) scaling for a fully coherent quantum computer where O q all the processing is done by the qubits. Unfortunately, the state preparation stage cannot be ignored as it takes (n2N) steps to reconstruct the necessary phases O q on an ensemble T2QC. Including this phase reconstruction step means that when using ensemble readout with V > 1, both the classical and quantum Fourier transforms take (N) operations but a classical computer requires (N) bits of storage, O O whereas the T2QC only requires (n ) qubits [VRM03]. O q

2.5.3 Simulating a cellular automata system on a T2QC

The trade-off between the number of nodes and the number of qubits per node in a T2QC can equally be thought of as analogues to a quantum/classical transition. This is most obvious if we consider the two theoretical limits, either an entirely coherent quantum computer (1 node, many qubits) or a classical processor using qubits for storage (many nodes, 1 qubit per node). These two limits are capable of simulating the quantum and classical worlds respectively. Applications of this sort of quantum/classical machine have been alluded to by several authors but as yet there have been no concrete demonstrations, either experimentally or theoretically. In this thesis we consider a novel approach to this issue, we combined the ideas of type-II quantum computing and cellular automata. We can look for cellular automata rules which simulate physical systems and then try to reformulate them 2.5. TYPE-II QUANTUM COMPUTERS 23 to run on a type-II quantum computer. One such example is probabilistic cellular automata rules where the central difficulty is one of random number generation. If a deterministic cellular automata rule is constructed with a finite (usually small) set of states, this can be realised with conventional logic gates. If the algorithm is inherently probabilistic, the implementation becomes considerably more complex. There are several algorithms which would provide interesting applications of a cellular automata but they are inherently probabilistic. Obviously, these algorithms require a source of random numbers or at least a random string of binary numbers to execute. This complicates the situation as most sources of (pseudo) random numbers require some form of real number arithmetic [Nie92], which increases the complexity of each node in the CA. The Metropolis Monte-Carlo algorithm for spin systems is an example of a probabilistic algorithm which several authors have tried to recast as a deterministic one in order to avoid this issue [Cre86, OP89, Her86], with varying levels of success. As a way out of this difficulty, following the arguments of Zurek, we can consider quantum mechanics as a source of random numbers which provides not just sub- jective random numbers but in fact objective random numbers [Zur05]. This leads directly from the inherent ‘unknowability’ of the result of a quantum measurement. If we now consider a qubit in a superposition state, the measurement of this qubit constitutes a random binary number with an infinite repeat length. The algorithm presented in chapter 10 takes advantage of this concept to implement the Monte- Carlo algorithm as a probabilistic cellular automata rule. This CA rule can then be implemented on a type-II quantum computer with only small numbers of qubits. 24 Chapter 2. BACKGROUND Chapter 3

CHARACTERISING A TWO-STATE HAMILTONIAN

The two-state quantum system is the work-horse for the study of few state quantum systems and provides the basis for most of quantum computing. In this chapter we consider the process of characterising such a system where the Hamiltonian is in general unknown. The various Hamiltonian parameters can be determined system- atically by analysing the time evolution of the state of the system. We also study the evolution of the system in the Fourier domain and derive analytic results for the Fourier spectrum as a function of the Hamiltonian parameters. In section 3.1 we review the work of Schirmer et al. [SKO04] and discuss the basic method for single-qubit characterisation in the time domain. In section 3.2 we consider an imperfect measurement process and develop an error model for this process, while section 3.3 extends the method to the Fourier domain. To use a two- state system as a qubit for quantum computing, complete controllability requires the ability to switch between two Hamiltonians. Section 3.4 covers the measurement of a second Hamiltonian and the relative phase between it and the initial Hamiltonian. The material in this chapter has previously been published in reference [CSG+05].

3.1 Characterising a two-state Hamiltonian

The Hamiltonian of an arbitrary two-level system can be written in terms of the Pauli matrices, d σ d H = · = | |(d I + d σ + d σ + d σ ), (3.1) 2 2 0 x x y y z z where d , d and d are real constants ( d 1) and d results in an unobservable x y z | i| ≤ 0 global phase factor which can be ignored. We can write an arbitrary state of the system as ψ = cos θ 0 + eiφ sin θ 1 (3.2) | i | i | i 25 26 Chapter 3. CHARACTERISING A TWO-STATE HAMILTONIAN

0 | i

s d

θ φ x

1 | i Figure 3.1: Bloch sphere representation of the state of a qubit (s) and its trajectory given an arbitrary Hamiltonian d. If the system is not in an eigenstate of the Hamiltonian, the state (given by the unit vector s) precesses around an axis deﬁned by d. The components of d are given by the Hamiltonian using Eq. (3.1) where the d gives the angular precession frequency around the vector (d ,d ,d )T . | | x y z

where 0 and 1 are the canonical states of the two-state system. The state of the | i | i system can then be represented on the Bloch sphere where its position in the sphere is the Bloch vector (s) where s 1, with a pure state having s = 1. The evolution | | ≤ | | of the Bloch vector due to some Hamiltonian (H) will be to precess around a unit vector (d ,d ,d )T with angular rotation frequency given by d . If the system is in x y z | | an eigenstate of the Hamiltonian, the Bloch vector is parallel to the axis of rotation and therefore does not precess, as expected. This process is illustrated in Fig. 3.1. For simplicity, we use polar coordinates to describe both the position of the Bloch vector and the Hamiltonian vector. In these coordinates, the Hamiltonian vector is given by d = d (d ,d ,d )T = d [sin θ cos φ, sin θ sin φ, cos θ]T . (3.3) | | x y z | | 3.1. CHARACTERISING A TWO-STATE HAMILTONIAN 27

As the complex phase (φ) is unobservable in a single two-state system, we set φ = 0, aligning the Hamiltonian with the x-axis, giving

d H = | |[sin θσ + cos θσ ]. (3.4) 2 x z

The evolution of the system under this Hamiltonian is given by the operator U(t)=

iHt e− which, using a generalised de Moivre formula [Mer98], can be rewritten as

idot/2 d t d t U(t)= e− I cos | | idˆ.σ sin | | . (3.5) 2 − 2 · µ ¶ µ ¶¸

If the system is initially in the state ψ(0) = 0 (θ = φ = 0) then (converting to | i | i polar coordinates) the evolution of the system is given by

We will take the observable as the projection onto the z-axis, which gives the expectation value σ = ψ(t) σ ψ(t) . (3.8) h zi h | z| i After cancelling the global phase and rearranging terms, this becomes

d t d t σ = cos2 | | + (cos2 θ sin2 θ) sin2 | | . (3.9) h zi 2 − 2 µ ¶ µ ¶

If we set d = ω, (i.e. the angular frequency of the precession given in units such | | that ~ = 1) this gives the evolution of the z-projection of the system as

z(t)= σ = cos ωt sin2 θ + cos2 θ. (3.10) h zi

Determining the parameters ω and cos2 θ gives the values of d , d and d . The | | x z process of characterising the Hamiltonian thus involves measuring z(t) and analysing 28 Chapter 3. CHARACTERISING A TWO-STATE HAMILTONIAN

Init. H(∆t) Meas.

Init. H(2∆t) Meas.

Init. H(3∆t) Meas.

Init. H(4∆t) Meas.

z(t)= σ h zi −1

Figure 3.2: To map z(t), the system must be repeatedly initialised, allowed to evolve under the system Hamiltonian and then measured. To map the time evolution of the system, the Hamiltonian step is applied for progressively longer time intervals (i∆t for i = 1, 2,...,n) where ∆t is the minimum controllable time interval and tob = n∆t is the maximum time over which the system is observed.

the time series to determine the appropriate parameters.

To measure z(t) we consider repeatedly initialising in a known state and then measuring the system after progressively longer time periods. Given enough measurements, the measured evolution zm(t) approximates the z-projection of the trajectory of the Bloch vector, z(t). Assuming this is an idealised projective measurement, the sinusoidal variation of the projection onto the measurement axis depends on both the magnitude and direction of the vector d and therefore on the parameters in the Hamiltonian. A schematic of this process is shown in Fig. 3.2 where the minimum controllable time interval is given by ∆t and the longest time the system is allowed to evolve is tob, giving the total number of time points Ns = tob/∆t. This process is then repeated Ne times to build up an ensemble average for each time point, giving a total of NT = Netob/∆t = NeNs measurements. This process is often termed a ‘Rabi oscillation experiment’ as this is the standard experimental method for demonstrating Rabi oscillations of a driven two-level atom, though the method applies equally to any two-level system undergoing coherent oscillations. 3.2. THE EFFECTS OF IMPERFECT MEASUREMENT 29

3.2 The eﬀects of imperfect measurement

To model the eﬀects of an imperfect measurement process, we consider the possibility of the measurement returning the incorrect result with probability η [0, 1]. This ∈ corresponds to a bit-ﬂip error (σx) occurring the instant before measurement with some probability η. Assuming the Bloch vector always starts at ψ(0) = 0 , z(t) | i | i should reach a maximum of one after each period. The measurement error will reduce this maximum, independent of the angle θ and can therefore be determined directly from zm(t).

Interestingly, a bit-ﬂip error before the evolution introduces the same eﬀect. This corresponds to imperfect initialisation, which leads directly from imperfect measurement if the measurement process is also used to initialise the system. To show this explicitly, we can consider the evolution of the system in the density matrix formalism. For some system evolution given by U(t) = exp( iHt), where H is given − by Eq. (3.4), the evolution of the z-projection is given by

z(t) = Tr[σzU(t)ρ(0)U†(t)], (3.11) where ρ(0) = 0 0 is the starting state, z(0) = 1. If we model an initialisation error | ih | as a bit ﬂip before the evolution with some probability η1, this gives the evolution of the ‘system+error’ as

pre z (t) = Tr[σ U(t)σ† ρ(0)σ U †(t)] = z(t). (3.12) err z x x −

Similarly we ﬁnd that applying the bit ﬂip error after the evolution with probability

η2 gives post z (t) = Tr[σ σ† U(t)ρ(0)U †(t)σ ]= z(t). (3.13) err z x x −

The two error locations are therefore equivalent as Tr[σz[U(t), σx]] = 0, for the Hamiltonian given in Eq. (3.4). The resulting system evolution with an error prob- 30 Chapter 3. CHARACTERISING A TWO-STATE HAMILTONIAN ability η is then given by

z (t) = (1 η η + η η )z(t)+ η zpost(t)+ η zpre(t) (3.14) η − 1 − 2 1 2 1 err 2 err = (1 η η + η η )z(t) η z(t) η z(t) (3.15) − 1 − 2 1 2 − 1 − 2 = (1 2η)z(t). (3.16) −

So measuring the maximum amplitude of the oscillations gives the total error 2η = η + η η η as the cumulative eﬀect of both the measurement and initialisation 1 2 − 1 2 errors.

3.3 Estimating the Hamiltonian using Fourier components

Once zm(t) is determined, one approach to extracting the Hamiltonian parameters is to ﬁt the data in the time domain, as originally proposed by Schirmer et al. [SKO04]. While this is suﬃcient for approximate estimates or data containing only a few oscillation periods, a more elegant method is to take the Discrete Fourier

Transform (DFT) of the data zm(t) and calculate the parameters from the Fourier coeﬃcients. As the evolution is only a function of one oscillation frequency, we can determine this frequency with higher precision than is the case in conventional signal analysis. Analysis in the Fourier domain also provides a convenient method for estimating the uncertainties in these parameter estimates, which is discussed in chapter 5.

As z(t) is a pure sinusoid, if zm(t) consists of an integer number of periods of oscillation, its discrete Fourier transform F (ν) = DFT[zm(t)] will take on a simple form consisting of δ-functions at ν = 0 and ν = ν where ν refers to the position ± p p of the peaks in frequency space. To indicate the diﬀerence between discrete and continuous variables we have used κ to indicate the channel number in the time domain and ν to indicate the channel number in the frequency domain, so t = κ∆t and ω = ν∆ω at the sampled points in time and frequency respectively. Using the 1 deﬁnition of the inverse discrete Fourier transform DFT− , zm(t) can be rewritten 3.3. ESTIMATING H USING FOURIER COMPONENTS 31 in terms of the discrete Fourier components for the zero [F (0)] and peak-frequencies

[F (νp)],

Ns/2 1 i2πνκ/Ns DFT− [DFT[zm(t)]] = F (ν)e

ν= Ns/2 X− i2πνpκ/Ns i2πνpκ/Ns = F (0) + F (ν )e + F ( ν )e− p − p

= F (0) + 2F (νp) cos(2πνpκ/Ns) z(t). '

In this way, the angle θ and the angular precession frequency ω can be determined directly from the Fourier spectrum without the need for ﬁtting the data in the time domain.

Including the eﬀects of an imperfect measurement using the model described in section 3.2, the following results can be derived,

2πν ω = p , (3.17) tob

F (0) cos θ = , (3.18) s1 2η − and 1 F (0) η = − F (ν ), (3.19) 2 − p which directly link the components of the Fourier transform to the system parameters, d, as d = ω, d = sin θ and d = cos θ. | | x z As we can only perform projective measurements onto one axis, many measurements are required to accurately determine zm(t) so the total number of measurements, NT , will typically be quite large. Once the time resolution and observation time are chosen, the measurements for each time point can be repeated until a suf- ﬁciently resolved peak is seen in the DFT spectrum. An example of this process is shown in Fig. 3.3 for progressively larger numbers of measurements at each time point. In this way, the number of measurements need not be chosen at the start, but the experiment repeated until a suﬃcient signal-to-noise ratio is obtained. 32 Chapter 3. CHARACTERISING A TWO-STATE HAMILTONIAN

(a) (a)

(b) (b)

(d) (d) Time (t) Frequency (ν)

Figure 3.3: The left hand plot shows a numerical example of a sampled time signal z(t) = [cos(2πt) + 1]/2 with the number of ensemble measurements Ne set to (a) Ne = 1, (b) 2, (c) 8 and (d) 500 at each time point, where each measurement is a projection onto the (1,-1) axis. The corresponding DFT for each signal is shown on the right for ν 0, illustrating the signal-to-noise improvement as more measurements are taken at≥ each time point.

3.4 Determining the relative phase angle

The process discussed so far is sufficient to characterise a single two-state Hamil- tonian, as dy can be arbitrarily set to zero. To provide a completely controllable two-state system, such as is needed for QIP, a second control Hamiltonian is required to implement all possible single-qubit rotations. If we consider characterising some initial Hamiltonian H = dr σ/2, we can use this to define the coordinate axes and r · then consider a second Hamiltonian, H = dk σ/2. This provides a second axis to k · rotate around which must also be characterised and the angle φ between these two axes must be determined. To measure this azimuthal angle, a different initialisation point must be chosen whose Bloch vector is linearly independent of the original initialisation point. A convenient choice is to rotate s around the first axis (dr) until in sits on the ‘equator’ defined by θ = π/2. The second Hamiltonian is then switched on instead and the qubit precesses around dk. The z-projection of this rotation can then be used to determine the angle φ between the two axes. As dr and dk have already been completely characterised, the entire process can be ‘boot-strapped’, progressively learning more information about the system. Of course, this process of measuring different Hamiltonians is equivalent to measuring the dependence of a system Hamiltonian on the ‘settings of a dial’ where each Hamiltonian corresponds to a different value of the experimentally controllable parameters. 3.4. DETERMINING THE RELATIVE PHASE ANGLE 33

To rotate s onto the equator, starting at s0 we apply dr for a time

1 cos(2θ ) + 1 t = arccos r , (3.20) eq ω cos(2θ ) 1 r · r − ¸

T which places the system in state s1 = [cos(β), sin(β), 0] with a phase angle β given by [SKO04] β = arctan[ sec(θ ) 2 cos(2θ )]. (3.21) − r − r p If we then use this as the new initialisation point, the z-component of the precession about dk is given by

z (t)= C[1 cos(ω t)] + D sin(ω t), (3.22) φ − k k where 1 C = sin(2θ ) cos(φ β) (3.23) 2 k − and D = sin(θ ) sin(φ β). (3.24) k − This procedure can only be applied if θ [ π , 3π ]. If θ or θ are not within this r ∈ 4 4 r k range, a more elaborate pulsing scheme is required. Once the two axes dr and dk have been characterised, measuring Eq. (3.22) allows both Hamiltonians to be completely reconstructed.

Using a similar method to the previous section, the parameters C and D can be determined from the components of the Fourier spectrum,

Ns/2 1 i2πνκ/Ns DFT− [DFT[zφ(t)]] = F (ν)e

ν= Ns/2 X− i2πνpκ/Ns i2πνpκ/Ns = F (0) + F (ν )e + F ( ν )e− R p R − p i2πνpκ/Ns i2πνpκ/Ns +iF (ν )e iF ( ν )e− I p − I − p = F (0) + 2F (ν ) cos(2πν κ/N ) 2F (ν ) sin(2πν κ/N ) R p p s − I p p s z (t). ' φ 34 Chapter 3. CHARACTERISING A TWO-STATE HAMILTONIAN

where FR and FI are the real and imaginary parts of the Fourier components. As the measurement error of the system has already been determined from the measurements of the other axes, the constants C and D from Eqs. (3.23) and (3.24) can be determined directly using 2F (ν ) C = − R p (3.25) (1 2η) − and 2F (ν ) D = − I p . (3.26) (1 2η) − These equations are valid if the signal consists of exactly an integer number of periods, though this will very rarely be the case. The difference in phase between the first and last sampled time point can be minimised (see section 5.2) but in realistic experiments there will always be some residual phase error. Any error induced in the magnitude of the Fourier components by this effect will be small, but the error induced in the complex phase (denoted χ so as not to be confused with the Hamiltonian angle θ) will not be negligible and must be corrected. We may do this by observing that in Eq. (3.22) the constant term and the negative amplitude of the cosine term must be equal. We can define the corrected complex angle χc so that this is the case using

F (0) χ = arccos − , (3.27) c 2F (ν ) · R p ¸ such that the corrected Fourier component

F (ν )= F (ν ) [cos(χ )+ i sin(χ )], (3.28) c p | p | c c is then used in Eqs. (3.25) and (3.26).

At this point, in order to keep track of the various sine and cosine terms, we will introduce the following notation. When dealing with an arbitrary angle Φ, we use

AΦ = cos(Φ) to refer to the cosine of the angle. Likewise, we deﬁne BΦ = sin(Φ) as the sine of the angle giving the relationship A = 1 B2 . Φ − Φ p As the value of θk has already been determined, φ can be found from either C 3.5. CHAPTER SUMMARY 35

or D, depending on the value of θk. For instance using

Aφ β = cos(φ β) − − = 2C/ sin(2θk) 3π = C/(Aθk Bθk ) θk > 8 , (3.29)

2 Bφ β = sgn(D) 1 Aφ β − − − q or

Bφ β = sin(φ β) − − = D/ sin(θk) 3π = D/Bθk θk < 8 , (3.30)

2 Aφ β = 1 Bφ β, − − − q depending on the value of θk, will minimise the eﬀects of noise. The phase angle φ is then given trivially by

φ = arccos(Aφ β)+ β, (3.31) − as expected.

3.5 Chapter summary

Measuring the time evolution of a two-state quantum system has been investigated (building on the work of Schirmer et al. ) as a systematic method for identifying an unknown Hamiltonian. Using closed form expressions for the evolution, the parameters of an unknown Hamiltonian can be measured, including the phase angle relative to some reference Hamiltonian. This ‘boot-strapping’ procedure can be used to progressively learn more about the system. The analysis of the system evolution can be performed equivalently in the Fourier domian, where the position and magnitude of the Fourier components provide the necessary parameter estimates. The Fourier domain proves advantageous for this analysis as the single frequency evolution allows the adverse effects of projective measurement and discrete time sampling 36 Chapter 3. CHARACTERISING A TWO-STATE HAMILTONIAN to be controlled in a way which is not possible in conventional signals analysis. We have also developed a model for an imperfect measurement channel and found expressions to both estimate and correct for its effect. In the following chapters, we will extend the Fourier analysis technique to include the effects of decoherence and provide uncertainty estimates for the measured parameters. Chapter 4

CHARACTERISING A TWO-STATE HAMILTONIAN IN THE PRESENCE OF DECOHERENCE

In the previous chapter, we developed a method for characterising the Hamiltonian of an unknown two state system. In this chapter we extend this method to allow the inclusion of non-Hamiltonian evolution and demonstrate how phenomenological decoherence parameters can also be extracted from the measured evolution of the system.

Throughout this and subsequent chapters we use the Lindbladian formalism, which is introduced in section 4.1, as this provides a convenient model of decoherence which can model a range of different decoherence channels. To illustrate the key ideas, we first consider the effect of a pure dephasing decoherence channel in section 4.2 and how this can be included in the characterisation process. This simple model is then extended to a more general decoherence model in section 4.3 and we derive analytic expressions for the Fourier transform of the time evolution of the system. Finally, in section 4.4, the effect of the imperfect measurement model introduced in chapter 3 is reconsidered when the system is subject to decoherence. The material in this chapter has appeared in reference [CGO+06].

For this discussion, we assume that the evolution of the system can be described using the master equation approach. More speciﬁcally, we consider a form of decoherence that permits the use of the Born-Markov approximations, which is a common ﬁrst approach when analysing solid-state systems. The technique presented here is also applicable to other models, provided they permit a master equation type description, as the methods of section 4.2 and 4.3 could equally be applied to other master equations.

37 38 Chapter 4. INCLUDING DECOHERENCE

4.1 Modelling the eﬀect of decoherence

To model the eﬀect of decoherence on the characterisation process, we use the Lind- bladian formalism to describe the evolution of the density matrix of the system, ρ(t). The time evolution of the system is then governed by the Liouville-von Neumann equation [NC00], dρ(t) i = [H,ρ(t)] + (ρ(t),L ), (4.1) dt −~ L i i X where ρ(t) is the density matrix of the system, H is the Hamiltonian and

1 (ρ(t),L )= L ρ(t)L† L†L ,ρ(t) , (4.2) L i i i − 2{ i i }

where Li is the Lindbladian operator corresponding to a particular decoherence channel. In general, a decoherence model can include several diﬀerent Lindbladian operators, each of which corresponds to a diﬀerent decoherence mechanism.

While the Lindblad formalism allows the inclusion of general forms of decoherence, it still assumes the Born (weak coupling) and Markovian (uncorrelated noise) approximations. These approximations are often made when analysing decoherence for a variety of quantum systems [BHPC03, HTW+06, BM03, BKD04]. There are, however, some systems where it is generally believed that the Markov approximation is not valid. A notable example of this are systems based on superconducting qubits [MCM+05, DWM04, APN+04, BCB+05], where the dominant source of noise is thought to be from small numbers of background charge fluctuators which strongly couple to the qubit. In this situation, it is often difficult or impossible to write down the evolution of the system in the form of Eq. (4.1) and therefore the analysis of the decoherence needs to be tailored to the particular system in question [MS04, MSS03, MP06, DL05, Wil05]. For simplicity, we will not treat systems of this form but merely point out that our analysis is only valid for systems whose evolution can be modelled using a master equation approach. Alternatively, our approach can be used to determine an effective phenomenological model for the system if Markovian evolution is assumed. 4.2. PURE DEPHASING 39

4.2 Pure dephasing

Initially we will only consider pure dephasing, as this is often considered to be the dominant form of decoherence for solid-state qubits [MSS01, KLG02, BM03], where the Lindbladian operator for pure dephasing is given by,

Lz = Γz σz. (4.3) p We will consider a more general model of decoherence in section 4.3. As in chapter 3, we use a Hamiltonian of the form

H = d[sin(θ)σx + cos(θ)σz], (4.4) though for brevity we use d = d /2 as the magnitude of the Hamiltonian vector. | | For the case of pure dephasing, we expect an approximately exponential decay in the oscillations given by the decoherence rate, Γz. As the original evolution consists of only one oscillation frequency, the exponential decay results in a broadening of the peaks in the Fourier spectrum into Lorentzian (Cauchy) distributions. It is instructive to look at how this behaviour is modified when θ = π/2, both in the 6 time and frequency domains, for reasons which will become apparent later. Solving Eq. (4.1) numerically for pure dephasing, the oscillations do decay exponentially when θ = π/2. When θ = π/2, the effect of dephasing is no longer purely 6 exponential decay, but shows a slow decrease towards the maximally mixed state z( ) = 0. Fig. 4.1 shows the evolution of the z-projection for (a) θ = π/2 and ∞ (b) θ = π/4 for several different dephasing rates. To characterise a system which is undergoing decoherence, we need to be able to account for these effects. Ideally we would like to solve Eq. (4.1) for an arbitrary Hamiltonian to obtain the time domain behaviour, though in general it is non-trivial to invert the evolution to compute the Hamiltonian parameters. Instead, we will look at the system behaviour in the Fourier domain. Fig. 4.2 gives the Fourier transform of the evolution shown in Fig. 4.1 to demonstrate the dependence on both the decoherence rate and the angle θ. 40 Chapter 4. INCLUDING DECOHERENCE

1 Γ/d=1/1000 (a) Γ/d=1/100 0.5 Γ/d=1/10 0 z(t)

−0.5

−1 1 Γ/d=1/1000 (b) Γ/d=1/100 Γ/d=1/10 0.5 z(t)

0 0 5 10 15 time [units of 2π/d]

Figure 4.1: The z-projection of the time evolution of a two state system after preparation in the z = 1 state, for several different dephasing rates. The magnitude of the Hamiltonian (d = 1) is kept constant while the angle between the components is varied. (a) For θ = π/2, the oscillations decay exponentially. (b) With θ = π/4, the system undergoes a different evolution, but still decays to the mixed state z( ) = 0. Note the difference in scales on the vertical axes in (a) and (b). ∞

Observing the peak positions under the influence of dephasing, we note that the peak position does not move appreciably. As the peak is approximately stationary, the magnitude of the Hamiltonian vector can still be determined from its position (at least to first order). As in the case for no decoherence, as θ is varied from π/2, the zero-frequency peak grows. The ratio of the area under the two peaks can be used to obtain a first order estimate of the value of θ, while the width of the peaks are strongly dependant on the dephasing rate.

In the case of negligible decoherence in chapter 3, the Hamiltonian parameters were determined from the Fourier transform of the system evolution and the resulting δ-function peaks. If we now consider the situation where the decoherence rate is non-negligible, but slower than the system oscillations (Γz < d), a broadening of the peaks in the Fourier spectrum is introduced. To ﬁnd the functional dependence of this broadening on the various system parameters, we need to solve Eq. (4.1) analytically. Starting from Eq. (4.1), we gain insight by using the conventional 4.2. PURE DEPHASING 41

(a) 600 Γ/d=1/1000 Γ/d=1/100 Γ/d=1/10 400

200 Fourier Coeff.

0 (b) Γ/d=1/1000 400 Γ/d=1/100 Γ/d=1/10 300 200

Fourier Coeff. 100 0 0 0.5 1 1.5 2 frequency [units of d]

Figure 4.2: The Fourier transform of the system evolution given in Fig. 4.1, in the presence of dephasing, plotted for ω 0. The magnitude of the Hamiltonian, d, is kept constant and results displayed for≥ (a) θ = π/2, and (b) θ = π/4. The peak is reduced when θ = π/2, as expected, and the zero-frequency component increases. As the decoherence6 rate increases, the peaks broaden and the frequency shifts slightly.

variable substitution

x(t) = [ρ01(t)+ ρ10(t)]/2 y(t) = [ρ (t) ρ (t)]/2i (4.5) 01 − 10 z(t) = ρ (t) ρ (t), 00 − 11 to rewrite the matrix equation as a set of three coupled diﬀerential equations [MS99], where ρij(t) are the components of the density matrix ρ(t). For the case of pure dephasing, this gives

dx(t) = d cos θ y(t) 2Γ x(t) dt − − z dy(t) = d[sin θ z(t) cos θ x(t)] 2Γ y(t) (4.6) dt − − z dz(t) = d sin θ y(t). dt −

The solution of Eqs. (4.6) in the time domain is tractable, but does not provide any useful insight due to its complexity. Instead we take the Fourier transform of the set 42 Chapter 4. INCLUDING DECOHERENCE of equations which results in a set of algebraic equations in terms of X(ω)= [x(t)], F Y (ω)= [y(t)] and Z(ω)= [z(t)], F F

iω X(ω) = d cos θY (ω) 2Γ X(ω) − z iωY (ω) = d[sin θ Z(ω) cos θ X(ω)] 2Γ Y (ω) − − z iω Z(ω) = d sin θY (ω)+ C . (4.7) − F

These can be solved to obtain an expression for Z(ω), where C is a constant of F integration arising from the initial condition, z(0) = 0. This gives the solution to 6 the Fourier Transform of z(t) as

C Z(ω)= 2 F 2 , (4.8) d (2Γz+iω) sin θ iω + 2 2 2 (2Γz+iω) +d cos θ where C is still unknown in general. F It is instructive to consider the case where θ = π/2, in which we find that setting C = 1 artificially and expanding to first order around ω = d, the real component F of Eq. 4.8 is given by Γ Re[Z(ω)] z , (4.9) ≈ (d ω)2 +Γ2 − z which is a Lorentzian centred about the frequency ω = d with width given by Γz. Returning to the general case, if we assume the system starts in the state x( ) = y( ) = z( ) = 0, the input state z(0) = 1 can be modelled as −∞ −∞ −∞ an impulse at t = 0, resulting in the term z(0)δ(0) being added to the equation for dz(t)/dt. Transforming to the frequency domain and taking into account the fact we are using the discrete Fourier transform gives

z(0) C = = Ns∆ω, (4.10) F ∆t

as the contribution from this impulse function, for Ns sample points. For the case of pure dephasing, irrespective of decoherence rate, the system must approach the perfectly mixed state in the long time limit, i.e. x( ) = y( ) = ∞ ∞ z( ) = 0. This means that the boundary condition requirements of the Fourier ∞ 4.3. INCLUDING A MORE GENERAL DECOHERENCE MODEL 43 transform are automatically satisfied. The steady state solution z( ) = 0 also ∞ means that issues of frequency resolution and phase matching, to be raised in chapter 5, are not as relevant. As long as enough data is gathered that the steady state limit is reached, the time evolution data can be ‘zero-padded’ to increase the frequency resolution. If the steady state limit is not reached, then the phase difference between the first and last time point will need to be minimised, as in the case of no decoherence, though a residual error will still be present due to the amplitude mismatch. Providing the necessary boundary conditions are met, we now have a general procedure for dealing with a system undergoing dephasing. We take the Fourier transform of the experimental data, as before, and measure the peak position to determine d approximately. We then use the analytic form given in Eq. (4.8) and perform a nonlinear fit on the data to obtain the system parameters d,Γz and θ. To obtain a more accurate fit, the fitting process can be repeated iteratively using the same equations and varying one parameter at a time. An initial estimate is obtained for each parameter and then the parameters are refitted in turn, until the estimates converge.

The parameters d, Γz and θ predominately control the peak position, height and width respectively and the interdependence of the parameters are second order eﬀects. We can see this in the numerical results shown in section 4.1 and it is this ﬁrst order independence of the parameters which insures good convergence, as the covariances between the parameters are small.

4.3 Including a more general decoherence model

To treat more general decoherence, we add terms which model spontanteous absorption (Γ+) and emission (Γ ) (e.g. thermal population transfer) given by Lindbladian − operators of the form L = Γ σ , (4.11) ± ± ± p where

σ σx iσy (4.12) ± ≡ ± 44 Chapter 4. INCLUDING DECOHERENCE are the usual annihilation and creation operators. The decoherence operator becomes

(ρ,Li)= (ρ,Lz)+ (ρ,L )+ (ρ,L+), (4.13) L L L − L i X containing the three forms of decoherence which, in general, will have three different characteristic rates. To apply the method given in section 4.1 to the more general case requires a variable substitution to match the boundary conditions. We define x0(t) = x(t) x( ), y0(t) = y(t) y( ) and z0(t) = z(t) z( ) and then solve − ∞ − ∞ − ∞ as before. The initial conditions must also be redefined such that x0(0) = x( ), − ∞ y0(0) = y( ) and z0(0) = z(0) z( ). Solving in the steady state limit, we get − ∞ − ∞ 2 d cos θ y( ) x( ) = ∞ (4.14) ∞ 4Γz +Γ+ +Γ − y( ) = K z( ) (4.15) ∞ ∞ Γ+ Γ z( ) = − − (4.16) ∞ Γ+ +Γ + d sin θK − where 2 d sin θ (4Γz +Γ+ +Γ ) K = − . (4.17) 2 2 2 4 d cos θ +(4Γz +Γ+ +Γ ) − The resulting solution in the Fourier domain is

(C +Γ+) [1 z( )] Γ [1 + z( )] d sin θ [L(ω)+ L∗( ω)] Z0(ω)= F − ∞ − − ∞ 2− 2 − (4.18) 2 d M sin θ iω +Γ +Γ+ + 2 2 2 − M +4 d cos θ where (C iω) [y( )+ i x( )] d z( ) sin θ L(ω)= F − ∞ ∞ − ∞ , (4.19) M 2 id cos θ − M = 2 iω +4Γz +Γ+ +Γ , (4.20) − and L∗(ω) denotes the complex conjugate of L(ω). As z( ) is a constant, Z(ω > ∞ 0) = Z0(ω > 0) and Z(0) = Z0(0)+ z( ). This solution is an algebraic combination ∞ of the free variables (d, θ,Γ and Γz) and can therefore be used as a ﬁtting function ± for the transform of the oscillation data, as outlined previously.

The effect of different decoherence channels can be difficult to discriminate in the Fourier domain, due to their similar action on the spectrum. Hence a multi- 4.3. INCLUDING A MORE GENERAL DECOHERENCE MODEL 45 parameter fit will exhibit large covariance terms, with a relatively flat potential surface of the fitting function that will not easily converge. In this case the underlying physics of the system should be used to connect the different decoherence rates and their asymptotic values, using thermodynamic or other physical arguments. In this way we can use the physics to provide additional constraints to improve the success of the fitting procedure, an example of this is now given for the model in question.

To illustrate how this process works, we consider the special case when the measurement (initialisation) axis is coincident with the axis in which the dephasing acts. In this situation we can measure the effect of the other (non-dephasing) decoherence terms separately and therefore reduce the number of free parameters. By repeating the experiments as detailed earlier, in the limit of either d =0or θ = 0, we build up a picture of the non-Hamiltonian evolution. As the system does not have a mechanism to move away from the z-axis, the influence of pure dephasing is effectively removed and the population decay results purely from the absorption and emission terms only. This situation corresponds physically to the limit of either no driving field (d = 0 in the rotating frame) or the large detuning limit (θ = 0) where the system eigenstates are coincident with the measurement basis (σz). The system evolution in this limit is shown in Fig. 4.3 for an example system where Γ /Γ+ = 5. − Notice that the path taken by the system is different depending on which state is used for initialisation, though the steady state limit, z( ), is the same for both. ∞ The steady state population is given by

Γ+ Γ z( )= − − , (4.21) ∞ Γ+ +Γ − which provides one equation for determining the two rates. Note Eq. (4.21) is just Eq. (4.16) with d = 0 or θ = 0. This means that by observing the long time behaviour only we can reduce the number of free parameters by one. We can use this result to deﬁne the ratio of Γ to Γ+ even in the presence of dephasing and − Hamiltonian evolution.

Observing the time behaviour of the total system, Eq. (4.1), with d = 0 or 46 Chapter 4. INCLUDING DECOHERENCE

0.8

0.6 z (t) 1 0.4

0.2

0 z(t)

−0.2

−0.4 z(∞) −0.6 z (t) −1 −0.8

−1 0 1 2 3 4 5 time [units of 1/Γ ] −

Figure 4.3: The evolution of a two-state system under the inﬂuence of both spontaneous absorption and emission, in the limit of large detuning (θ 0). For this → example, the emission rate is ﬁve times that of spontaneous absorption (Γ /Γ+ = 5). − The path taken by the system depends on which initial state is used, though the asymptotic behaviour is the same. The two paths are labelled z 1 and z1 depending − on whether the system is initialised in the ground, z(0) = 1, or excited state, z(0) = 1, respectively. −

θ = 0 provides us with another handle, as shown in Fig. 4.3. If we label the trajectories taken by the system from the two initial states as z 1 and z1 for the − ground and excited states respectively, we can then ﬁt the curves to determine both decay rates. Alternatively, plotting the diﬀerence between the trajectories gives a simple expression,

z1(t) z 1(t) = 2 exp[ t(Γ+ +Γ )], (4.22) − − − − which can be easily fitted to determine the sum of the rates. This can then be used with Eq. (4.21) to determine both rates independently without the need to fit a double exponential. In practice, the experiment could be conducted by binning each measurement result based on the previous measurement and therefore the initialisation state. This would save time in the initialisation phase as single qubit operations could be minimised. Using this type of auxiliary experiment, the number of free parameters in the expression for Z(ω) can be reduced, resulting in better convergence during the fitting 4.4. IMPERFECT MEASUREMENT WITH DECOHERENCE 47 process. This in turn gives higher precision estimates for the system parameters, given the same number of measurements.

4.4 Imperfect measurement with decoherence

In section 3.3 we showed that the peaks in the Fourier spectrum can be used to determine an eﬀective error rate for an imperfect measurement process. In the case where the system is undergoing decoherence, there are no longer only two frequency components. In this situation an equivalent analysis can be performed by summing over all Fourier components. To prove this, we show that the initial state of the system, z(0), can be determined from the Fourier transform of the data. The deﬁnition of the discrete Fourier transform is given by

N 1 − [z(t)] = z(κ ∆t)ei2πκν/N , (4.23) F κ=0 X where N is the number of time or frequency channels, κ denotes the channel number of the time series, ν denotes the frequency channel number and both ν and κ are integers. If we compute the sum of the discrete Fourier spectrum over all frequency channels, this gives

N 1 N 1 N 1 − − − [z(t)] = z(κ ∆t)ei2πκν/N . (4.24) F ν=0 ν=0 κ=0 X X X Interchanging the order of the summations gives

N 1 N 1 N 1 − − − [z(t)] = z(κ ∆t) ei2πκν/N , (4.25) F ν=0 κ=0 ν=0 X X X as z(κ ∆t) is independent of ν. If we consider the inner summation term, evaluating the real and imaginary parts separately gives

N 1 N 1 N 1 − − − ei2πκν/N = cos(2πκν/N)+ i sin(2πκν/N). (4.26) ν=0 ν=0 ν=0 X X X 48 Chapter 4. INCLUDING DECOHERENCE

When κ = 0, the cosine term is

N 1 − cos(2πκν/N) = N, (4.27) ¯ ν=0 ¯κ=0 X ¯ ¯ whereas, when κ > 0 the summation over an¯ entire period cancels out for suitable values of N, giving N 1 − cos(2πκν/N) = 0. (4.28) ¯ ν=0 ¯κ>0 X ¯ ¯ Similar, the imaginary sine term is equal to zero¯ for all values of κ. Putting this together we ﬁnd that the inner summation is equal to

N 1 − i2πκν/N e = Nδκ,0 (4.29) ν=0 X i.e. the Kronecker delta function. This then gives us

N 1 N 1 − − [z(κ ∆t)] = N z(κ ∆t)δ = Nz(0), (4.30) F κ,0 ν=0 κ=0 X X or N 1 1 − z(0) = [z(κ ∆t)]. (4.31) N F ν=0 X The initial state of the system is therefore given by the sum over the discrete Fourier transform of the evolution, divided by the number of time points. If the system is undergoing decoherence, the result is the same as long as the boundary conditions are still satisﬁed. If z( ) = 0 then the variable substitution discussed in section 4.3 ∞ 6 can be used and the sum computed over the Fourier transform of z0(t).

The sum over the Fourier transform therefore depends on the cumulative eﬀect of both imperfect initialisation and measurement where this error is equivalent to multiplying the initial state by (1 2η), as discussed in section 3.2. This assumes − that the decoherence channel commutes with the Hamiltonian under the trace of the observable, which is true for decoherence channels composed of σz and σx, for the Hamiltonian considered thus far. 4.5. CHAPTER SUMMARY 49

4.5 Chapter summary

In this chapter we have extended the technique of Hamiltonian characterisation to model the effects of decoherence. Using the Lindbladian decoherence model, the evolution of the system undergoing both Hamiltonian and non-Hamiltonian evolution can be modelled. Solving in the Fourier domain, we obtained closed form expressions for the spectrum of the system evolution which can be used as fitting functions to obtain estimates for the system parameters. We have also extended our measurement error model and shown that the sum of the Fourier spectrum provides an estimate for this error mode. The methods discussed in this chapter are very general and may be applied to other master equation models which are not of the Lindblad form. The complexity of the decoherence model which can be employed is ultimately restricted by the need to fit the measurement data. If too many degrees of freedom are introduced into the model, the fitting procedure will not converge sufficiently to provide usable parameter estimates. In this situation, we find that using the physics of the system and performing other auxiliary experiments can provide additional constraints on the parameters and therefore improve the estimates. As this technique provides estimates for phenomenological decoherence parameters, the accuracy of these estimates are ultimately limited by the accuracy of the decoherence model itself. In the next chapter we build on these ideas and develop of method for obtaining uncertainties for the system parameter estimates. 50 Chapter 4. INCLUDING DECOHERENCE Chapter 5

UNCERTAINTY ESTIMATION WHEN CHARACTERISING A TWO-STATE HAMILTONIAN

The characterisation method discussed so far provides estimates for both the Hamil- tonian and decoherence parameters of a two-level system. For most practical applications, if we wish to estimate the parameters of a two-state system, we also need to know the uncertainty in those estimates. This chapter considers how to include uncertainty estimates for the key parameters and how these uncertainties propagate through to uncertainties in the final Hamiltonian. In section 5.1 we consider the simplest case of uncertainty arising from the finite number of measurements used to estimate the system parameters. We ignore the effects of decoherence and just consider the uncertainty in both peak position and height, in the Fourier domain. Section 5.2 includes a more rigorous discussion on obtaining accurate estimates for the oscillation frequency, while section 5.3 extends this analysis to the phase angle determination process. The cumulative effect of these uncertainties on the final Hamiltonian is derived in section 5.4. Section 5.5 considers the case where the decoherence of the system cannot be ignored and therefore the data must be fitted using the equations derived in chapter 4. The accuracy of the uncertainty relations developed in this chapter are confirmed using example simulations. In section 5.6 we generate synthetic measurement data (using arbitrarily chosen Hamiltonians) and then use this data to estimate the original system parameters. In section 5.6.1 and 5.6.2, the distribution of estimates and uncertainties is computed from many example runs (with and without decoherence) and these results are compared to those expected based on elementary statistics. The scaling of the uncertainties, with increasing number of measurements, is investigated in section 5.7 and the implications this has for single qubit rotations in quantum information processing are discussed in section 5.8. The material presented

51 52 Chapter 5. UNCERTAINTY ESTIMATION in this chapter has been published in reference [CSG+05] and [CGO+06]. Throughout this chapter, we usex ˆ to indicate the estimate obtained for some true value x in order to distinguish between the true value and its estimate where necessary. The uncertainty δx refers to the predicted standard deviation of this estimate and therefore in the ideal situation x 3δx xˆ x + 3δx, with 99.7% − ≤ ≤ conﬁdence.

5.1 Estimating the uncertainty in the measured quantities

Determining the system parameters from the components of the Fourier spectrum leads us to a straightforward way of calculating the uncertainty from the spectral noise. If we consider an example Fourier spectrum, such as that shown in Fig. 5.1, we can define the noise spectrum n(ν) to be the parts of the Fourier spectrum which do not include F ( ν ) and F (0), the peaks of interest. This is a good approximation ± p when the total observation time, tob, constitutes an integer number of periods and therefore F ( ν ) and F (0) approach δ-functions. As we are primarily interested in ± p characterising two-state systems to use as qubits, we assume that many oscillations can be performed before the effects of decoherence become important. If environment decoherence is negligible over the period of interest, the noise spectrum will originate primarily from the fact that each measurement provides a binary result. This discretisation or ‘projection’ noise [IBB+93] appears as a white- noise floor in the the Fourier spectrum. In this situation, the noise due to the discrete measurement of the system will be a limiting factor in the analysis, though other factors like noise in the control Hamiltonian will also contribute. The uncertainty in the oscillation frequency determination will be primarily limited by the precision in the time control of the measurements. Ideally, the uncertainty in the angular frequency measurement should be of the same order as the time resolution in the measured signal (δω/ω ∆t/t ). The uncertainty in the ≈ ob Hamiltonian angle δθ and the measurement error δη will be governed by the noise 5.1. ESTIMATING THE UNCERTAINTIES 53

0.5

0.45

0.4 F(−ν ) F(ν ) p p 0.35

0.3

)| signal peaks ν 0.25 F(0) |F( 0.2 noise floor 0.15 standard deviation δ F noise floor n(ν) 0.1

0.05

0 −80 −60 −40 −20 0 20 40 60 80 channel number ν

Figure 5.1: Example Fourier transform of a simulated measurement record, showing the Fourier component peaks clearly above the noise ﬂoor. The measurement record consists of Ns = 500 time points with Ne = 2 ensemble measurements, giving NT = 1000 total measurements.

level in the Fourier spectrum. Typically, the fractional uncertainty in ω will be an order of magnitude smaller than for θ or η as finding ω only requires finding the peak location whereas the other parameters depend on the peak height which is directly affected by the spectral noise.

The uncertainty in the Fourier peaks is given by the standard deviation (SD) of the noise spectrum so for simplicity we will deﬁne δF = SD[n(ν)]. Once we have the uncertainty in the frequency δω and the Fourier spectrum δF , using conventional uncertainty analysis [Kir94], we can derive the expressions for the uncertainty in the calculated values. Throughout this discussion we will use the standard error propagation method [Ku69] where the variance of some function w = f(x, y) is given in terms of the variances var(ˆx) and var(ˆy) and the covariance cov(x, y) between x and y. In its simplest form, the variance of a function can be calculated using

∂F 2 ∂F 2 var(w ˆ) = var(ˆx)+ var(ˆy) ∂X ∂Y · ¸ · ¸ ∂F ∂F +2 cov(ˆx, yˆ), (5.1) ∂X ∂Y · ¸ · ¸ 54 Chapter 5. UNCERTAINTY ESTIMATION for small variances in the measured parameters. In the situation under consideration, there is a correlation between the error in F (0) and that in F (νp) as this error comes from the shared white-noise ﬂoor of the Fourier spectrum. This means the covariance is not zero and is approximately equal to the variance of the noise signal itself. Using this approach, the uncertainty in the measurement error (η) and the Hamil- tonian angle (θ), from Eq. (3.18) and (3.19), can be estimated using the following equations, 3 δη = δF, (5.2) 2

F (0) δF 2 δη 2 δA2 = + θ 1 2η 2F (0) 1 2η − "µ ¶ µ − ¶ # 1 2η F (0) + − − δF 2 (5.3) (1 2η)3 ¯ ¯ ¯ − ¯ ¯ ¯ and ¯ ¯ 2 1/2 1 δθ = (1 A )− δA = δA , (5.4) − θ θ ˆ θ ¯sin θ¯ ¯ ¯ ¯ ¯ ˆ ¯ ¯ ¯ ¯ where Aθ = cos θ. ¯ ¯ This process results in an estimate and its associated uncertainty for the angular frequency ω, rotation axis θ and the measurement error η. A simplistic error analysis is given here to illustrate the ideas. The use of more sophisticated techniques such as maximum likelihood estimation should provide tighter bounds on the estimated parameters for a given set of data [JKMW01, DPS00, FH01]. For large numbers of measurements, where the centre-limit-theorem applies, these techniques become equivalent asymptotically.

5.2 Determining the precession frequency accurately

Performing a discrete Fourier transform (DFT) on the measured evolution immediately places some constraints on the selection of the measurement parameters. In order to satisfy the Nyquist sampling criteria, at least two sample points for 5.2. DETERMINING THE PRECESSION FREQUENCY ACCURATELY 55 every period of oscillation are required to avoid aliasing. This means that some estimate for the oscillation period Tpredict must be known in order to guarantee that

∆t < Tpredict/2, though in practice the period of oscillation will usually be known approximately on theoretical or experimental grounds. Conventional DFT theory states that the frequency resolution (∆f) of a DFT signal is the inverse of half the total time of the signal, ∆f = 2/tob [Bra00]. This means that to resolve the frequency signal, we need to observe at least two complete oscillation periods, though typically many more periods will need to be observed to obtain a clearly deﬁned peak in the frequency spectra. For an arbitrary signal the frequency resolution of the spectra limits the precision with which one can determine the frequency (f ∆f). The more periods observed the more accurately the deter- ± mined frequency of oscillation. Ultimately this will be restricted by the decoherence time of the system, as decoherence reduces the amplitude of the oscillations for long observation times.

To use Eqs. (3.17)-(3.19), we require that the observation time tob is an integer number of periods. To ensure this, we need to know the precession frequency to the same precision as the time control (∆ω/ω = ∆f/f ∆t/t ). Conversely, if ≈ ob we can guarantee that we have an integer number of periods, this will yield the corresponding frequency. The DFT of a pure sinusoid has the special property in that it only approaches a δ-function when the time signal consists of an integer number of periods (there is no phase difference between the start and end of the signal) [Bra00]. If there is some phase difference, ∆ϕ, then the DFT has ‘leakage’ into the other channels, resulting in a overall spread of the signal throughout the spectrum. This effect is demonstrated in Fig. 5.2 for example sinusoids having various values for the phase difference, ∆ϕ = ϕ(0) ϕ(t ), between the start and end points in the time signal. − ob Using this information, we can locate the ‘minimum-phase-point’ (MPP) where the difference in phase between the start and end of the signal is minimised. This amounts to selecting only an integer number of periods of the signal. As the period of the signal is not known beforehand, the easiest method is to record the data and then reprocess it later to ignore some of the data points. While this results in 56 Chapter 5. UNCERTAINTY ESTIMATION

(a) (a)

(b) (b)

Time (t) Frequency (ν)

Figure 5.2: The left hand plot shows time signals which are truncated at various time points to produce a net phase diﬀerence of (a) ∆ϕ = π, (b) ∆ϕ = π/2 and (c) ∆ϕ = 0 between the start and end of the signal. The corresponding DFT for each signal is shown on the right, where the peak approaches a δ-function only for ∆ϕ 0. ≈ throwing away some information, the lost data consists of, at most, one period.

An eﬀective way of locating the MPP is to compare the magnitude of the channel comprising the central frequency peak F (ν ) and its adjoining channels F (ν 1) p p − and F (νp + 1). When the leakage is minimised, the ratio of the central channel to its neighbours should be a maximum. An example test function which was found to perform well with varying levels of noise is

2F (ν ) F (ν 1) F (ν + 1) P (t )= p − p − − p , (5.5) p F (ν 1) + F (ν + 1) p − p where once again F (ν) is the normalised DFT of the original signal from zm(0) to z (t ) where t is the MPP and t T t t . An example plot m p p ob − predict ≤ p ≤ ob of P (tp) is shown in Fig. 5.3. A clear peak is observed at the point where the phase of the sinusoid (ϕ) is an integer multiple of 2π, i.e. ϕ(0) = ϕ(tp = 2πm) for some integer m. Once the MPP has been determined, the frequency is given by

ω = 2πνp/tp where νp is the peak channel number and tp is the MPP. The advantage of this method is that the MPP can usually be determined to an accuracy of close to ∆t/tob and the full-width-half-maximum (FWHM) of the function P (tp) gives an estimate for the uncertainty of the resulting frequency. We therefore deﬁne

δω = 2π/FWHM[P (tp)] so that the resulting uncertainty provides a more realistic estimate than that assumed based on the time resolution. 5.3. ESTIMATING THE UNCERTAINTY IN φˆ 57

φ(0)=φ(2πm) ) p P(t FWHM

t p

Figure 5.3: The test function P (tp) used to locate the point at which there is zero phase diﬀerence between the ﬁrst and last sample point. The amount of the time signal to use in the DFT is given by tp and the uncertainty is given by the FWHM of P (tp).

5.3 Estimating the uncertainty in φˆ

The uncertainty in the phase estimate φˆ will depend on the uncertainty in both the original axes characterisation and the noise in the Fourier spectrum used to compute C and D, as deﬁned in section 3.4. The uncertainty in the parameters C and D can be calculated using

3 2 2C 2 δC2 = δF 2 + δη2 (5.6) 2(1 2η) (1 2η) ¯ ¯ ¯ ¯ ¯ − ¯ ¯ − ¯ ¯ ¯ ¯ ¯ and ¯ ¯ ¯ ¯ 2 2 2D 2 δD2 = δF 2 + δη2 (5.7) (1 2η) (1 2η) ¯ ¯ ¯ ¯ ¯ − ¯ ¯ − ¯ ¯ ¯ ¯ ¯ where δF and δη are deﬁned¯ in section¯ 5.1. Here,¯ we have¯ ignored the covariance terms to simplify the analysis. The contribution due to correlated errors is small as the calculation of φ depends on three sets of measurements (dr, dk and Aφ β) which − are independent of each other.

As the rotation about the axis dr discussed in section 3.4 can only be performed to the same accuracy as the axis itself is characterised, there will also be some uncertainty in the azimuthal angle estimate βˆ. This can be approximated by setting 58 Chapter 5. UNCERTAINTY ESTIMATION

δθ δβ, which gives the uncertainty r ≈

Bβ δAβ = δAθr , (5.8) Bθr

where Bβ = sin(βˆ) as deﬁned earlier and therefore Aβ δAβ = Bβ δBβ.

We can then deﬁne the uncertainty in Aφ β in terms of C or D as −

2 2 2 2 2 δC δAθk δBθk δAφ β = Aφ β + + (5.9) − − C A B "µ ¶ µ θk ¶ µ θk ¶ # or 2 2 2 2 δD δBθk δAφ β = Aφ β + . (5.10) − − D B "µ ¶ µ θk ¶ # Writing the cosine of φˆ as

Aφ = cos(φˆ)= AβAφ β BβBφ β, (5.11) − − − gives the uncertainty relationship

2 2 2 2 2 2 Bφ βAβ 2 2 BβAφ β 2 − − δAφ = Aφ β + 2 δAβ + Aβ + 2 δAφ β (5.12) Ã − Bβ ! Ã Bφ β ! − − for the uncertainty in φˆ which is the phase diﬀerence between dr and dk.

5.4 Estimating the uncertainty in the Hamilto- nian parameters

Once the estimates θˆr, θˆk, φˆ,ω ˆr andω ˆk and have been found, the Hamiltonians can be estimated using the following equations,

ωˆ Hˆ = r (B σ + A σ ) r 2 θr x θr z

= Hˆr,xσx + Hˆr,zσz (5.13) 5.5. UNCERTAINTIES WITH DECOHERENCE 59 and

ωˆ Hˆ = k (B A σ + B B σ + A σ ) k 2 θk φ x θk φ y θk z

= Hˆk,xσx + Hˆk,yσy + Hˆk,zσz, (5.14)

where Hj,i is the i-th component of the j-th Hamiltonian. The uncertainty in each of the components of Hr are given by

2 δH δB 2 δω 2 r,x = θr + r (5.15) ˆ B ωˆ Ã Hr,x ! µ θr ¶ µ r ¶ and 2 δH δA 2 δω 2 r,z = θr + r . (5.16) ˆ A ωˆ Ã Hr,z ! µ θr ¶ µ r ¶

For Hk the component uncertainties are

2 δH δB 2 δω 2 δA 2 k,x = θk + k + φ , (5.17) ˆ B ωˆ A Ã Hk,x ! µ θk ¶ µ k ¶ µ φ ¶

2 δH δB 2 δω 2 δB 2 k,y = θk + k + φ (5.18) ˆ B ωˆ B Ã Hk,y ! µ θk ¶ µ k ¶ µ φ ¶ and 2 δH δA 2 δω 2 k,z = θk + k . (5.19) ˆ A ωˆ Ã Hk,z ! µ θk ¶ µ k ¶

5.5 Uncertainties with decoherence

Up until this point we have ignored the effects of decoherence in the uncertainty analysis as we are primarily concerned with characterisation of qubits, which implies long decoherence times. We may now consider incorporating uncertainty estimates into the decoherence analysis of chapter 4 and show how similar analysis can be performed, even when observing the system over time scales commensurate with the decoherence time of the system. The most straight forward method of including the effects of decoherence is to 60 Chapter 5. UNCERTAINTY ESTIMATION use the expressions developed in chapter 4 as a fitting function and then performing a non-linear fit in the Fourier domain. This provides estimates for the parameters d, θ and φ as well as any parameters introduced in the decoherence model. Using conventional nonlinear fitting routines to fit the measured data has the advantage that these routines also provide uncertainty estimates based on the goodness-of-fit. These uncertainties can be used to directly estimate the uncertainty in the final Hamiltonian parameters using the same relations derived in section 5.4 for the case of no decoherence. In general, these uncertainty estimates will not be independent as there will be some covariance between parameters and therefore any analysis should consider these effects.

5.6 Example simulations

To illustrate these ideas and determine the accuracy of the parameter estimate and its uncertainty, we simulated the measurement procedure on an arbitrary example system, Hr = 0.1σx + 0.05σz. At each point in time, we use this Hamiltonian to synthesise binary measurement data with the expected distribution and then we calculate the average measurement value. This produces a discrete approximation to the expectation value at an arbitrary time point, while allowing control over both the time resolution and the number of ensemble measurements.

Using an observation time tob = 500 and progressively larger numbers of measurements, the increase in precision can be observed. In Fig. 5.4, the components

Hr,x and Hr,z are plotted for increasing numbers of measurements. The error bars are given by 3δH (the 3-sigma conﬁdence interval) and the true value is shown as a solid line. As the number of measurements increases, the uncertainty reduces and the estimated values converge to the true value, as expected. The complete process is then simulated using a second example Hamiltonian (Hk = 0.6σx +0.45σy +0.1σz) and similar results are obtained but with increased uncertainty as the components of Hk rely on the measurements of both Hr and Hk, so there is more scope for accumulated errors. 5.6. EXAMPLE SIMULATIONS 61

Figure 5.4: An example of the systematic reduction in the uncertainty of the Hamil- tonian parameters as the number of measurements is increased. The error bars are given by three times the uncertainty estimate for each point and the solid line gives the ‘true’ value (Hr,x = 0.1, Hr,z = 0.05). The estimates are seen to converge to the true value as the number of measurements are increased.

5.6.1 Accuracy of the uncertainty estimate

In order to compare the uncertainty calculated using the equations in section 5.4 with the expected spread of the data, we repeated the simulations of the example system many times with the same number of measurements. By looking at the spread of the resulting estimates from many experiments and comparing this to the derived uncertainty from one experiment, we can conﬁrm that the uncertainty provides a good bound. Providing a good error bound on the Hamiltonian parameters alleviates the need to perform characterisation many times to obtain good statistics.

To measure the distance between the real Hamiltonian vector d and its estimate dˆ we use the following distance metric,

d dˆ = | − | (5.20) D d | | 62 Chapter 5. UNCERTAINTY ESTIMATION and a measure of the uncertainties is

We simulated the characterisation procedure for the example system using tob = 5 500, Ns = 10 and Ne = 50 with a measurement error probability of 10% (η = 0.1). Fig. 5.5(a) shows a histogram of for H over 5000 simulated runs, the D r average uncertainty δ over 5000 runs is also shown. For this example 98.4% of D the simulation runs lie within 3δ , illustrating that the uncertainty provides a good D bound on the estimated parameters.

The ﬁdelity for H shows a similar distribution, though the absolute uncer- D k tainty is greater for a given number of measurements as more steps are required to determine the azimuthal angle φ, than to determine θk by itself. Fig. 5.5(b) shows the equivalent histogram for determination of the Hamiltonian Hk over 5000 simulated runs. Three times the average uncertainty (3δ ) includes 98.7% of the data, D showing an excellent agreement with a normal distribution. The intervals for both and are slightly too small, as a 3-sigma interval should contain approximate Dr Dk 99.7% of the data. This discrepancy is due to the eﬀect of correlated errors between the Fourier components δF and the uncertainty in the MPP location (δω). In general, as the noise level in the Fourier spectrum increases, the width of the peak P (tp) will also increase. This results in a small correlation between the uncertainties inω ˆ and θˆ which has not been taken into account.

For a given set of experimental data, the width of P (tp) and the standard deviation of the noise ﬂoor of the Fourier spectrum will decrease as the number of measurements increases. The relationship between these errors can then be determined and will (in general) be non-trivial. The covariance can then be calculated, the result of which would be to add an additional term to Eqs. (5.13)-(5.19), increasing the overall uncertainty. This additional term will be small as the fractional uncertainty inω ˆ is typically much smaller than in θˆ which implies the covariance between them will also be small, relative to the other uncertainties.

The measurement error estimate (ˆη) is found to be very well behaved, with 99.5% 5.6. EXAMPLE SIMULATIONS 63

900 (a) 800

700

600

3δ r 500 D

400

300 Number of Occurrences 200

100

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Dr 900 (b) 800

700

600

3δ k 500 D

400

300 Number of Occurrences 200

100

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Dk

Figure 5.5: The distribution of for the estimated (a) Hr and (b) Hk over 5000 simulated runs. For these simulations,D (a) 98.4% and (b) 98.7% of the estimates are found to lie within the average uncertainty interval (3δ ). The absolute uncertainty D in Hk is greater than for Hr as more steps are required, giving a larger accumulated error. 64 Chapter 5. UNCERTAINTY ESTIMATION

800

700

600

500 3δη +3δη 400 −

300 Number of Simulations 200

100

0 0.097 0.098 0.099 0.1 0.101 0.102 0.103 η (estimated)

Figure 5.6: The distribution of the estimated measurement error (ˆη) for 5000 simulated runs. For this simulation, 99.5% of the estimates lie within the uncertainty 3δη. ± of the estimates lying within the error bounds, which is very close to the value expected for a 3-sigma conﬁdence interval. A histogram ofη ˆ is shown in Fig. 5.6 with 3δη labelled for 5000 runs, each run consisting of N = 5 105 measurements. T × From these simulations, we can see that the derived uncertainty relations do indeed provide good estimates for the uncertainty in the Hamiltonian estimation. This allows the uncertainties to be computed from one set of experimental data, rather than requiring many runs to obtain the necessary statistics.

5.6.2 Estimating the uncertainty with decoherence

To illustrate the characterisation of a system undergoing decoherence, we simulate the same example Hamiltonian H = 0.1σx + 0.05σz and include a pure dephasing term with decoherence rate Γ/d = 0.1. We numerically solve the system evolution for tob = 250 and then include the effect of finite measurement by simulating projective measurement, using Ns = 500 and Ne = 100. The resulting Fourier spectrum is then fitted using the Levenberg-Marquardt nonlinear regression algorithm [Mar63, Pre96] to perform a non-linear fit in the Fourier domain, see Fig. 5.7. We obtain rough 5.6. EXAMPLE SIMULATIONS 65

(a) 1

0.5

0 z(t)

−0.5

−1 0 1 2 3 4 5 time [units of 2π/d] (b) 0.02

0.015

0.01

0.005 Fourier Coeff.

0 0 0.5 1 1.5 2 frequency [units of d]

Figure 5.7: Simulated data for evolution due to the example Hamiltonian, H = 0.1σx + 0.05σz. (a) The time series data for Ne = 100 (points) which approximates the ensemble average of the evolution (solid line). (b) The Fourier transform of the time domain data (points) plotted with the ﬁtted function (solid line) using the estimated parameters in Table 5.1.

estimates for the various parameters and then perform an iterative ﬁtting process where each parameter is varied in turn. Generally, convergence is achieved within 2-3 iterations, although this depends on the number of measurements and ultimately on the quality of the experimental data.

The numerical values obtained from the data shown in Fig. 5.7 are given in Table 5.1. The true value (x), the estimate (ˆx), the 3-sigma conﬁdence interval (δx) and the fractional uncertainty (δx/xˆ) are given for d, θ and Γz.

In order to investigate the dependencies between the various parameters, we can use the covariances computed from the non-linear ﬁtting routine to plot the conﬁdence interval for each pair of parameters. These ‘error ellipses’ are show in Fig. 5.8 for each pair of the three parameters, where the 3-sigma contour is plotted with the true value and its estimate. As the major and minor axes of the ellipses are similar, we can use the outer boundaries (outlined in red) to determine the uncertainty for each parameter to simplify the analysis, though this overestimates these uncertainties. 66 Chapter 5. UNCERTAINTY ESTIMATION

x xˆ δx δx/xˆ d 0.2236 0.2228 0.0014 0.0063 θ 1.1071 1.1149 0.0115 0.0103 Γz 0.0224 0.0219 0.0007 0.0320 Table 5.1: Example values from a simulated run of the ﬁtting procedure discussed above, using the data shown in Fig. 5.7. The true value (x), its estimate (ˆx), the uncertainty (δx) and the fractional uncertainty (δx/xˆ) are given for the three system parameters d, θ and Γz with Ns = 500 and Ne = 100.

0.025 0.228 0.228 0.024 0.226 0.226 0.023 0.224 0.224 0.022 z d d Γ 0.021 0.222 0.222

0.02 0.22 0.22

0.019 0.218 0.218

1.061.08 1.1 1.121.141.16 1.061.08 1.1 1.121.141.16 0.02 0.022 0.024 θ θ Γz

Figure 5.8: Error ellipses for the ﬁt used to obtain the parameters in Table 5.1. The three dimensional ellipsoid has been projected down on to the three pairwise combinations of the parameters. The blue ellipse corresponds to the 3-sigma conﬁdence interval whereas the red box corresponds to the outer boundaries used to compute the uncertainties in Table 5.1. The black circle indicates the parameter estimate while the black cross shows the true value.

5.7 Scaling behaviour of the uncertainty

The usefulness of this technique is ultimately governed by how many measurements are required to obtain a given precision in the ﬁnal Hamiltonian estimate. This scaling behaviour limits the minimum time required to characterise an unknown system to any arbitrary precision.

5.7.1 Scaling with no decoherence

To investigate the uncertainty scaling, the example system used in section 5.6 with no decoherence was characterised with progressively larger numbers of measurements. The average of the resulting estimated uncertainty δ is plotted in Fig. 5.9 Dr for several diﬀerent values of the measurement error (η). For increasing numbers of measurements, the Hamiltonian estimate gets progressively more accurate, as 5.7. SCALING BEHAVIOUR OF THE UNCERTAINTY 67

−2 10 N−1/2 η=0.0 η=0.1 η=0.2 r −3 D

δ 10 Average

−4 10

4 6 8 10 10 10 Number of Measurements (N)

Figure 5.9: The average uncertainty δ of the estimate for the Hamiltonian H = Dr 0.1σx + 0.05σz as a function of total number of measurements. Each data point is the average of 10 simulation runs. The solid line shows 1/√NT where NT is the total number of measurements. As the total number of measurements increases, the overall precision to which the Hamiltonian is known increases. For a random measurement error, the achievable precision is reduced but still asymptotes to a scaling of one over the square-root of the number of measurements.

expected. This uncertainty scaling with total number of measurements (NT ) is approximately proportional to 1/√NT with the achievable precision reduced by the eﬀect of the measurement error. This constant factor is eﬀectively a ‘penalty’ which depends on the measurement error (η) but is largely independent of the number of measurements.

From this type of analysis we can estimate how many measurements are required to achieve a certain precision in the final result. Assuming all other factors are negligible, the achievable precision scales as one over the square-root of the number of measurements. In order for this method to be effective, the Hamiltonian must be constant in time, or more precisely, the fields controlling the Hamiltonian must be stable to higher precision than that required for characterisation. If the Hamiltonian has some random fluctuations, this will contribute to the noise level in the Fourier spectrum. The magnitude of this ‘jitter’ can be found by performing this analysis 68 Chapter 5. UNCERTAINTY ESTIMATION with progressively larger numbers of measurements. If the uncertainty in the parameters is found to asymptote to some constant value, this provides an estimate for the stability of the Hamiltonian. Alternatively, this ‘jitter’ can be considered a form of decoherence which again can be estimated by determining when the system is no longer well described by delta functions.

5.7.2 Scaling with decoherence

As in the case of no decoherence, we are ultimately interested in how the parameter uncertainties scale with increasing number of measurements. Fig. 5.10 shows the fractional uncertainty (1-sigma level) for the individual components of the Hamilto- nian, the uncertainty measure δ and the decoherence rate of the example system, Dr as a function of the number of measurements. Similar to the case for no decoherence, the frequency uncertainty is predominantly controlled by the time resolution and is typically much smaller than the other uncertainties. Using a constant number of time points (N = 1 106) and increasing the number of ensemble measurements s × (Ne) we find that the fractional uncertainty scales proportional to 1/√NT . The fractional uncertainty is approximately a factor of 3.5 times larger than for no decoherence. This represents the penalty for fitting including the decoherence terms using the non-linear fitting procedure. The uncertainty does not change appreciably when the decoherence rate changes, with the curves for Γz/d = 0.1, Γz/d = 0.01 and

Γz/d = 0.001 giving identical behaviour. The limiting factor in both the fitting procedure and the uncertainty analysis is, ‘how appropriate is the decoherence model?’ If the model used is not suitable, this will be apparent as the fitting procedure will not satisfactorily converge, even after many iterations. This can be most easily determined by inspection of the Fourier spectrum shape, when compared to that of the closest fitting parameters. If the model is found to be the limiting factor in the parameter estimation, then a more sophisticated model is required which may require additional experiments. An example of this type of additional experiment is the process discussed earlier for determining spontaneous absorption and emission terms independently from the effects of dephasing. In situations where the Born and/or Markov approximations 5.8. IMPLICATIONS FOR SINGLE-QUBIT ROTATIONS IN QIP 69

−1 10 δHz/Hz δHx/Hx δΓz/Γz δ D 1/√N −2 10

−3 10 Fractional Uncertainty

−4 10 4 5 6 7 8 10 10 10 10 10 Number of Measurements (N)

Figure 5.10: The uncertainty estimate as a function of total number measurements for an example Hamiltonian undergoing pure dephasing in the bare qubit basis. Each data point is the average of 10 simulation runs. The fractional uncertainty is plotted for each of components of the Hamiltonian and the dephasing rate Γz, where Γz/d = 0.1 for this example. The average uncertainty measure, δ r, is also plotted for comparison with Fig. 5.9. The scaling is approximately proportionalD to 1/√NT and the absolute fractional uncertainty is approximately independent of the decoherence rate. break-down, it may not be possible to model the decoherences using a closed-form expression for the time evolution, such as Eq. (4.1). In this case, a more sophisticated analysis would be required to determine the exact decoherence processes.

5.8 Implications for single-qubit rotations in QIP

To be able to perform single-qubit rotations of the type required for quantum computing applications, a certain level of accuracy is required. The threshold theorem

5 for quantum error correction states that if a physical error rate of p 10− can be ≤ achieved then concatenated quantum error correction protocols can be implemented successfully for arbitrary precision computation [Pre98]. This physical error rate gives the probability of an error due to decoherence of the system, where the process of error correction projects a (in general) continuous error into a discrete error with some probability. The errors introduced due to inaccurate characterisation will also 70 Chapter 5. UNCERTAINTY ESTIMATION contribute, though in a less predictable way. Typically, gate operations are assumed

6 to have a precision of 10− or better, however the previous analysis show that this would require 1012 measurements during characterisation. For a typical measurement readout time of 1µs this gives an initial characterisation time of approximately 12 days. This time estimate turns out to be a naive view, as the precision of the gate operations is not equivalent to the probability of a discrete error due to decoherence. For a single gate rotation around an ideal angle θ, the true rotation will be around an angle θ(1+²) and therefore the probability of a discrete error p (²)2 where ² δθ/θ. ∝ ≈ Given the previous discussion on the scaling of δθ with number of characterisation measurements N, the probability of discrete error on a single gate operation actually

1 6 12 scales proportional to N − , which requires only 10 rather than 10 measurements. As well as errors induced by inaccurate knowledge of the Hamiltonian angle (θ), errors can also be introduced due to an inaccurate rotation frequency or ‘over rotation error’. In general, this will have a similar effect to an angle characterisation error as (for small errors) they are equivalent. In addition, for the characterisation process discussed so far, the percentage uncertainty in the rotation frequency is typically an order of magnitude smaller than the uncertainty in the Hamiltonian angle which means that angle errors are the dominant source of gate error. For multiple gate operations, the probability of a discrete error scales as np where n is the number of gate operation time steps and therefore the number of possible error locations [KLZ98], assuming that errors in different qubits are uncorrelated. In the worst case, the rotation error accumulates as n² which gives p = np (n²)2, the T ∝ total probability of error for n possible error locations. This means it is possible (in the worst case) for the uncertainty in the angle to accumulate over multiple rotations. This will not always be the case as certain rotations (such as a 2π rotations) are less susceptible to characterisation errors than others and it is also possible to get error cancellation. Several techniques exist for dealing with systematic characterisation errors of this kind [VC04, WL03], much of which has recently regained interest for QIP applications [Jon03, CLJ03]. One such technique, which has been known in the NMR 5.9. CHAPTER SUMMARY 71 literature for some time, is composite pulsing [EBW90]. This involves carefully constructing a pulse sequence for a given rotation in order to reduce characterisation errors in both the angle (off-resonant errors) and the rotation frequency (pulse length errors). Recent work by Brown et al. [BHC04] has shown that in-fact, systematic characterisation errors can be eliminated to arbitrary order using strings of composite pulses. For a single imperfect gate with fractional error ², the resulting gate error can be reduced to (²n) for arbitrary n using a composite pulse O sequence whose length scales as n3. Using this or similar techniques, we can imagine a trade-off between long initial characterisation time (large number of characterisation measurements) and longer composite pulse sequences for our gate operations

(slower operating speed). In addition, by choosing ﬁne time sampling (large Ns) we can obtain very precise frequency estimates at the expense of poor angular resolution due to small numbers of ensemble measurements. The imprecise angular estimate could then be accounted for using composite pulsing. Similarly, poor time resolution and large numbers of ensemble measurements will give accurate angle estimates at the expense of rotation frequency resolution. There may also be situations where it is advantageous to precisely characterise some gates and/or qubits but not others.

5.9 Chapter summary

In this chapter, a method has been developed for computing the uncertainties associated with estimating an unknown Hamiltonian. Using this analysis, the uncertainty bounds on the parameter estimates can be calculated from a single set of experimental data. This alleviates the need to repeat the procedure many times to obtain an accurate statistical analysis. We have derived a series of expressions using error propagation theory which apply for systems with and without decoherence. Using numerical simulations, we have compared the distributions obtained from many example runs to those expected from the elementary statistics analysis and have found good agreement. The scaling behaviour of this procedure has also been investigated, providing an estimate for the number of measurements required to accurately characterise an unknown system. This is important for applications such as QIP, where 72 Chapter 5. UNCERTAINTY ESTIMATION it is vital that accurate estimates are obtained for all the system parameters. Chapter 6

EXPERIMENTAL DEMONSTRATION

In this chapter we present experimental results obtained in collaboration with Dr. Fedor Jelezko from the University of Stuttgart. The design, methodology and analysis of the experiment are due to the author. The data collection was performed by both Dr. Jelezko and the author, in Dr. Jelezko’s laboratory at the No. 3 Physical Institute in Stuttgart. These results demonstrate the use of the characterisation techniques discussed in previous chapters and represent the ﬁrst experimental implementation of these methods. As a test system we used the electronic spin levels in the Nitrogen-Vacancy (NV−) colour centre in diamond. This centre provides an ideal test system as it has been well characterised spectroscopically [vOMG88, HMF93a, HMF93b], provides a clean two-level system and displays long decoherence times [JGP+04b, KCC+02, KCB+03]. The NV centre has been extensively studied both theoretically [NKT+01, NKJ+03, MHS06] and experimentally [vOMG88, HMF93a, HMF93b, KCB+03, JPG+02]. It has been suggested for a range of quantum information processing applications and therefore much is also known about its coherence and controllability properties, both using ensemble [CK01, KCC+02] and single centre [JGP+04b, JGP+04a, JTG+01] measurements. For a review of QIP applications for the NV centre, see [WJ06] or [GOD+06]. There have been a number of experiments using the magnetic sublevels of the NV centre which demonstrate both one- and two-qubit operation using microwave driven transitions [JGP+04b, JGP+04a] as well as recent experiments demonstrating coupling between the spin states of NV centres and neighbouring single Nitrogen spins [GDP+06, HMEA06]. To demonstrate the characterisation procedure experimentally, we will use these microwave transitions as they are precisely controllable. The magnetic sublevels of the NV centre are well described by conventional atom-

73 74 Chapter 6. EXPERIMENTAL DEMONSTRATION photon interaction theory and therefore the mapping between experimental parameters and the effective Hamiltonian is well known. We can therefore use the NV centre to compare the Hamiltonian parameters we obtain from experiment with those expected from theory and confirm the accuracy of the characterisation process. This is in contrast to most solid-state systems, where the effective Hamiltonian is strongly fabrication dependent and therefore needs to be measured either directly or indi- rectly for every device. The NV centre is also an excellent candidate for testing the characterisation process as it has recently been used to demonstrate quantum process tomography [HTW+06] and therefore affords a more direct comparison between the methods. In section 6.1 we describe the physics of the NV centre in diamond and the effective two-level system used in the results presented throughout this chapter. The experimental setup is discussed in section 6.2 including the process used to identify a single NV centre. In section 6.3 we present the experimental data which demon- strates Rabi oscillations, while the effect of the hyperfine splitting is discussed in section 6.4. The effective Hamiltonian is then calculated for a range of different microwave excitation frequencies and the dependence on the experimental parameters is measured in section 6.5. Throughout this chapter, the energy of the various transitions and Hamiltonian components are measured in units of frequency, as per convention in the spectroscopy literature.

6.1 The NV centre

The NV centre in diamond consists of a substitutional nitrogen adjacent to a vacancy defect. Throughout this discussion we only consider defects having a charge state

14 of 1 (NV−) and in which the nitrogen isotope is N. Defects of this type can − be readily selected experimentally, due to their unique optical and nuclear spin properties. The crystal structure for the NV centre is illustrated in Fig. 6.1. The orbital ground state of the centre is a singlet (A), whereas the orbital excited state is a doublet (E). These orbital states are further split into magnetic sublevels [MHS06]. The transition between the orbital ground and excited states is optically active and 6.1. THE NV CENTRE 75

Figure 6.1: The crystal structure of the NV centre showing the substitutional nitrogen adjacent to the vacancy defect in the unit cell of a diamond lattice.

it is this transition which we will use for measurement and readout. While there is still some debate about the exact structure of the excited levels and the various mechanisms, the physics of the ground state levels and the key processes are well understood. A brief discussion of the pertinent details is presented in this and following sections.

The electronic ground spin state of the centre is a spin triplet (3A) which is split into a singlet m = 0 and doublet m = 1 , separated by 2.88 GHz due | s i | s ± i to crystal field splitting. The degeneracy between the doublet states is lifted in a magnetic field and it is this configuration (B~ > 0) we will use as our example two- level system, where the two levels consist of one of the m = 1 states and the | s ± i m = 0 state. The energy level structure is illustrated in Fig. 6.2 with the two- | s i level system illustrated. The nuclear spin of the Nitrogen introduces an additional hyperfine splitting into the ground state energy levels. Inclusion of the hyperfine effects are discussed in section 6.4. 76 Chapter 6. EXPERIMENTAL DEMONSTRATION

3E Fine Structure m s = 1 Two level

514 nm 637 nm ± System

3A ms =0 B~ ﬁeld

Figure 6.2: The energy levels of the NV centre illustrating the ground (3A) and excited (3E) states. The transition between these levels is optically active with a wavelength of λ 637nm. Alternatively, driving the system with a shorter wavelength (λ< 637)≈ results in a transition to higher levels, followed by a non-radiative transition down to the 3E state. The ground spin state of the centre is split due to zero ﬁeld splitting, while under nonzero magnetic ﬁeld the m = 1 states are split. | s ± i For these experiments, we will use the magnetic spin levels ms = 0 and ms = 1 as our two level system. | i | − i

6.1.1 Eﬀective Hamiltonian

The Hamiltonian for the NV centre is given by [CK01, HMF93a]

1 H = (g β )B S + D S2 S2 (6.1) e e · z − 3 · ¸ 1 (g β )B I + P I2 I2 − n n · z − 3 · ¸ 1 + + +A SzIz + A (S−I + S I−) k 2 ⊥ where D is the crystal field splitting term, A and A are the hyperfine interaction k ⊥ terms in the directions parallel and perpendicular to the applied field. The nuclear quadrapole interaction is given by P while ge and βe are the electron (and gn and

βn are the nuclear) g-factor and Bohr magneton respectively. For this discussion, we will use the values for these quantities as given in reference [HMF93a, HMF93b], which are reproduced in Table 6.1. Throughout this analysis, we will assume that the magnetic ﬁeld is orientated along the z-axis (the [111] crystal direction).

While the Hamiltonian given in Eq. 6.1 has been shown to provide a very good description for the magnetic sublevels of the NV centre [JGP+04b, HMF93a], the 6.1. THE NV CENTRE 77

D 2.88 GHz A 2.3 MHz k A 2.1 MHz ⊥ P -5.04 MHz geβe 28.03 GHz/T gnβn 3.07 MHz/T Table 6.1: Experimentally measured parameters for the Hamiltonian given in Eq. 6.1 for the magnetic spin levels of the NV centre, as given in reference [HMF93a]. experiments presented in section 6.5 demonstrate that the eﬀective system Hamil- tonian can be measured directly from the Rabi oscillation experiments alone.

6.1.2 Pumping and readout

The transition from the 3A ground state to the 3E excited state of the NV centre is an optically active transition. Laser pumping this transition results in population transfer to the excited state, which then decays radiatively to the ground state. If this process is performed resonantly with only one of the magnetic sublevels, the entire transition is spin conserving and therefore the emission of photons can be used to readout the spin state. Alternatively, driving the system with a higher energy laser results in population transfer to higher lying states, followed by a non-radiative transition down to the 3E state, followed by spontaneous emission. The resulting intensity of the emitted light from the optical cycle is dependent on the spin state with transitions involving the m = 1 having substantially lower emission rates | s ± i than those involving the m = 0 state. Observing the intensity of the emitted | s i light during the pumping cycle therefore allows the spin state of the system to be determined, though the presence of non spin conserving processes eventually polarises the system. One explanation for this spin polarisation is that the system can also decay non-radiatively from the excited state to a metastable singlet state, 1A. At room temperature, the system can move from the 1A state back to the 3E state by thermal activation which does not conserve spin. This allows eﬃcient spin polarisation by illuminating the centre with a laser of suﬃcient energy to promote the electron to the 3E state which then decays preferentially (via the singlet state) to the m = 0 state. | s i 78 Chapter 6. EXPERIMENTAL DEMONSTRATION

Using this method, spin polarisations of 60 80% have been demonstrated in NV − centres [JGP+04b, HSM06]. At low temperatures, its has also been suggested that the transition out of the 1A state is driven by the excitation laser, again resulting in polarisation of the spin state [KCC+02]. We can use these asymmetric effects to both initialise and readout the state of the NV centre. At room temperature, the decay path via the metastable state quickly repolarises the system, which reduces the available photons coming from the optical transition. The emitted light can still be used to determine the spin state of the system but due to the poor collection efficiency, this is not a ‘single-shot’ readout mechanism. The results presented in the following sections were obtained at room temperature using this readout mechanism and therefore depend strongly on the fluorescence level. Performing the same experiments using a resonant laser excitation would allow a ‘single-shot’ readout process, improving the signal-to-noise ratio and therefore the performance of the method presented in this chapter.

6.2 Experimental setup

The experimental setup used during these experiments is essentially identical to that used in reference [JGP+04b]. The microwave field signal is applied to the NV centre via a short cut loop connected to a 40W travelling wave tube amplifier. A confocal microscope was used to collect the emitted photons for readout and to physically select an NV centre of interest. The diamond sample used was a type 1b diamond nanocrystal with naturally abundant NV centres. In order to find a useful NV centre, the sample was scanned while being illumi- nated with 514 nm laser light in order to optically pump the system from the 3A to the 3E via the non-radiative decay pathway. The emitted florescence was measured at each point in order to provide a map of all optically active colour centres in that region of the sample [JTG+01]. A centre was then selected with the confocal microscope and an anti-bunching experiment was performed to verify that the emission did indeed originate from a single colour centre [JTG+01]. Fig. 6.3 shows the florescence map for the diamond sample used in these experiments with the NV centre of 6.3. OBSERVATION OF RABI OSCILLATIONS 79

Figure 6.3: Confocal microscope image of diamond surface under laser illumination. The colour centre circled in red was used for all of the following measurements. interest circled in red. Using Optically Detected Magnetic Resonance (ODMR) techniques [JTG+01], the energy separation between the levels was measured and this was used to conﬁrm that the colour centre was a NV colour centre and that the nitrogen atom in the centre was in fact 14N.

6.3 Observation of Rabi oscillations

Once a single NV centre was selected, a Rabi oscillation experiment was performed using the sequence of operations illustrated in Fig. 6.4 to map out the oscillations as a function of delay time τ. Initially the microwave generator frequency (fMW ) was tuned to the resonant frequency (fres = 2733.3MHz) as determined by ODMR, resulting in the Rabi oscillation data presented in Fig 6.5. Each data point consists of 2.4 106 readout cycles as each cycle does not provide an unambiguous × state determination. For this readout mechanism, the ﬂuorescence level cannot be calibrated absolutely and hence depends on the optical setup.

6.4 Including the hyperﬁne interaction

While the electronic ground state of the NV centre is well described as a two-level system, the hyperﬁne coupling to the nitrogen nuclear spin splits the energy levels, 80 Chapter 6. EXPERIMENTAL DEMONSTRATION

Laser

Microwave τ

Detection

Initialisation Evolution Measurement

Figure 6.4: The pulse sequence applied to obtain the Rabi oscillation data. Flourescence Level [Arb. Units]

0 0.05 0.1 0.15 0.2 0.25 Time [µs]

Figure 6.5: Example Rabi oscillation data for the on-resonance case, fMW = fres. The solid curve is a theoretical model with the system parameters determined using the method detailed in section 6.5. There is no beating due to the hyperfine interaction as the system is tuned to the central line, so the effects of the other two hyperfine lines cancel at each point in time. Note that the fluorescence level cannot be used to determine the state absolutely as the readout is not ‘single-shot’. as indicated in Fig. 6.6. This means the qubit is no longer composed of one pair of levels but in fact can consist of one of three different pairs, depending on the spin state of the nitrogen nucleus, as 14N has a net isospin I = 1. The difference in energy between these configurations is well described by Eq. 6.1. For magnetic field strengths of B 103mT, the individual spin states are not mixed by the hyperfine ¿ interaction and therefore the splitting of the levels is given approximately by the strength of this interaction, A . An example of an ODMR spectra for a NV centre k consisting of a 14N is shown in Fig. 6.7 with the hyperfine splitting labelled. As these experiments where conducted at room temperature, k T g β B so B À n n the nuclear spin states are equally populated. The Rabi oscillations are therefore 6.5. HAMILTONIAN CHARACTERISATION 81

Hyperﬁne Structure 0 1 1 − − +1 c b a 0 0 B~ 1 ﬁeld +1− ms mI

Figure 6.6: Splitting due to the hyperfine interaction at finite magnetic field, 0 < B < 102.8mT, with the spin allowed transitions labelled in blue. For our purposes, we will treat transition c as our target transition to use as a qubit. Note that the spacing between levels is not to scale. well described by the average of the three possible nuclear spin configurations. This results in beating effects on a time scale given by the strength of the hyperfine coupling. An example experimental trace is shown in Fig. 6.8, showing the modulation of the oscillations and the theoretical curve, demonstrating the effects of the nuclear spin. If the electronic states of the NV centre are to be used for QIP applications, the effect of this interaction could be eliminated using pulse sequences which refocus the system.

6.5 Hamiltonian characterisation

In order to construct a fitting term, as per the method of chapters 3-5, we must include the effects of the hyperfine interaction. We can include the effect of dephasing by adding an exponential decay term. While this provides only an approximate description of the process, this is more than sufficient for the precision of the data being analysed. The dephasing time for this experiment was found to be T2 = 3.2µs, using a separate Hahn echo experiment. The decoherence parameters could also have been obtained from the Rabi oscillations using the methods of chapter 4 but this would require much higher quality data. The fitting function we used consists of the average of three delta functions, 82 Chapter 6. EXPERIMENTAL DEMONSTRATION

2.3 MHz 2.3 MHz

Flourescence Level [Arb. Units] a c b

2794 2796 2798 2800 2802 2804 2806 2808 2810 Microwave Frequency [MHz]

Figure 6.7: ODMR spectra for an example NV centre showing the hyperﬁne splitting due to the 14N nucleus. The transitions a, b and c correspond to the three spin allowed transitions illustrated in Fig. 6.6.

positioned at frequencies separated by the strength of the hyperﬁne interaction. These delta functions are then convolved with a Lorentzian, whose width is given by the measured decoherence rate. The ﬁtting term used is therefore,

C 1 1 1 Z (ω)= F + + (6.2) NV 3 Γ2 +(ω ω )2 Γ2 +(ω ω )2 Γ2 +(ω ω )2 · − 1 − 2 − 3 ¸ with 1/Γ = 3.2µs and C is a scaling parameter which depends on the maximum F ﬂuorescence level. The resonant frequency of each transition is given by

2 2 ω1 = Ω + (∆ A ) − k q 2 2 ω2 = Ω + (∆) (6.3)

2 2 ω3 = pΩ +(∆+ A ) k q where the hyperﬁne coupling is A = 2.3MHz, Ω is the transition strength and ∆ is k the detuning from the target transition (labelled c in Fig. 6.6). We can then write the eﬀective two-state Hamiltonian for our target qubit as

Ω ∆ H = σ + σ , (6.4) 2 x 2 z 6.5. HAMILTONIAN CHARACTERISATION 83 Flourescence Level [Arb. Units]

0 0.05 0.1 0.15 0.2 0.25 Time [µs]

Figure 6.8: Example Rabi oscillation data for a system detuned from resonance, fMW = fres 40MHz. The beating of the oscillations is due to the hyperfine interaction with− the nitrogen nucleus. The decay envelope of the oscillations is enitrely due to this beating as the dephasing time of the system is 3.2µs, considerably longer than the period of observation. The solid line corresponds to a theoretical curve using the system parameters determined by fitting in the Fourier domain. where Ω = d sin θ and ∆ = d cos θ using the notation of chapter 3. | | | | The Levenberg-Marquardt non-linear fitting routine was used to obtain estimates for Ω and ∆ from the experimental data and the uncertainty in these estimates was found using the methods of chapter 5. Fig. 6.9 shows the time evolution data for the microwave field tuned to resonance (∆ = 0) and the Fourier transform of the data. The solid line indicates the result of fitting with Eq. 6.2. In order to test the characterisation method, we performed the Rabi experiment at various different microwave frequencies (fMW ) and compared the detuning measured from the characterisation process with that expected based on the actual detuning. As the absolute fluorescence level is a function of the experimental setup, the fluorescence levels which correspond to 0 and 1 were chosen to minimise the least-squared error across all values of detuning. Fig. 6.10 shows the measured detuning ∆ as a function of actual detuning f f , with the error bars indicating MW − res the 1-sigma confidence interval. The solid line indicates the relationship expected theoretically for detuning as measured from the central line of the hyperfine split resonance lines. The dashed lines indicate the detuning from the other two hyperfine split transitions. Note that the Rabi data is never exactly ‘on resonance’ as the mi- 84 Chapter 6. EXPERIMENTAL DEMONSTRATION

0.5

−0.5 Amplitude [Arb. Units] −1 0 0.05 0.1 0.15 0.2 0.25 Time [µs]

0.5

0.4

0.3

0.2

0.1

0 0 10 20 30 40 50 60 70 80 Fourier Amplitude [Normalised] Frequency [MHz]

Figure 6.9: Example Rabi oscillation data at resonance, fMW = fres (top). The Fourier transform of this data (bottom) with the ﬁt obtained using the Levenberg- Marquardt algorithm and Eq. 6.2. crowave source is always detuned from at least two of the three possible transitions, due to the system averaging over the three spin states of the nitrogen atom. The response of the system is approximately linear, with the divergence from the theoretically expected response well within the uncertainty estimates. The uncertainties increase as the detuning gets larger due to the reduction in the visibility of the original oscillations. As the detuning increases, the eﬀective Hamiltonian is dominated by the σz term and therefore the oscillations reduce in amplitude, resulting in greater relative uncertainty in all of the estimated parameters.

In Fig. 6.11 we plot the value of Ω obtained from the characterisation procedure for the same data plotted in Fig 6.10. Ideally this should be constant as the microwave intensity was not changed during the experiment and we see that it is indeed well described by a constant response, the dashed line indicating a weighted line of best ﬁt giving Ω = 27.43MHz.

While the general behaviour of the experimental data agrees with that expected from the theory (within the error margins estimated), there is a systematic oﬀset for larger detunings which is most obvious in the residuals for both Fig. 6.10 and Fig. 6.11. This is attributed to drift in an analogue sweeper which was used for these 6.5. HAMILTONIAN CHARACTERISATION 85

55 Residuals 1 50 0

45 −1

−2 40 −3

) [MHz] 35 −4 ∆ −5 30 0 10 20 30 40 50

Effective Detuning ( 15

0 0 10 20 30 40 50 Detuning (f −f ) [MHz] res MW

Figure 6.10: Effective detuning as measured from Rabi oscillation data, as a function of detuning as measured from the microwave generator. Error bars indicate the 1- sigma confidence level obtained from the fitting process. Notice that the errors become larger for large detuning as the visibility of the oscillations is reduced. The solid line shows the expected linear behaviour while the dashed lines indicate the detuning from the other two hyperfine levels. The insert shows the residuals of the data when compared to the expected behaviour which show a systematic trend above 20MHz. Also, note that the effective detuning should not reach zero as the system is always detuned from at least two of the three hyperfine levels. experiments. This was later confirmed using seperate measurements and found to be as large as 1-3 MHz over a period of hours. As the measurements shown in Fig. 6.10 were taken consecutively over a period of two days, this would more than account for the drift seen in the measured detunings. Another source of error is variation of the flourescence level over time, due to movement of the physical position of the confocal microscope system. As we have chosen the 0 and 1 readout levels based | i | i on the flouresence level of the on resonance experiment, any shift in the absolute level will have an effect on the inferred detuning. If the experiment were to be repeated using single-shot readout mechanism, discrimination between the 0 and 1 levels | i | i could be achieved with much greater precision and therefore this problem would be reduced. In addition, with finer time sampling, longer observation times and lower temperature measurement, the noise in the Fourier spectrum and therefore the error bars could be reduced so that more subtle effects such as decoherence, relaxation 86 Chapter 6. EXPERIMENTAL DEMONSTRATION

35 ) [MHz] Ω 30

Residuals 1 20 0

−1

Effective Transition Strength ( 15 −2

−3 0 10 20 30 40 50 10 0 10 20 30 40 50 Detuning (f −f ) [MHz] res MW

Figure 6.11: Effective interaction strength (Ω) as measured from Rabi oscillation data, as a function of detuning as measured from the microwave generator. Error bars indicate the 1-sigma confidence level obtained from the fitting process. Notice that the errors become larger for large detuning as the visibility of the oscillations is reduced. The dashed line shows a weighted linear fit to the data, giving a value of Ω = 27.43MHz while the insert shows the residuals of the fit. and hyperfine effects could be distinguished.

6.6 Chapter summary

Using the NV colour centre in diamond as an example system, the characterisation methods developed in chapter 3-5 have been demonstrated experimentally. These methods were found to be effective even with noisy data, consisting of only a few oscillation periods. In addition, by comparing the response of the effective Hamilto- nian as a function of the experimental controls, we were able to detect drifts in the controls which were of the same order as those introduced by hyperfine coupling of the NV centre. We have confirmed the usability of our techniques in the laboratory and demonstrated high precision system characterisation, with the accuracy of the analysis being limited by the present experimental setup. Chapter 7

TWO-QUBIT CHARACTERISATION

One of the key requirements for a physical system to be used for quantum information processing applications is that the system must have a controllable two-qubit coupling [NC00, DiV00]. This is typically realised by an interaction between a pair of two-level systems which act as qubits. It is this interaction which leads to entanglement and the ‘spooky action at a distance’ eﬀects which give quantum computers their power. While some systems have a well known and characterised native two- qubit interaction, this is not generally the case. Particularly in solid-state systems, the interaction Hamiltonian is often a function of many control and fabrication parameters [KHD02, SSHG+05, VYW+00, KGS+04, WHKG04]. As such, the form of the Hamiltonian can vary from device to device and even vary within diﬀerent sections of a single device. This means characterisation of some sort is critical in order to control the interaction and produce accurate gate operations for quantum computing applications. In this chapter we consider how to characterise the Heisenberg Hamiltonian as this is a commonly discussed interaction for generating entangling gates for QIP applications. The material presented in previous chapters has been focused on the general problem of characterising an unknown two-state system. In this chapter we focus more closely on the use of characterisation to generate accurate two-qubit gates and its implications for quantum computing, though much of the discussion applies equally to other applications. In section 7.1 we consider the general problem of characterising higher Hilbert-space dimensional systems. We then study the general Heisenberg type Hamiltonian in section 7.2 as an example of a higher dimensional Hamiltonian of direct interest for quantum computing applications. In section 7.3 we present the concept of mapping the entanglement generated as a function of time as an alternative method of determining the Hamiltonian components. A method for

87 88 Chapter 7. TWO-QUBIT CHARACTERISATION determining uncertainties in the characterisation process is presented in section 7.4 for two-qubit interactions and this is used to link the issues of characterisation with those of fault tolerance and quantum error correction. The eﬀects of single qubit terms in the Hamiltonian and imperfectly prepared input states are discussed in sections 7.5 and 7.6 respectively. The contents of this chapter has previously been published in [CDH06] and [DCH06].

Throughout this chapter we will use the notation, XX = σ σ , ZI = σ σ x ⊗ x z ⊗ 0 etc. for brevity when discussing two-qubit interactions, where σi are the usual Pauli operators.

7.1 Characterisation of higher dimensional systems

Earlier, we developed methods for characterising an unknown two-state system. We now extend these methods to consider higher dimensional systems, either to interactions between two-state systems or single particle multi-level systems. A general procedure for n-dimensional systems is straightforward and proceeds as follows. Consider an arbitrary Hamiltonian H which can be expressed in diagonal form as

Hd = V HV †, (7.1) where V is a diagonalisation matrix, the diagonal form of the Hamiltonian is then

H = diag λ ,λ ,...,λ , (7.2) d { 1 2 n}

where λk is the kth eigenvalue. As in previous chapters we only consider Hamilto- nians which are constant in time so that the state of the system, given some initial state ψ(0) , can be written as | i

iHt ψ(t) = U(t) ψ(0) = e− ψ(0) . (7.3) | i | i | i 7.1. CHARACTERISATION OF HIGHER DIMENSIONAL SYSTEMS 89

If we take the probability of measuring the system in the state φ φ at time t as | ih | Pφ(t), the evolution of this measurement probability is then given by

2 iHdt 2 P (t)= φ ψ(t) = φ V †e− V ψ(0) . (7.4) φ |h | i| h | | i ¯ ¯ ¯ ¯ Making the following substitutions,

α V ψ(0) (7.5) | i ≡ | i and β V φ , (7.6) | i ≡ | i gives

iHdt 2 iλ1t iλ2t iλnt 2 P (t)= β e− α = β diag e− ,e− ,...,e− α . (7.7) φ h | | i h | { }| i ¯ ¯ ¯ ¯ Expanding this expression¯ in terms¯ ¯ of the components of α and β gives¯ | i | i

n 2 iλkt Pφ(t)= αkβk∗e− , (7.8) ¯ ¯ ¯k=1 ¯ ¯X ¯ ¯ ¯ which, after evaluating the modulus-squared¯ gives ¯

n n 2 2 2 2 i(λk λl)t P (t) = α β + α β e− − φ | k| | k| | k| | l| k=1 Ã l=k ! 6 Xn Xn n = α 2 β 2 + 2 α 2 β 2 cos[(λ λ )t]. (7.9) | k| | k| | k| | l| k − l Xk=1 Xk=1 Xl>k This is the general expression for evolution due to the Hamiltonian H. Note that the evolution simply consists of a constant term and a series of oscillating terms at frequencies given by the energy difference between each pair of eigenvalues. These oscillating components can be identified in the Fourier spectrum of the time evolution data and therefore used to reconstruct the eigenspectrum. This process is completely analogous to conventional spectroscopy, though in this case coherent evolution and readout are all that is needed, rather than applying external driving fields and observing the resultant absorption or emission spectrum. 90 Chapter 7. TWO-QUBIT CHARACTERISATION

Looking at Eq. 7.9 we can see that by careful choice of the input states and measurement bases, we can simplify the evolution of the system by setting particular values of αk and βk to zero. This allows the isolation of each of the oscillatory components in turn and in principle reconstruction of the original Hamiltonian. The difficulty with this process is that the diagonalisation matrix V effectively performs a basis rotation on both the input and measurement bases, which (if completely unknown) makes it extremely difficult to identify which components are being observed in any given trace. For Hamiltonians of known structure this process is considerably simplified, even if the magnitude of the various components is not known. This is exactly the case of interest for Hamiltonian characterisation as we are considering systems of known behaviour (hence their use as a controllable quantum system) but whose parameters vary due to fabrication or other variations. In this case, the structure of the Hamiltonian can be used to provide a model for the diagonalisation matrix and therefore to help select input and measurement basis states appropriately. It should be noted that we are assuming a certain level of controllability in preparation of either input or measurement basis states. If this controllability is not available, the amount of information which can be obtained about the system through observation of coherent oscillations is severely limited. In practice this should not prove too restrictive as the systems of typically interest are those which can be controlled with reasonably high precision.

7.2 The Heisenberg Hamiltonian

To investigate the process of characterising a specific multi-particle system, we will consider an interaction of the Heisenberg type. This interaction is often suggested for quantum computing applications for generating entangling gates. In an implementation of a quantum computer consisting of nominally identical qubits, the physical interaction between any given pair of qubits is similar, so we expect the structure of the Hamiltonian to be similar across a given device. On the other hand, the size of the various couplings are a strong function of the fabrication process and will therefore vary from qubit to qubit. In these situations, not only is it important to 7.2. THE HEISENBERG HAMILTONIAN 91 identify the size of the relative components, but for scalable systems this characterisation must be done in an efficient manner. In this case, we require a process which can be largely automated, requires no physical modification to the original fabricated qubit system and ideally uses only the physical infrastructure present in the quantum computer.

The general Hamiltonian for qubits interacting via a Heisenberg interaction can be written as

H = c1XX + c2YY + c3ZZ, (7.10) where c R are the coupling strengths of the various components of the Hamil- i ∈ tonian. When c1 = c2 = c3 = d, this is the conventional (isotropic) Heisenberg interaction of the form H = d(XX + YY + ZZ), which is typical of spin state qubit coupling, whereas for c1 = c2 = J and c3 = 0, this is commonly termed the XY interaction. If c3 = J and c1 = c2 = 0, this is the interaction due to an Ising type coupling (H = JZZ), common in pseudo spin schemes. From the point of view of two-qubit gate design, for an Ising interaction it is not important which of the three terms is non-zero as the Hamiltonians JXX, JYY and JZZ are locally equivalent [ZW05].

For this analysis we consider general Heisenberg type Hamiltonians with c1, c2 and c3 treated as parameters to be determined. Many solid-state quantum computing proposals rely on this type of interaction e.g. [AAB+03, BB03, NC00], as the general Heisenberg case covers a large class of quantum systems including real spin (i.e. exchange coupling) systems [Kan98, LD98, FRS+03, VYW+00, DBK+00, ML05] and pseudo-spin systems such as charge based designs [MSS01, HDW+04].

As in earlier chapters, we only consider Hamiltonians which are piecewise constant in time. In addition, we assume controllability of any single qubit terms (i.e. from previous single qubit characterisation) such that they can be turned oﬀ during the two-qubit interaction, or alternatively that the single-qubit terms commute with the rest of the Hamiltonian. The restrictions imposed by this assumption are discussed in section 7.5 but correspond to those required for most quantum computing proposals. As we are interested in reducing the systematic errors introduced by imperfect characterisation (rather than random errors caused by interaction with the 92 Chapter 7. TWO-QUBIT CHARACTERISATION

++ + + Input State PZZ (t) PZZ−(t) PZZ− (t) PZZ−−(t) 00 cos2[(c c )t] 0 0 sin2[(c c )t] | i 1 − 2 1 − 2 11 sin2[(c c )t] 0 0 cos2[(c c )t] | i 1 − 2 1 − 2 01 0 cos2[(c + c )t] sin2[(c + c )t] 0 | i 1 2 1 2 10 0 sin2[(c + c )t] cos2[(c + c )t] 0 | i 1 2 1 2 ( 0 + 1 ) 1/4 1/4 1/4 1/4 | i | i ( 0 + 1 ) ⊗ | i | i ( 0 + 1 ) 1/4 1/4 1/4 1/4 | i | i ( 0 1 ) ⊗ | i − | i Table 7.1: The analytic form of the various measurement probabilities for a system evolving due to a general Heisenberg Hamiltonian, given diﬀerent initial states. After evolution, the system is measured in the ZZ basis. environment), we have assumed that the eﬀect of decoherence is negligible within the observation time though this can, in principle, be treated with the methods of chapter 4.

To begin, we analytically derive the evolution of the system described by the Hamiltonian given in Eq. (7.10), from some initial state ψ(0) to the state ψ(t) | i | i at some later time. To measure this evolution, we use a similar method to that discussed in chapter 3 where the system is repeatedly initialised in the state ψ(0) , | i allowed to evolve for a time n∆t and then measured. For the case of a Hamiltonian of the form given in Eq. 7.10, we can calculate the evolution of the measurement results in the computational (ZZ) basis. Table 7.1 shows the probability of measuring one of the four possible results ( 00 , 01 , 10 and 11 ) as a function of time for several | i | i | i | i ++ + + diﬀerent input states. These probabilities are labelled PZZ , PZZ−, PZZ− and PZZ−− respectively. We would like a set of input states which allows the measurement of each of the parameters in the Hamiltonian separately. For the example shown in

Table 7.1 this is not the case as the evolution is a function of just c1 and c2. For other input states we ﬁnd that the evolution is, in general, a function of all three parameters in various combinations.

Alternatively, if we consider the evolution of the system measured in the XZ basis, we ﬁnd simpler relationships between the system parameters. Measuring in 7.3. MAPPING THE ENTANGLEMENT 93

++ + + Input State 2PXZ (t) 2PXZ−−(t) 2PXZ−(t) 2PXZ− (t) 2 2 ++ 00 cos [(c c )t] sin [(c c )t] 2P (t) 2P −−(t) | i 1 − 2 1 − 2 XZ XZ 2 2 ++ 01 cos [(c + c )t] sin [(c + c )t] 2P (t) 2P −−(t) | i 1 2 1 2 XZ XZ 2 2 ++ ( 0 + 1 ) ( 0 + 1 ) cos [(c c )t] sin [(c c )t] 2P −−(t) 2P (t) | i | i ⊗ | i | i 2 − 3 2 − 3 XZ XZ 2 2 ++ ( 0 1 ) ( 0 + 1 ) cos [(c + c )t] sin [(c + c )t] 2P −−(t) 2P (t) | i − | i ⊗ | i | i 2 3 2 3 XZ XZ Table 7.2: The analytic form of the time evolution for the various measurement probabilities (in the XZ basis) for particular input states. The other input states obtained by applying bit ﬂips to each of the input states result in swapping the sine and cosine terms. this basis is equivalent to applying a Hadamard rotation to one of the qubits before measuring the system in the ZZ basis. This requires that we have good control over single qubit rotations or at least control over rotations around one axis. In this situation we ﬁnd more useful relationships for the evolution which are given in Table 7.2 for several example input states, labelling the four measurement proba- ++ + + bilities in the XZ basis as PXZ , PXZ−, PXZ− and PXZ−− respectively. The evolution for these input states is only ever a function of two of the three parameters in the Hamiltonian which means the parameters can be determined exactly using three of the four input states. Each of these input states can be created using Hadamard rotations, again requiring the precision characterisation of only one rotation axis for each qubit. As the evolution is a simple sinusoidal variation where the frequency of oscillation gives the value of each parameter, the values of c1 to c3 can be determined with high precision using a Discrete Fourier Transform of the time evolution data for three input states.

7.3 Mapping the entanglement generated by the Heisenberg Hamiltonian

The difficulty with the process of measuring a two-qubit interaction is that the time evolution is both a function of the single and two qubit terms in the Hamiltonian. In the case of the Heisenberg Hamiltonian, we have simplified this process by re- stricting the class of interactions that we consider. Alternatively, we can look at 94 Chapter 7. TWO-QUBIT CHARACTERISATION the entanglement generated by the interaction. By definition, if the entanglement changes with time, then a two qubit interaction must be present, since local operations alone cannot generate a change in entanglement1. This also allows a direct link to the design of entangling gates as it is the amount of entanglement generated which is of interest. Through the concept of ‘local-equivalence’ if is often sufficient to realise a maximally entangling gate, with single qubit unitary rotations being used to then convert the available gate into the gate required for the circuit of interest [ZVSW03, ZW05]. This leads us to employ the idea of using the change in entanglement to analyse the interaction and isolate the effect of the terms of interest in the Hamiltonian. The entanglement of the state generated by this evolution can be quantified using the squared concurrence [Woo98]

2 2 C = ψ∗ YY ψ , (7.11) |h | | i| where C2 varies between 0, when the qubits are unentangled, to 1 when they are maximally entangled. One method of measuring the concurrence is to measure the system in the ZZ and XZ bases. For brevity, we write the probability of

λ1λ2 measuring the ith qubit in the λi eigenstate of the αi operator as Pα1α2 , where λ = 1 and α = X, Z. In terms of these quantities the squared concurrence is i ± i given by: [HZWL03, SH00]

2 + + ++ ij C = 4 P − P − + P −−P 2 P cos(A + B) (7.12)  ZZ ZZ ZZ ZZ − ZZ  ij sY   where ++ + 2P P −− P − cos(A)= XZ ZZ ZZ (7.13) − −+ 2 PZZ−−PZZ− and p + + 2P − + P −− + P − 1 cos(B)= XZ ZZ ZZ . (7.14) + ++ − 2 PZZ−PZZ p 1While a change in entanglement can be used to infer the existence of two-qubit interaction terms, it cannot be used to exclude the presence of single qubit terms within the Hamiltonian. 7.3. MAPPING THE ENTANGLEMENT 95

Input State ψ(0) C2(t) | i ψ 00 sin2[2(c c )t] | 1i | i 1 − 2 ψ 01 sin2[2(c + c )t] | 2i | i 1 2 ψ ( 0 + 1 ) ( 0 + 1 ) sin2[2(c c )t] | 3i | i | i ⊗ | i | i 2 − 3 ψ ( 0 + 1 ) ( 0 1 ) sin2[2(c + c )t] | 4i | i | i ⊗ | i − | i 2 3 Table 7.3: The analytic form of the entanglement generated by Eq. (7.10) for four diﬀerent input states.

In Table 7.3 we consider the time evolution of the entanglement given four different initial states ( ψ to ψ ). In each case the evolution is a simple sinusoidal | 1i | 4i function with frequency given by the combination of two of the three parameters in the Hamiltonian given in Eq. (7.10). Using the set of input states ψ to ψ , the evolution of the system due to the | 1i | 4i Heisenberg Hamiltonian results in a significant simplification of Eqs. (7.12)-(7.14). + + For instance, if the systems starts in state ψ , then P −(t) = P − (t) = 0 for | 1i ZZ ZZ ++ all time, whereas starting with ψ gives P (t) = P −−(t) = 0. In fact, for ψ | 2i ZZ ZZ | 3i and ψ , P (t) = 1/4 for all four measurement results and therefore these states | 4i ZZ need not be measured at all. These relations drastically reduce the number of measurements required to determine the concurrence and are true for any value of the coefficients of Eq. (7.10), as they lead directly from the symmetries of this Hamiltonian. The input states considered here are either the computational states or can be reached from the computational states using a Hadamard rotation on both qubits. As the frequency of oscillation in each case is a linear combination of the coefficients ci, determining the frequencies for evolution from the four starting states determines all the parameters including their signs. The choice of which input states to use is largely arbitrary, depending on which frequency components are to be measured and which states can be prepared most easily. The four states discussed here are chosen purely for the fact that they can be prepared from the computational states using only Hadamard gates. A side effect of using the Fourier transform and the squared concurrence is that it removes any sign information, hence the need for four states in general. If the sign 96 Chapter 7. TWO-QUBIT CHARACTERISATION of all the coefficients are known beforehand, or can be determined with a minimal number of measurements, then any three of these input states are sufficient for complete characterisation.

The Fourier transform of the oscillation data gives the system parameters but, in contrast to the single qubit case discussed in earlier chapters, these depend on the peak positions in frequency space, rather than the peak amplitudes. While the oscillation frequencies present in the concurrence evolution are also present in the original probability evolution, the use of entanglement as a measure means the evolution is invariant under interchange of qubits and unaﬀected by the inclusion of single qubit terms which commute with the two-qubit interaction.

At this point an obvious question is, can we use other measures of entanglement or is concurrence somehow special? As we are only considering pure states, all bipartite entanglement measures are equivalent and so the difference comes down to implementation. In order to measure the Hamiltonian components accurately, it is important that the entanglement measure we use does not artificially introduce spurious frequencies into the evolution. This immediately rules out any entropic measure which depends on a function of the form f(x) = x log(x) because if x(t) varies sinusoidally, the logarithm of this function contains an infinite number of higher order harmonics. These higher order harmonics complicate the frequency analysis and prevent unambiguous discrimination of the Hamiltonian components. Most common entanglement measures are in some way related to the von Neumann entropy (i.e. x(t) ρ(t)) and therefore suffer from this problem. These include ⇔ the entropy of entanglement [BDSW96], the entanglement of formation [Woo98] and logarithmic negativity [Ple05]. Interestingly though, using the square of the negativity itself as an entanglement measure results in equivalent expressions to those obtained with the square of the concurrence.

Another consideration is how easily can the required measurements be performed experimentally, as most measures of entanglement require the complete reconstruction of the density matrix or at least a partial reconstruction. The advantage of using concurrence is that it has a closed form which requires only two measurement channels, as shown in Eqs. (7.12)-(7.14). In fact this is the minimum number of 7.4. UNCERTAINTY ESTIMATION AND GATE ERRORS 97

ψ | 1i 1 C2 0.5

0 ψ 0 10 20 30 40 50 60 | 2i 1 C2 0.5

0 ψ 0 2 4 6 8 10 12 14 16 18 20 | 3i 1 2 C 0.5

0 ψ 0 5 10 15 20 25 30 35 40 45 | 4i 1 2 C 0.5

0 0 2 4 6 8 10 12 14 16 18 Time [Energy−1]

Figure 7.1: Plot of the sampled entanglement, C2(t), as a function of time (in inverse units of the Hamiltonian energy) for the input states given in table 7.3, for an example Hamiltonian H = 1.2XX + 0.6YY + 1.4ZZ. Each time point is the average of Ne = 10 measurements and there are Ns = 200 time points. In each case the observation time has been chosen to obtain approximately the same number of sample points per oscillation period, independent of input state. measurement channels required to characterise a Heisenberg type Hamiltonian with arbitrary coeﬃcients.

7.4 Uncertainty estimation and gate errors

To illustrate our analysis procedure visually, Fig. 7.1 shows the evolution of the entanglement for an example Hamiltonian given Ne = 10 entanglement measurements at each time point. Fig. 7.2 shows the Fourier transform of this data, illustrating the peaks clearly above the noise ﬂoor. From this example we see that even though the oscillations in the time domain are not well resolved, in the frequency domain the peaks can clearly be seen above the discretisation (or projection) noise, which is due to the use of a ﬁnite number of projective measurements. As this characterisation process ultimately relies on accurate determination of the oscillation frequency, many of the existing techniques for frequency standards are directly applicable [HMP+97, WBIH94]. Rather than perform the extensive 98 Chapter 7. TWO-QUBIT CHARACTERISATION

ψ | 1i ψ2 0.25 | i ψ3 |ψ i | 4i 0.2

0.15

Fourier coefficient 0.1

0.05

0 0 0.5 1 1.5 Frequency [Energy]

Figure 7.2: Discrete Fourier transform of the data shown in Fig. 7.1 for diﬀerent input states. From the position of the peaks, the values of the Hamiltonian parameters can be determined. uncertainty analysis of chapter 5, we will just look at the scaling behaviour and use this to estimate the ﬁnal uncertainty. In practice, the analysis developed in chapter 5 to determine the frequency uncertainty is directly applicable to this problem.

Ultimately, there are two parameters to be chosen, the number of discrete time points, Ns, and the number of ensemble measurements, Ne. The minimum number of discrete time points is governed by the Nyquist criteria, giving N 2t /t s ≥ ob osc where tosc is the period of oscillation and tob is the maximum time over which the system is observed. To reduce the frequency uncertainty, tob should be as large as possible, though this will be limited by the decoherence time of the system. As we have a single frequency oscillation, the uncertainty in the frequency determination can be reduced by having large numbers of ensemble measurements on the last few time points and using this to estimate the phase of the oscillation.

In the ideal case (where Ns is large), only two measurements are necessary at all time points with the exception that Ne measurements are taken at the ﬁnal two points, giving a total number of measurements NT = 2Ns + 2Ne. This is in contrast to the example given in Fig. 7.1, where the same number of measurements are taken at each time point. The error in the phase determination on the ﬁnal two points 7.4. UNCERTAINTY ESTIMATION AND GATE ERRORS 99

is given by the projection noise and scales as 1/√Ne, giving the uncertainty in the frequency as [HMP+97] 4π δω = . (7.15) tob√Ne The fractional uncertainty in the frequency is therefore given by

δω 4 , (7.16) ω ≥ Ns√Ne which is both a function of the number of ensemble measurements and the number of time points.

When constructing gates for quantum computing applications, it is important to link the uncertainties in the characterisation process to typical error models to determine the probability of a gate error produced by an uncertainty in the measured system Hamiltonian. To do this, we define an imperfect gate operation U U U im ≡ ² such that U is the required gate operation followed by some error gate U². Given 1 Uim, the effective error gate is U² = UimU − . The effective error probability is then defined as 1 p = 1 Tr[U ] 2 . (7.17) eff − 16 | ² |

When characterising a quantum system, if the Hamiltonian is found to deviate from the form expected on theoretical grounds by an amount such that the error introduced by this deviation is larger than that due to characterisation uncertainties, we then use the measured Hamiltonian (rather than the theoretical one) to construct the gate. For many Hamiltonians, a two-qubit gate can be constructed using canonical decomposition, which uses at most, three applications of the Hamiltonian together with single qubit rotations [ZVSW03, ZW05]. As our procedure measures the various terms in the Hamiltonian directly, it allows the construction of a pulse sequence to perform the required two-qubit gate, even when the Hamiltonian diﬀers greatly from the theoretically expected form. Using this type of gate construction, the error rate of the gate is now governed by the characterisation uncertainties alone.

To make this more concrete, we can calculate the peﬀ for two common examples of native gates, assuming they are generated from an ideal Hamiltonian (i.e. 100 Chapter 7. TWO-QUBIT CHARACTERISATION theoretical). The analysis is similar for the case of a well characterised but non- ideal Hamiltonian, though there is a cumulative eﬀect if the two-qubit interaction is applied multiple times.

For an ideal Ising Hamiltonian (c1 = c2 = 0, c3 = J), the native gate is the CNOT gate, which can be constructed by applying the Ising Hamiltonian for a time tgate = π/(4J) combined with appropriate single-qubit rotations. Consider an example where characterisation is performed on the system, resulting in c1 = c2 = 0 and c = J ², with ² = δω/ω the uncertainty in the peak position. We then take an 3 ± imperfect gate generated by a pulse of length t = π(1+ ²)/(4J), which is equivalent to imperfect knowledge of J. This gives the eﬀective error probability,

π² π² 2 p = sin2 , (7.18) eﬀ 4 ≈ 4 ³ ´ ³ ´ for small ², assuming errors in the single qubit rotations are negligible. Similarly, for an ideal isotropic Heisenberg Hamiltonian (c1 = c2 = c3 = d), the native entangling gate is the square-root-of-swap (√SWAP). Following the same procedure (assuming that the characterisation procedure leads to a common uncertainty ² in the peak positions) we obtain an equivalent expression,

3 π² p = sin2 . (7.19) eﬀ 4 4 ³ ´

In Fig. 7.3, peff is plotted for both the Ising and Heisenberg Hamiltonians for two different values of Ns and compared to the conservative (and often assumed) fault- 4 tolerant threshold of 10− . This is the effective error rate, below which arbitrary quantum computation can be performed using concatenated error correction codes, as discussed in section 5.8 of chapter 5. The larger the value of Ns, the more precise the initial estimate when Ne = 1. As Ne increases, the uncertainty scales as

1/√Ne, as expected. This allows us to calculate directly the time needed to initially characterise the system to obtain a given gate error rate. For instance, if Ns = 10 4 time points are chosen, then a conservative estimate of NT = 10 measurements are needed to reduce the error rate to below that required to satisfy the fault- tolerant threshold, again neglecting the eﬀects of single qubit errors. If more time 7.5. EFFECT OF SINGLE QUBIT TERMS IN THE HAMILTONIAN 101

−1 −2 10 10

N =10 s −3 10 ) ω / ω

−2 −4

δ 10 10

−5 10 N =100 s Ising (CNOT) eff

−3 −6 p 10 10 Fractional Uncertainty (

−7 10 Ising (CNOT) Heisenberg (√SWAP) Fault Tolerant Threshold −4 −8 10 10 1 2 3 4 5 10 10 10 10 10 Number of Measurements (N ) T

Figure 7.3: The uncertainty in the Hamiltonian parameters as a function of the total number of measurements NT = 2Ns + 2Ne, obtained from Eq. 7.16. The curves are plotted for initial values of Ns = 10 and Ns = 100, for increasing Ne. The right hand axis shows the effective probability of a discrete gate error (peff ) for the Ising case (the Heisenberg case differs by a factor of 3/4). points are used, the required number of measurements reduces accordingly, though this is limited by the requirement that at least two measurements are required at each time point to measure the concurrence. These estimates for the number of measurements required should be compared to the case considered in chapter 5

4 8 4 where N = 10 10 was required to achieve a probability of error, p = 10− . T − eﬀ This discrepancy stems directly from the fact that the components of the Heisenberg Hamiltonian can be determined from the peak position in Fourier space, whereas the single-qubit characterisation required both the peak height and position.

7.5 Eﬀect of single qubit terms in the Hamilto- nian

Throughout this chapter, we have assumed that the unknown Hamiltonian took the form

H = H2q + H1q, (7.20) 102 Chapter 7. TWO-QUBIT CHARACTERISATION

where H2q is given by Eq. (7.10) and H1q are single qubit terms such that [H2q,H1q]= 0. This restriction allows us to factor the evolution into separate single- and two- qubit evolution (U = U2qU1q) where the single qubit evolution U1q does not change the entanglement of the system. While this may at first appear restrictive, it actually includes several Hamiltonians of interest to solid-state quantum computing. This includes the effective exchange interaction between phosphorous donor spins in silicon [WHP02] and the magnetic dipolar interaction between deep donors in silicon [dDD04]. For both these examples, the commutation relation holds, irrespective of the value of the various coupling parameters. A notable exception is the standard two-qubit interaction model for superconducting qubits [MSS01]. In this case, not only is characterisation difficult, but gate design requires approximate and numerical methods [ZW05, SSHG+05]. In general, for a Hamiltonian of arbitrary form, the eigenstates and therefore the evolution frequencies are non-linear functions of all the system parameters.

7.6 Eﬀect of imperfectly prepared input states

We can also consider the eﬀect of errors in the single qubit rotations used to prepare the input states given in Table 7.3. In general the system evolution is a function of six frequencies given by the sum and diﬀerences of c1, c2 and c3 and we have chosen the input states to isolate each frequency in turn. Taking an imperfect input state which is close to one of the states given in Table 7.3, e.g.

1 ψ (0) ψ (0) = ( 00 + √η 01 ), (7.21) | 1 i → | 1 iimp √1+ η | i | i for some small error probability η, and expressing the evolution of the concurrence in a series expansion about η, gives

2 2 C (t) = sin (2ω1, 2t)(1 2η) − − 1 + [cos(4ω1, 3t) cos(4ω2, 3t) (7.22) 2 − − − + cos(4ω t) cos(4ω t)] η + (η2) 1,3 − 2,3 O 7.7. CHAPTER SUMMARY 103

where ωi, j = ci cj. The evolution now contains oscillating terms at the other five ± ± system frequencies with amplitude η as well as the original evolution at a frequency given by c c and amplitude 1 2η. This means small errors in the input state 1 − 2 − do not affect the Hamiltonian parameter estimates, as they are derived from the position of the peak in frequency space, not its height. If the input state is completely unknown, the six frequency components are still present but there is now ambiguity as to which peak corresponds to which frequency. The inclusion of imperfect alignment of the measurement bases has an similar effect to imperfect state preparation, with the amplitude of the undesirable frequency components now being related to the extent of the misalignment. In this discussion we have not considered the possibility of non-Heisenberg terms, such as XZ or Y X as this complicates the situation considerably, again introducing ambiguity into the frequency spectrum. The effect of these terms is equivalent to a series of single qubit gates before and/or after the evolution [ZVSW03, ZW05] and requires more sophisticated analysis [DCH06]. However, an upper bound on the size of these terms is again given by the projection noise and so scales as 1/√Ne.

7.7 Chapter summary

In this chapter we have discussed how the characterisation methods developed in proceeding chapters can be extended to the case of interactions between two two- level systems. The general Heisenberg Hamiltonian was investigated as an example of special interest to quantum computing. The evolution of a system under this Hamiltonian was derived for various input states and measurement bases. The concept of mapping the entanglement was then introduced as an alternative method of determining the relevant system parameters. This was shown to have advantages when using an arbitrary interaction for generating an entangling gate, due to its in- sensitivity to local operations and direct connection to gate decomposition methods. A method of estimating the uncertainties in this process was presented and its implications for designing gates compatible with fault-tolerant operation of a quantum computer was investigated. This allowed us to estimate the number of measure- 104 Chapter 7. TWO-QUBIT CHARACTERISATION ments required to characterise a two-qubit interaction with suﬃcient accuracy to achieve fault-tolerant quantum computation with concatenated quantum error correction. Finally, the eﬀect of single qubit terms in the Hamiltonian and imperfect state preparation were investigated, with imperfect initialisation resulting in extra frequency components in the evolution. The type of characterisation procedure discussed in this chapter is of fundamental importance in experiments using two-qubit interactions, especially in the solid-state where precision control or uniformity of the Hamiltonian terms cannot be assumed a priori. Chapter 8

BURIED DONOR CELLULAR AUTOMATA

In the following chapters, rather than considering imperfect manufacture, we consider quantum systems which display limited coherence. Speciﬁcally we look at applications using systems which can be considered coherent for short periods of time or that display limited spacial coherence, e.g. small numbers of qubits.

In this chapter we explore an alternative quantum-dot cellular automata (QDCA) architecture using single buried dopants in semiconductors. Isolated donors constitute a logical miniaturisation of quantum-dots as they provide a very strongly confined potential and well characterised energy levels. In section 8.1 we briefly discuss the operation of a conventional QDCA cell. In section 8.2 we show how recent work on constructing a charge-based qubit for quantum computing using phosphorous donors in silicon [HDW+04, DHJ+03] can be applied to a system of dopants arranged in the layout of a QDCA, which we refer to as Buried Dopant Cellular Automata or BDCA. In section 8.3 we obtain numerical estimates for the energy levels and interaction strengths for the case of phosphorous donors in silicon (Si:P), though the concepts are generally applicable to other dopants. Using these estimates for the coupling strength, an effect model is developed in section 8.4 to investigate the switching of these cells. Conventional QDCA rely on incoherent evolution (governed by the T1 relaxation time) to mediate transitions between the logical states. For phosphorous donors in silicon we consider possible relaxation mechanisms and estimate the resulting switching time in section 8.5. In the following chapter we extend these results by investigating the possibility of using coherent evolution to mediate the transition between logical states and show that this provides a substan- tial increase in switching speed. The material in both this chapter and chapter 9 has previously appeared in reference [CGW+05a] and [CGW+05b].

105 106 Chapter 8. BURIED DONOR CELLULAR AUTOMATA

Figure 8.1: Two possible states for a basic QDCA cell where the 0 and 1 states constitute the ground or ‘computational’ states and ‘e’ labels the position of the electrons.

8.1 Quantum-dot cellular automata

The simplest QDCA is a cell composed of four quantum-dots containing two mobile electrons which can move between the dots via tunnel junctions. The electrons tend to occupy diagonally opposite sites to minimise the energy due to the Coulombic interaction. These two ground (or computational) states are labelled zero and one, see Fig. 8.1, where ‘e’ indicates the position of the electrons. The next highest ener- getic states are non-computational states and ideally are only transiently populated during correct operation.

If two dots are placed next to each other, one cell influences the state of the other cell via capacitative coupling. In an array of cells, when the first cell is switched from one computational state to the other, the rest of the chain relaxes to minimise the energy of the total system. The result of this relaxation is to transfer the state information of the initial cell along the chain without net electron flow and minimal energy dissipation. The speed at which this switching occurs is governed by the incoherent tunnelling rate of the junctions, the inverse of which is referred to as the T1 or relaxation time. Classical information processing can be performed in this scheme, as shown in Fig. 8.2 for a QDCA wire and inverter [TL94]. It is also possible to realise universal classical computation using the majority gate and therefore other non-trivial circuits of interest, such as a full-adder, using this scheme [SOA+99b]. 8.2. BURIED DONORS AND THE HYDROGENIC APPROXIMATION 107

Figure 8.2: Layout for a QDCA wire (a) and inverter (b) which demonstrate information transfer and binary inversion respectively after Tougaw and Lent [TL94].

8.2 Buried donors and the hydrogenic approximation

The use of buried donors in a semiconductor matrix has been discussed for charge- based quantum computing [HDW+04, CBB+03, SPP+03]. While the advantages of semiconductor fabrication and gate control are well known, the fast dephasing and relaxation effects mean that charge-based quantum computing using buried donors is still technically difficult. On the other hand the QDCA architecture is not as seriously affected by dephasing or relaxation, as the system is always in the ground state when it is measured. The basic idea is to construct an array of four ionised donors which contains two ‘free’ electrons, therefore mimicking the layout of a conventional QDCA. This configuration also represents a limit in terms of miniaturisation for this form of nano-computing as each potential well is created by only one donor atom. This type of device could be fabricated by either direct atomic placement [SCS+03, Tuc01] or ion-implantation [DHJ+03, SPP+03, JYH+05]. In order to provide a model for the Si:P donor system, we will use the effective mass or ‘hydrogenic’ approximation in which the outer shell electron of a phosphorous donor in silicon can be treated as a hydrogenic orbital with the energies and distances scaled appropriately. A more complete treatment of effective mass theory 108 Chapter 8. BURIED DONOR CELLULAR AUTOMATA for shallow donors is given by Kohn [Koh55], though the key equations are repeated here. Eq. (8.1) and (8.2) give the scaling factors for the effective Bohr radius (aB∗ ) and effective energy (E∗) of the donor electron in terms of the effective mass of the donor electron and the dielectric constant of the substrate,

me aB∗ = ²r aB, (8.1) m∗

m∗ 1 ∗ E = 2 E. (8.2) me ²r The advantage of this approach is that solving a hydrogenic system with a small number of electrons is more tractable than a full electronic calculation of the phosphorous donor within a silicon lattice. These results also generalise to other shallow donor systems. We will concentrate on phosphorous donors in silicon and therefore quote the appropriate energy levels, where the conversion is 13.6eV in a hydrogenic system is approximately equal to 20meV for Si:P. The effective Bohr radius of the phosphorous electron is approximately 3nm for an effective electron mass m∗ = 0.2me and the silicon dielectric constant ²r = 11.7. The hydrogenic approximation provides good qualitatively agreement for Si:P but it does underestimate the donor energy level, which is found experimentally to be 45meV (compared to 20meV as stated earlier). While this discrepancy can be addressed with more sophisticated techniques, the hydrogenic approximation is accurate enough for our purposes as we are interested in the energy difference between electronic configurations, rather than the absolute magnitude.

8.3 Eﬀective Hamiltonian

To provide a convenient formalism, we construct an effective Hamiltonian using a pseudo-spin approach to describe the BDCA system. By defining each pair of phosphorous donors and their shared electron as a single pseudo-spin object, we can define two states, top (T) and bottom (B), which specify the position of the electron. Each BDCA cell then consists of a pair of these objects, where the computational states are T B 0 and BT 1 respectively, as shown in Fig. 8.3(a). | i ≡ | i | i ≡ | i 8.3. EFFECTIVE HAMILTONIAN 109

Figure 8.3: (a) The ground states of the buried donor BDCA cell where the positions of the electrons ‘e’ are designated by top (T) and bottom (B). These computational states are referred to as T B and BT , and are assigned the logical values of 0 and 1 respectively. (b) The excited| i or| ‘non-computational’i states are labelled T T and BB respectively and correspond to the ﬁrst excited state of the system. | i | i

In this way, the labelling for the non-computational states is T T and BB , | i | i as shown in Fig. 8.3(b). Initially we will assume that the electrons cannot move laterally so each electron is bound to its particular donor pair, as indicated by the ellipses in Fig. 8.3. This corresponds to a situation where the direction of tunnelling is controlled by conﬁning potentials or the geometry of the cell. A more complete justiﬁcation for this assumption is given below.

Once the position of the electrons is encoded using this pseudo-spin approach, an eﬀective Hamiltonian can be developed using the Pauli spin representation,

H =∆ σ(1) +∆ σ(2) + µ σ(1) + µ σ(2) + J σ(1) σ(2) + J σ(1) σ(2) eﬀ 1 z 2 z 1 x 2 x xx x ⊗ x yy y ⊗ y + J σ(1) σ(2) + J σ(1) σ(2) + J σ(1) σ(2), (8.3) zz z ⊗ z xz x ⊗ z zx z ⊗ x

(i) where σα (α = x, y, z) is the Pauli operator applied to the ith donor pair and the coeﬃcients ∆, µ and J are eﬀective parameters to be determined from the hydrogenic model, as outlined below.

The canonical hydrogenic Hamiltonian [Sla63] of the two electron/four donor 110 Chapter 8. BURIED DONOR CELLULAR AUTOMATA system is given (in scaled atomic units) by

2 2 1 1 1 1 1 1 1 1 1 H = O1 O2 2( + + + + + + + ), (8.4) − − − r1a r2a r1b r2b r1c r2c r1d r2d − r12 where rij is the separation between the ith electron and the jth donor (j = a, b, c, d) and r12 is the separation between the electrons. Numerically evaluating this Hamil- tonian in the basis of the four states ( T B , BT , T T and BB ) enables the | i | i | i | i elements of the matrix H = ψ H ψ (8.5) ij h i| | ji to be determined. The elements of Eq. (8.5) are then equated to the coeﬃcients in Eq. (8.3) to determine an eﬀective Hamiltonian. The basis of states is represented using a linear combination of atomic orbitals (LCAO). We use the anti-symmetric spatial wavefunction for the H2 molecule as our basis wavefunction,

1 ψ(~r1, ~r2, R~ 1, R~ 2) = [φ(~r1, R~ 1)φ(~r2, R~ 2) φ(~r1, R~ 2)φ(~r2, R~ 1)], (8.6) | i 2(1 S2) − − p where φ(~r , R~ ) = exp( ~r R~ ) (8.7) i j −| i − j| is the hydrogenic wavefunction (in reduced units) for an electron at position ~ri associated with a donor at position R~ (i.e. r = ~r R~ ) and j ij | i − j|

~ ~ 2 R~ 1 R~ 2 R1 R2 S = e−| − | 1+ R~ 1 R~ 2 + | − | (8.8) Ã | − | 3 ! is the normalisation constant [Sla63]. Using this as our basis, we can express the states T B , BT , T T and BB in terms of four corresponding wavefunctions for | i | i | i | i the various donor positions. The spin-orbit coupling for donor electrons in silicon is known to be very small [App64] and therefore spin effects may be neglected as the spin and charge degrees of freedom are assumed to be decoupled at all times. The elements of Eq. (8.5) are then used to obtain estimates for the numerical coefficients in the effective Hamiltonian (Heff ) for a square BDCA cell of side length R. Using the effective model of Eq. (8.3), we can estimate the energy difference 8.3. EFFECTIVE HAMILTONIAN 111

(E E ) between the excited (non-computational) and ground (computational) ex − gs states of one cell as a function of system size. This energy gap gives an estimate of the temperature at which the system must be operated to ensure that the non- computational states are not thermally excited. The energy diﬀerence for a range of separations is plotted in Fig. 8.4. The points are found using the LCAO approach while the line is an approximation found using electrostatic arguments and the geometry of the system and has the form

(2 √2)E∗a∗ E E = − B , (8.9) ex − gs R

where R is the side length of the BDCA cell in nm and aB∗ = 3nm, E∗ = 20meV are the effective Bohr radius and effective ground state energy respectively. This approximation is found to be valid in the region where the electron wavefunctions do not strongly overlap with each other, deviating from the numerical results for R . 10nm. Assuming a Boltzmann distribution for the energy states at finite temperature, the occupation probability (P ) of the excited states ( T T and BB ) is given by ex | i | i

∆E P e− kB T , (8.10) ex ≈ where E E =∆E > 0 is the energy difference between the ground and excited ex − gs states. For a separation of 15nm, we require an operating temperature of < 3K to achieve >99% occupation of the computational states. For an operating temperature of 100mK, the occupancy of the computational states is approximately 100%. For the rest of the discussion we assume a square cell of side length 15nm. As this is equivalent to a cell size of approximately 5 Bohr radii, the electrons can be said to be well localised and overlap effects are not significant. To simplify our analysis, we set the σ , σ σ and σ σ type terms to zero and then x x ⊗ x y ⊗ y reintroduce them later in a systematic fashion. This corresponds to a situation where a surface gate potential is used to control overlap of the electron wavefunctions and therefore control the tunnelling rate between donors. For large cell sizes, this is a good approximation to the physical situation as the tunnelling rate without an 112 Chapter 8. BURIED DONOR CELLULAR AUTOMATA

Figure 8.4: Energy difference between the ground (Egs) and first excited (Eex) states of a square BDCA cell made up of phosphorous donors in silicon. The energy difference is computed for various donor separations (R) by numerically integrating the Schrödinger equation. The points are full quantum mechanical calculations using the LCAO approach and the solid line is the energy difference determined analytically from simple electrostatic arguments, Eq. (8.9). applied barrier bias will be very low. For small cell sizes there will be wavefunction overlap even without a barrier gate. In this case a confining potential can be used to localise the electrons and provide more control over the tunnelling characteristics. Given these approximations, many of the terms in Eq. (8.3) are approximately zero (with no gate voltages applied) and the only significant term for R = 15nm is

Jzz = 1.21 meV, which is due to the Coulombic repulsion between the electrons.

8.4 BDCA switching

To study how a BDCA cell would switch from one computational state to the other, we consider the eﬀect of control gates and nearby cells. We propose a structure where pairs of donors are positioned in a line with a surface ‘barrier’ gate constructed between them which is used to control the tunnelling rate between the pairs of donors. At the end of the chain, ‘symmetry’ gates allow the system to be switched 8.4. BDCA SWITCHING 113

R Symmetry Gate e e SET

Barrier Gate R

Symmetry Gate e e SET

Figure 8.5: Simpliﬁed layout of a BDCA chain, where the cell size is labelled R and the cell spacing (S) is the distance between the centre of neighbouring cells. The circles represent the position of the donors. The position of the electrons ‘e’ is shown for the ground state conﬁguration which corresponds to some non-zero bias on the control gates with the labelled polarity. The position of the electrons are measured with single-electron transistors (SET).

Figure 8.6: Illustration of the BDCA model used in this and the subsequent chapter with the electron densities plotted for the ground state, given the gate polarities shown. The symmetry and barrier surface gates are used to perform switching and control the tunnelling rate respectively. The charge density of the electrons is computed from the wavefunction corresponding to one of the computation states of the cells. from one state to another, depending on the bias applied. An illustration of this concept is shown in Fig. 8.6 while Fig. 8.5 gives a schematic view of our model with the cell size R and cell separation S labelled. Sensitive electrometers, such as single-electron transistors (SETs) [GD92], are used to measure the position of the electron at the other end of the chain. Most of the following section refers to a single BDCA cell made from two pairs of donors for simplicity, but the discussion applies equally to a long chain of donor pairs. The eﬀect of the barrier (B) and symmetry gates (S) for this cell can be included by adding terms to the eﬀective Hamiltonian, Eq. (8.3), of the form

E (t)σ(1) E (t)(σ(1) + σ(2)), (8.11) S z − B x x 114 Chapter 8. BURIED DONOR CELLULAR AUTOMATA

where the magnitude of the coeﬃcients (ES and EB) are controlled by the voltages applied to the gates.

The symmetry gates localise the system in one of the two computational states, based on the gate’s polarity. The voltages applied to each symmetry gate have equal magnitude but opposite sign to ensure a symmetrical effect on the chain. The barrier gate controls the tunnelling rate between pairs of donors by repelling or attracting the electron clouds. If there is a large enough separation between donors, the barrier gate is required to allow tunnelling by reducing the potential barrier that the electron feels. This justifies our initial assumption that when the separation is large enough tunnelling only occurs between donor pairs and not along the BDCA chain. The use of compensating gates could also be used to confine the electrons and therefore prevent tunnelling in unwanted directions.

The effect of these surface gates has been modelled as pure σx and σz terms in the Hamiltonian due to the symmetry of the system. We estimate the surface gate voltages to be 100 1000mV depending on the donor depth and the presence of an − oxide barrier layer, based on estimated gate voltages for charge-qubits [HDW+04, LGD+05]. Assuming arbitrary high precision in donor placement, the barrier gate has an equal effect on donors either side of the barrier and can be considered as a pure σ component of the Hamiltonian. The barrier gate would also induce σ σ x x ⊗ x and σ σ style coupling but this is expected to be small compared to the J and y ⊗ y ZZ pure σx coupling. In addition, the barrier gate can also have a negative bias applied to improve the localisation of the computational states during readout, though this is not directly modelled here. While the symmetry gates would not be pure σz (having some residual σx effect due to wave-function overlap) this could also be compensated for by using the barrier gate or additional compensation gates.

In Fig. 8.7 the eigenspectrum for a single BDCA cell is plotted as a function of the symmetry gate potential (ES) for zero barrier gate potential. The computational states ( T B and BT ) are localised even for a very small symmetry potential. Also | i | i note that the two computational states are degenerate when there is no potential diﬀerence applied to the control gates, as expected. 8.5. INCOHERENT SWITCHING 115

Figure 8.7: Eigenspectrum for a four donor cell as the symmetry potential (ES) is swept from -2 to 2 (meV) with no applied barrier potential (EB = 0) and a cell size R = 15nm.

8.5 Incoherent switching

If we allow the system to evolve via incoherent relaxation (in direct analogy with the quantum-dot schemes), the transition from a high to a low energy eigenstate of the system is mediated by phonons in the lattice. For the moment we will ignore the eﬀect of the barrier gate and assume the electrons are well localised. This situation is valid over time scales greater than the dephasing time of the donor wavefunction or where the barrier gate has a negative bias producing a high potential barrier. To obtain an estimate for this rate, we assume that the electrons within a BDCA cell can relax independently and we therefore use a similar approach to that used to estimate the relaxation rate for charge-qubits based on buried donors [BM03, FF04]. Following the approach used by Barrett and Milburn [BM03] and Bockelmann and Bastard [BB90], we write the relaxation rate due to thermal phonons,

2 3 64D q [nB(E, Tph)+ α][1 sinc(qif R)] Γ = if − , (8.12) ph ~ 2 2 4 πρ cs[(qif aB) + 4] 116 Chapter 8. BURIED DONOR CELLULAR AUTOMATA

where α = 1 for emission and 0 for absorption of a phonon, qif is the wavenum- ber of a phonon with a magnitude equal to the energy diﬀerence between the states ( q = E/~c ), R is the separation between the donors and n (E, T ) = | if | s B ph 1 [exp (E/k T ) 1]− is the Bose occupation function for a bath of phonons at B ph − temperature Tph. We have ignored eﬀects due to coherent tunnelling and used the following parameters [BM03] for Si:P where D = 3.3eV is the deformation potential,

3 3 1 ρ = 2329kgm− is the density of silicon, c = 9.0 10 ms− is the speed of sound in s × silicon and aB = 3nm is the effective Bohr radius of the donor electron. To estimate the incoherent switching time, we calculate the energy levels for a BDCA cell (the target cell) as a function of the state of the neighbouring cell. In the incoherent limit, we can assume that each electron is well localised and so the effect of the neighbour cell is to lift the degeneracy of the computational states of the target cell. For a cell of side length R = 15nm and cell spacing S = 30nm, a well localised neighbour induces an energy splitting of 1.64meV between the computational states of the target cell. This is calculated using the difference in electrostatic repulsion between the target cell and its neighbour. The splitting caused by the neighbouring cell can be modelled as a bias on the symmetry gate of the target cell. In this case the equivalent symmetry gate potential is ES = 0.82meV. The resulting energy levels can be read from Fig. 8.7. Using these energy levels we can calculate the relaxation rate from the first excited state to the ground state ( BT T B ) and | i → | i estimate the switching time of the system. Relaxation in this system is phonon mediated and therefore acts on each electron separately. This means the direct transition from state BT to state T B is sup- | i | i pressed and instead the system must relax via a co-tunnelling (two electron) process which requires absorption and emission of phonons to reach the ground state. The two possible (first order) decay paths are illustrated in Fig. 8.8 for the energy levels corresponding to a symmetry potential of 0.82meV. To estimate the total relaxation rate, we add the co-tunnelling rates [GD92] for the two possible paths,

(1) (1)~ (2) (2)~ ΓA ΓE ΓA ΓE ΓTotal = + , (8.13) E TT E BT E BB E BT | | i − | i| | | i − | i| 8.5. INCOHERENT SWITCHING 117

|BB〉 2.03 meV

(2) ΓA

|ΤΤ〉 0.39 meV (1) ΓA |ΒΤ〉 -0.39 meV (1) (2) ΓE ΓE

|ΤΒ〉 -2.03 meV

Figure 8.8: Energy level diagram for a single BDCA cell in the presence of a well localised neighbouring cell, where the cell size R = 15nm and there is a cell spacing S = 30nm. The direct transition ( BT T B ) is suppressed as the interaction is phonon mediated and must therefore| proceedi → | viai single-electron transitions. The two (ﬁrst order) decay paths from one computational state to the other are illustrated with their associated transition rates for absorption (A) and emission (E).

(1,2) where ΓA,E are defined in Fig. 8.8 and E k is the energy of the k th state. For | i | i these energy levels and an operating temperature of 3K, the calculated relaxation rate is ΓTotal = 1.1MHz, which gives a switching time of 0.9µs. This is almost two orders of magnitude slower than the estimated maximum switching rates of 90MHz for Al/Al-oxide QDCA structures [OAT+99] at 70mK. This is to be expected as there are no defined tunnel junctions in the buried donor case. The switching time is shown in Fig. 8.9 for a range of operating temperatures and cell sizes (R) with the spacing between neighbouring cells given by the cell centre-to-centre distance S = 2R. While the switching rate varies with cell size, temperature effects dominate in this regime as the system requires enough thermal energy to mediate the two- electron transition. At higher temperatures, faster switching is expected but the occupation of the ground state is reduced, according to Eq. (8.10), resulting in an overall loss of fidelity.

While the switching time is slow compared to modern microelectronics, it does demonstrate that if the system is initially setup in some unknown excited state it will decay to a known state, i.e. the ground state. The switching time of the system could also be improved by reducing the barrier potential seen by the donors which results in more overlap between the donor sites. Though this technique will be limited by 118 Chapter 8. BURIED DONOR CELLULAR AUTOMATA

Figure 8.9: Incoherent switching time calculated for a range of cell sizes (R) with the spacing between the neighbouring cells given by the cell centre-to-centre distance S = 2R. Higher operating temperatures result in faster switching times but also result in higher excited state populations, reducing the overall fidelity. the fast dephasing of the charge system which tends to localise the electrons. In chapter 9 we consider an alternative switching scheme using coherent evolution of the system to perform the required switching within this dephasing time. Circuits based on QDCA were originally suggested as a possible replacement for standard CMOS (Complementary metal-oxide-semiconductor) computing technology. At present, this possibility still seems remote, due to a range of technical issues, including the difficulty of miniaturisation and the high precision control and fabrication required. Constructing a QDCA circuit using atomically placed donors suggests a solution to both the miniaturisation and fabrication precision problems. Unfortunately, it also introduces other problems such as the difficulty of maintaining localisation and readout of the state of the system. Given these fabrication difficulties it is, at present, unlikely that an entire CMOS based electronic device could be replaced by QDCA circuits. On the other hand, QDCA circuits still show promise for niche applications where they are integrated into conventional circuitry. In this case, simple computational tasks could be performed by QDCA, which have a lower thermal budget and smaller space require- 8.6. CHAPTER SUMMARY 119 ments, in order to improve the efficiency of the entire device.

8.6 Chapter summary

In this chapter we have presented a new architecture for realising a quantum-dot cellular automata using single phosphorous donors in silicon. We have explicitly calculated the interaction strengths and resulting energy levels for this system using the hydrogenic approximation for shallow donors. Using a pseudo-spin model, we have investigated the switching behaviour of this system when operating via incoherent relaxation and ﬁnd the switching speeds to be of the order of microseconds for a cell size of 15nm. In chapter 9 we investigate the possibility of coherent switching, which results in a considerably faster switching speeds. 120 Chapter 8. BURIED DONOR CELLULAR AUTOMATA Chapter 9

COHERENT BURIED DONOR CELLULAR AUTOMATA

In this chapter we extend the results of the previous chapter by investigating an alternative ultra-fast (picosecond) switching mechanism by incorporating coherent evolution between the system eigenstates. The use of coherent evolution is central to the use of buried donors to construct a quantum-dot cellular automata (QDCA) but is also applicable to other solid-state systems such as quantum dots or superconducting structures. This constitutes an alternative switching mechanism for QDCA schemes where coherence can be maintained long enough for the cell to be switched from one classical state to another without the need for the long coherence times typically required for quantum computing applications. In section 9.1, the basic concept of adiabatic switching is discussed and a model developed to study this effect for buried donor cellular automata or BDCA cells. The relevant time-scales are calculated numerically in section 9.2 while in section 9.3 we investigate the effect of decoherence on the switching behaviour. Scaling behaviour of a chain of BDCA cells is treated in section 9.4, including the effects of large numbers of cells on both the switching fidelity and maximum switching speed.

9.1 Coherent switching

An alternative to the incoherent switching discussed in the previous chapter is to use the high tunnelling rates of charge-based quantum-computing schemes to perform coherent switching of the BDCA chain. As we have relatively strong coupling between electrons in this system, we can consider adiabatic evolution as a mechanism to switch from one computational state to the other. This is analogous to the technique of Rapid Adiabatic Passage for electromagnetically induced population transfer of atoms and molecules [VHSB01]. While this method has similar

121 122 Chapter 9. COHERENT BURIED DONOR CELLULAR AUTOMATA advantages to the adiabatic switching discussed by Lent and Tougaw [LT97], the method discussed here is entirely coherent and therefore does not rely on dissipation to ensure the ground state is always occupied. To study adiabatic switching, we use the model introduced in chapter 8 with cell size R = 15nm but we now consider varying the barrier and symmetry potentials,

EB and ES in Eq. 8.11. The eﬀect of the barrier potential on the eigenspectrum of a BDCA cell is shown in Fig. 9.1, with EB = 1.2meV. Modifying the barrier within each donor pair delocalises the electron and increases the coupling between the donors. When E = 0, there is an energy gap between the ground and ﬁrst excited B 6 states at the point of zero symmetry potential, at which point the eigenstates include contributions from all four basis states, not just the computational states.

When ES = 2meV and EB = 1.2meV the computational ground state population has been reduced to 80% (compared to without barrier gate induced coupling between the donors). For example, the lowest energy eigenstate of the system is

0.39 T T + 0.89 T B + 0.14 BT + 0.17 BB for ES = 2meV ψ gs = | i | i | i | i − (9.1) | i  0.17 T T + 0.14 T B + 0.89 BT + 0.39 BB for E = 2meV  | i | i | i | i S  when written out in the position basis, for EB = 1.2meV. For these eigenstates, the system can no longer be considered to be only in the computational states T B or | i BT , resulting in a loss of fidelity during operation. | i The energy gap between the ground and first excited state allows adiabatic evolution to be used to shift the population from one state to the other, provided that the adiabatic criteria is satisfied. The adiabatic criteria can be stated as [Mes65]

e ∂H g h | ∂t | i 1, (9.2) e H e g H g 2 ¿ |h | | i−h | | i| where g and e are the ground and ﬁrst excited states respectively. We therefore | i | i want to control the symmetry and barrier gates in such a way as to maximise the energy gap (allowing fast adiabatic operation) but maintain good localisation for readout. The pulse scheme given in Fig. 9.2 can be applied to achieve the energy level split- 9.1. COHERENT SWITCHING 123

Figure 9.1: Eigenspectrum for a four donor cell as the symmetry potential (ES) is swept from -2 to 2 (meV) with an applied barrier potential (EB) of 1.2meV

ting while still ensuring the computational states are highly populated for readout. This involves applying a voltage pulse to the barrier gate to varying the tunnelling rate in a Gaussian fashion, while simultaneously switching the control gates from one polarity to the other. The following functions were used for the barrier and symmetry gate terms in Eq. 8.11,

(t t /2)2 E (t)= B exp − p (9.3) B max − 2ς2 · ¸

(t tp/2) ES(t)= Smaxerf − , (9.4) − √2ς · ¸ where tp is the total time over which the pulse sequence is applied and ς is the standard deviation of the pulse, which was set to ς = tp/6. Bmax and Smax are the maximum barrier and symmetry potentials respectively. The resulting eigenspectrum is shown for this pulse sequence in Fig. 9.3, as a fraction of the pulse time tp. The degeneracy of the ﬁrst two states is lifted but the computational states are still strongly populated (> 99.999%) before and after the application of the pulse scheme as E 0 at t = 0 and t = t . B ≈ p 124 Chapter 9. COHERENT BURIED DONOR CELLULAR AUTOMATA

Figure 9.2: The symmetry (ES) and splitting (EB) potentials which are applied to the system to achieve adiabatic evolution, where tp is the time over which the pulse is applied and the standard deviation of the pulses (ς) is set to tp/6. Bmax and Smax are the maximum barrier and symmetry potentials and are used to control the amount of tunnelling and localisation respectively.

9.2 Time dependant behaviour

As we are only considering a relatively small basis of states, we can numerically solve the density matrix master equation to study the time dependence of the system including decoherence. The equation of motion is

i ρ˙ = [H,ρ]+ [ρ], (9.5) −~ L where the Liouvillian ( [ρ]) describes the decoherence of the system. Integrating L Eq. (9.5) in the limit of no decoherence ( [ρ] = 0) gives the pure state of the L system as it evolves over time. Fig. 9.4 shows the state population for the pulse sequence given in section 9.1 over a pulse time tp = 100ps with Smax = 2meV and B = 1.2meV. The system is initially in T B and is then adiabatically switched max | i to BT while only transiently occupying the non-computational states. | i To determine the ﬁdelity of transfer, we plot the ﬁnal occupation probability of each state as a function of total pulse time (t ) assuming we start in state T B , p | i 9.3. THE EFFECT OF DEPHASING 125

Figure 9.3: Eigenspectrum for a four donor cell as the pulse scheme given in Eq. (9.3) and (9.4) is applied to the barrier and symmetry gates respectively, where Smax = 2meV and Bmax = 1.2meV. as shown in Fig. 9.5. Three distinct regions can be identiﬁed. For pulse times of less than 0.1ps, the pulse sequence is applied too quickly for the system to evolve, which is to be expected as the pulse time is much less than the tunnelling rates of the system. Pulse times of greater than 20ps satisfy the adiabatic criteria, Eq. (9.2), and the system moves smoothly from one computational state to the other with a ﬁdelity of 99.95%. Between these regions we see that after switching, the non- ≥ computational states are occupied with varying probabilities.

9.3 The eﬀect of dephasing

To study the eﬀects of decoherence, we introduce [ρ] = 0 in Eq. (9.5). As we have L 6 shown previously that the relaxation rate (1/T1) due to phonons is expected to be of the order of microseconds, we only consider a phenomenological Γ2 = 1/T2 (pure dephasing) rate. We simply model this as a decay of the oﬀ-diagonal terms of the density matrix, [ρ]=Γ [ρ diag(ρ)]. (9.6) L 2 − 126 Chapter 9. COHERENT BURIED DONOR CELLULAR AUTOMATA

Figure 9.4: Population of states as a function of time showing complete population transfer from the state T B to state BT while only transiently populating the non- computational states (|BBi and T T| ).i The time over which the pulse sequence is | i | i applied is tp = 100ps and the eﬀects of decoherence are ignored.

Fig. 9.6 shows the probability of successful transfer from one computational state to the other as a function of total pulse time and dephasing time T2. The region of 99% transfer is enclosed by the dotted contour line on the plot. The region in the ≥ top right corner corresponds to the system dephasing faster than it is being switched, resulting in a loss of fidelity. The final state in this region is a uniform statistical mixture of the computational and non-computational states. From Fig. 9.6 we see that with finite dephasing, there is a window within which coherent transfer can still occur. For dephasing of 500ps and a total pulse time of 20ps, a transfer probability of >99% can be achieved.

9.4 Scalability of the buried donor scheme

The buried donor scheme can be scaled by adding more pairs of donors in a similar fashion to that used for the quantum-dot system to form a line of cells. The incoherent switching time for a line of cells is predicted to scale approximately linearly based on simulations of quantum-dot systems [LT97]. To compare this to the scaling 9.4. SCALABILITY OF THE BURIED DONOR SCHEME 127

Figure 9.5: Final population of states as a function of total pulse time (tp), ignoring the eﬀects of decoherence. High ﬁdelity transfer ( 99.95%) between computational states is observed for pulse times greater than 20ps.≥ For pulse times of less than 0.1ps the system does not have time to evolve from its initial state. Between these times, the non-computational states are partially occupied.

of the coherent scheme discussed in this chapter, we estimate the time to adiabatically switch a chain of buried-donor cells. As shown in Fig. 8.5, this conﬁguration involves a ‘strip’ barrier gate running the length of the chain and a pair of symmetry gates at one end. The switching of the chain is achieved by applying the same pulse sequence given earlier, Eq. (9.3) and (9.4), to coherently follow the adiabatic path from one computational state to the other. This is incorporated into an eﬀective Hamiltonian of the form

N N 1 − (1) (i) (i) (i+1) Heff = ES(t)σz + EB(t) σx + JZZ σz σz , (9.7) i=1 i=1 X X where N is the number of donor pairs (or N/2 BDCA cells) and we have only included the effects of nearest neighbour coupling. The interaction term JZZ is approximated using Eq. (8.9) where J = (E E )/2 for a given cell size R. The minimum ZZ ex − gs evolution time which satisfies the adiabatic criteria will increase with the number of donor pairs and is controlled by the scaling behaviour of the energy gap between 128 Chapter 9. COHERENT BURIED DONOR CELLULAR AUTOMATA

Figure 9.6: Probability of successful transfer as a function of both total pulse time and dephasing time. The region of 99% successful transfer is enclosed by the dotted contour line in the bottom right≥ corner.

the ground state and the ﬁrst excited state, Egap. The energy gap is limited by the height of the potential barrier which the electrons see, which is controlled by

EB. This model assumes that the entire chain is in a pure state throughout the transfer and therefore the transfer must be fast compared to the dephasing time of the system.

To simplify the analysis we will ignore decoherence and only consider situations where the minimum energy gap occurs at the midpoint of the pulse sequence, where

ES = 0. This puts a limit on the maximum barrier coupling which can be applied in order to introduce an energy gap. Fig. 9.7 shows the maximum coupling (Bmax) which still maintains the minimum energy gap at the centre of the eigenspectrum. The scaling behaviour of the maximum allowable barrier coupling is modelled by ﬁtting to an exponential decay,

B 0.49 exp( N/4.2) + 1.32, (9.8) max ≈ −

where N is the number of donor pairs and Bmax is the maximum barrier coupling in meV.

The exact behaviour of the system for large numbers of donors is computationally 9.4. SCALABILITY OF THE BURIED DONOR SCHEME 129

Figure 9.7: Scaling behaviour for the maximum barrier coupling (Bmax) which can be applied while still maintaining the minimum energy gap at the centre of the eigenspectrum (the degeneracy point). Bmax is calculated for increasing numbers of donors pairs and then ﬁtted to an exponential function.

expensive to calculate. We can obtain an estimate for the minimum pulse time (tp) which still provides high fidelity ( 99.95%) transfer by observing the scaling of the ≥ adiabatic time, (1) 6√2S e σz g t = κ max h | | i, (9.9) adiab √π E 2 | gap| with increasing number of donor pairs, where κ is a scaling constant used to compare the minimum evolution time with the previous calculations. This equation is derived by assuming that the adiabatic criteria, Eq. (9.2), must be less than 1/κ to achieve adiabatic evolution and noting that the time derivative of Eq. (8.11) simplifies considerably at the degeneracy point (the centre of the eigenspectrum, Fig. 9.3). At this point the majority of the Hamiltonian is constant in time and the time derivative of the symmetry bias, Eq. (9.4), gives the numerical prefactors of Eq. (9.9). Setting κ = 20ps corresponds to the required pulse time to achieve 99.95% fidelity when ≥ switching a single cell comprised of 2 donor pairs with a cell size of R = 15nm, as shown in Fig. 9.5. Fig. 9.8 shows tadiab for up to 12 donor pairs (6 QDCA cells) calculated by diagonalising the effective Hamiltonian of the system, Eq. (9.7), for 130 Chapter 9. COHERENT BURIED DONOR CELLULAR AUTOMATA

Figure 9.8: Scaling behaviour of tadiab as a function of the number of donor pairs in a BDCA chain, for various cell sizes (R). This gives an estimate for the scaling of the minimum allowable evolution time for high ﬁdelity transfer. several diﬀerent cell sizes. As the cell size is increased, the Coulombic coupling between pairs of donors is reduced and so the minimum switching time increases.

While tadiab may underestimate the minimum allowable evolution time, we expect the scaling behaviour of the system to be similar. The scaling behaviour is approximately linear for these system parameters, though it will ultimately be restricted by the decoherence time of the system. This could be improved with a stronger bias ﬁeld or by bringing the donors closer together.

Based on simulations of incoherent adiabatic switching [LT97], we expect the coherent control techniques discussed in this chapter to be applicable to more complex logic structures. It should be noted however that large scale QDCA structures require staggered gate sequences where one region of the computation is allowed to proceed while other regions are held in a particular state. The extension of this idea to coherent switching is an area for further study, as the inherent time reversibility of quantum mechanics makes this type of latched operation more diﬃcult to realise. The operation of multi-bit QDCA gates using coherent switching also needs to be investigated as these too would need to be constructed in such a way as to be 9.5. CHAPTER SUMMARY 131 completely reversible. The adiabatic evolution discussed here could be applied to many types of coherent systems based on quantum-dots, not just those based on buried donors. The adiabatic pathway allows for fast switching and high ﬁdelity with minimal requirements on gate timing and accuracy and in this way is similar to Coherent Transfer by Adiabatic Passage [GCHH04].

9.5 Chapter summary

The use of coherent evolution to provide fast and controllable switching of BDCA cells was investigated for the case where quantum coherence can be maintained throughout the switching process. This was found to provide a fast and effective switching mechanism, with a cell of side length 15nm having a coherent switching time of 20ps and a fidelity of greater than 99.95%. The effects of dephasing on this process was investigated and found to have minimal effect as long as the dephasing time is approximately 10-100 times greater than the switching time. The scaling behaviour of the system was investigated for a simple line of cells and found to scale approximately linearly with the number of cells. 132 Chapter 9. COHERENT BURIED DONOR CELLULAR AUTOMATA Chapter 10

SIMULATING THE ISING MODEL ON A TYPE-II QUANTUM COMPUTER

In this chapter, the implementation of the Metropolis Monte Carlo algorithm is investigated as a possible application of a distributed quantum/classical computer, or type-II quantum computer (T2QC). The Ising model is used as an example of a physical system which can be simulated in this way. The operation of the type- II quantum computer is discussed in section 10.1. An algorithm is presented in section 10.2 for simulating the Ising model which can run on this type of hybrid computer. The necessary gate operations for simulating the Ising model are presented for one and two dimensions in sections 10.3 and 10.4 respectively, while the classical communication step is discussed in section 10.5. Section 10.6 discusses the implementation of this algorithm on architectures with and without ensemble readout and computes the required gate accuracy. The eﬀect of ensemble readout is then studied in section 10.7 and compared to that with single shot readout. The contents of this chapter has previously been published in reference [CHP04].

10.1 Type-II quantum computers for simulating quantum cellular automata

A type-II quantum computer (T2QC) is an alternative use of controllable quantum coherent systems, consisting of an array of small quantum information processors connected by classical communication [Yep01c]. Fig. 10.1 shows conceptually the design of a T2QC where each node contains a small number of qubits (10’s or fewer) and the nodes are connected via classical communication channels. Each node is initialised into some quantum state and then the entire computer undergoes some

133 134 Chapter 10. SIMULATING THE ISING MODEL ON A T2QC

Classical Communication Channels

Qubits Nodes

Figure 10.1: Design of a Type-II Quantum Computer, composed of many small conventional quantum computers global unitary operation. The state of each node is then measured and the results used to re-initialise the neighbouring nodes for the next computational step. These are known as the collision and streaming steps respectively, using the lattice-gas terminology. This form of quantum computer is able to solve certain fluid dynamics problems which are difficult to solve on a classical computer [Yep01b] [Yep01a] [VVY03]. While this is a useful application in its own right, it also provides an important ‘stepping stone’ towards globally phase coherent quantum computers of the type required to run Shor’s factoring algorithm [Sho97] and other important quantum computing algorithms. It is this possibility of using few qubit devices to perform meaningful computing tasks which has resulted in much interest in applications such as type-II quantum computers. The type-II architecture is also well suited to a range of other physical problems. The possibility of performing quantum operations on a distributed lattice of nodes immediately brings to mind the concept of a cellular automata (CA) or more specifically its quantum generalisation, a quantum cellular automata or QCA. The trade off between the number of nodes and the number of qubits per node can be thought of as the quantum/classical boundary. As we increase the number of qubits per node to that of the many 1000’s required for complex quantum simulation tasks, we approach the regime of complete quantum coherence. Taking the other extreme, of one qubit per node, results in a purely classical automata as the coherence is lost at every streaming step. In between these two extremes lies a region where some advan- 10.2. METROPOLIS SIMULATION OF THE ISING MODEL 135 tages are to be gained by using quantum coherence, though this improvement will obviously not be as great as that obtained from a fully coherent quantum computer.

One such system is the Metropolis Monte-Carlo (MC) method [MRR+53] for simulating the Ising model. It has been previously suggested that this problem can be simulated using a set of cellular automaton rules on a lattice [Vic84]. Using this approach, the problem is well suited to a type-II quantum computer as it requires many lattice points but only nearest neighbour interactions. While there are several other deterministic or semi-deterministic schemes available to simulate the Ising model with simple cellular automata [Cre86] [OP89] [Her86], they are only approximations to the Metropolis solution. This is because the random number generation required by the Metropolis algorithm is diﬃcult to implement without ﬂoat-point numbers or at least very large integers. A cellular automaton rule typically requires only small numbers of bits or integers. The major advantage of using a T2QC to implement the Metropolis algorithm is that the random number generation can be included using quantum superposition and projective measurement.

In this chapter, the evolution rules are reviewed for the Metropolis algorithm and then the necessary unitary evolution is determined using the quantum circuit formalism. The probabilistic aspect of the algorithm is incorporated in the form of the weighted coin-toss application of quantum superposition. The necessary quantum evolution circuits are derived for both the one-dimensional (1D) and two-dimensional (2D) Ising models. The implementation of these circuits is discussed for technologies where only the resulting binary state can be measured and also where the ensemble of states can be measured.

10.2 Metropolis simulation of the Ising model

The Ising model consists of a regular array of spins si which take one of two values 1, 1 . The energy at each site i is given by {− }

E = J s s , (10.1) i − i j j=i X6 136 Chapter 10. SIMULATING THE ISING MODEL ON A T2QC where the sum is over nearest neighbours. The use of periodic boundary conditions is assumed, though fixed boundaries can be easily implemented. As the spins can take only two values (assuming there is no external field) the energy at each site can only take a finite number of values, the number of which depend on the dimension of the lattice. For the case of a 1D chain of Ising spins, the change in energy can take three distinct values, ∆E = 0J, 4J. The value of J is assumed to be constant ± across the lattice and for most of the discussion, it is assumed to be positive (J > 0). This situation corresponds to the ferromagnetic case, though the generalisation to the anti-ferromagnetic case (J < 0) is straightforward. Metropolis Monte-Carlo [MRR+53] is a popular method of simulating the Ising model for a given temperature. The Metropolis algorithm involves flipping a spin at random and calculating the corresponding change in energy (∆E). This spin ∆E flip is then accepted with probability given by p = min 1,e− kB T , where T is cl { } the temperature of the lattice and kB is the Boltzmann constant. Throughout this discussion it is assumed that kB = 1 which results in temperature being expressed in units of J. As the energy levels given by Eq. (10.1) are discrete, the Metropolis algorithm is easily converted to a set of cellular automaton rules. In order to implement this, the same evolution rule must be applied to the entire lattice simultaneously. However, if the Metropolis algorithm is applied to every site simultaneously the ‘feedback catastrophe’ results, as pointed out by Vichniac [Vic84]. Instead, the rule must be applied to (at most) every second site in a checkerboard configuration [CJR79]. This can be achieved by either storing two sublattices or using a parity bit which controls whether the evolution rule is applied or not [Cre86]. As the quantum circuit formalism is universal and encompasses classical boolean logic [BBC+95], type-II quantum computers can be designed to simulate any deterministic or probabilistic cellular automata. The probabilistic aspect can be incorporated through the use of a qubit in a superposition state to provide a weighted coin-toss probability. The Ising model algorithm is presented as an example of this idea, although the process can be applied to any cellular automata rule in general. The appropriate evolution rules required to implement the Metropolis algorithm 10.3. 1D ISING MODEL 137 are given here using the conventional quantum circuit notation [BBC+95] [NC00] which is briefly reviewed in appendix A. For each circuit, S is designated the | ii si+1 on-site Ising spin (s ) where S = , S0 is the new on-site spin after application i i 2 | ii of the quantum circuit and Scl0 is the classical measurement result. This allows a simple correspondence between the Ising states and the qubit encoding, giving ‘spin-up’ (+1) as binary state 1 and ‘spin-down’ ( 1) as state 0. −

10.3 1D Ising model

For a particular on-site spin Si of the 1D Ising model, the neighbouring spins are designated A and B, where the mapping between Ising spins on a lattice and nodes of a T2QC is illustrated in Fig. 10.2. As the spins only take discrete values the change

si−1 si si+1 A S B

(a) (b)

Figure 10.2: Mapping between sites on an Ising spin lattice (a) and nodes of a T2QC (b), showing the nearest neighbour assignments for a one dimensional lattice. in energy due to a single spin flip can only take a finite set of values (∆E = 0J, 4J), ± as long as there is no global field. The Metropolis method can be implemented for this system using the circuit given in Fig. 10.3, where P is a qubit initialised in the | i 4J state P = √P 1 + √1 P 0 with P = e− T giving the appropriate probability | i | i − | i for the given temperature and coupling strength. The circuit also has an ‘ancilla’ qubit which is initialised with the state 0 and is used for temporary storage during | i the computation. The general algorithm steps are as follows

1. Initialise on-site spin for each node, S , based on the initial state of the Ising | ii lattice

2. Initialise input states A and B based on the value of the neighbouring spins using the streaming rule (see section 10.5) 138 Chapter 10. SIMULATING THE ISING MODEL ON A T2QC

0 S S | i | i 0 A Scl | i B | i 0 | i P | i

Figure 10.3: Quantum circuit for the evolution of the on-site spin S for the 1D Ising model | i

3. Apply the quantum circuit gate sequence to the entire computer simultaneously

4. Measure resultant spin for each node

5. Use resultant spin to re-initialise neighbouring nodes

6. Repeat until equilibrium is reached

The parameters of interest (mean magnetisation for instance) can be calculated once the output of each node is measured. In Fig. 10.3 the circuit requires 5 qubits and 4 multi-qubit gates, though it is possible to reduce the number of qubits by rewriting the circuit in terms of two-qubit gates and single qubit rotations [BBC+95]. The choice of which two-qubit gates to use for this decomposition is largely a matter of which gates are most suitable for a given implementation. In this form the circuit will require at least 2d + 1 qubits for a d-dimensional lattice to represent the on-site spin and its nearest neighbours. A simpler version of this circuit has been presented by Miakisz et al. [MPS04] in the context of quantum game theory. Their circuit is reproduced in Fig. 10.4 and requires four qubits, two CNOT gates and one multiqubit gate, in addition to single qubit rotations. The truth-table given in Fig. 10.1 shows that the circuit given in Fig. 10.3 or 10.4 is equivalent, once measured, to the classical Metropolis method. This table gives the output of the quantum circuit S0 for each input state ASB as well as the | i | i classical result Scl0 and its probability of acceptance pcl. All but two of the entries in 10.3. 1D ISING MODEL 139

0 S X S | i | i 0 A X Scl | i B X | i P | i

Figure 10.4: Simpliﬁed circuit for the 1D Ising model, due to Miakisz [MPS04]

ASB S0 S0 p | i | i cl cl √P + √1 P 1 P |↓↓↓i | ↑i − | ↓i ↓ −P ↑ 1 |↓↓↑i | ↑i ↑ 1 |↓↑↓i | ↓i ↓ 1 |↓↑↑i | ↓i ↓ 1 |↑↓↓i | ↑i ↑ 1 |↑↓↑i | ↑i ↑ 1 |↑↑↓i | ↓i ↓ √P + √1 P 1 P |↑↑↑i | ↓i − | ↑i ↑ −P 4J ↓ Note: P = e− T

Table 10.1: Truth-table for the quantum circuit in 10.3, showing the equivalence to the classical Metropolis algorithm once the superposition state S0 is measured | i

this table are exactly equivalent to the classical Metropolis result, where a spin flip is accepted as long as the change in energy is zero or negative. The first (last) entry differs in that the result is a superposition state with a probability P of obtaining a spin-up (spin-down) result and a corresponding probability 1 P of obtaining − the opposite result. This is equivalent to the random number generation step of the Metropolis algorithm where a configuration is rejected or accepted based on a weighted probability. 140 Chapter 10. SIMULATING THE ISING MODEL ON A T2QC

A | i B | i C | i D | i 0 | i 0 | i 0 | i 0 | i 0 S S | i | i 0 P1 Scl | i P2 | i 0 | i

Figure 10.5: Quantum circuit for 2D Ising model 10.4 2D Ising model

Using the same approach, the Metropolis method for the 2D Ising model can be implemented using the circuit given in Fig. 10.5.

Inputs A-D are the neighbouring spins and P1 and P2 are the probabilities P1 = 4J 8J e− T and P2 = e− T respectively. These correspond to qubits initialised in the states P = √P 1 +√1 P 0 and P = √P 1 +√1 P 0 . The mapping between | 1i 1| i − 1| i | 2i 2| i − 2| i Ising spins and T2QC nodes is illustrated in Fig 10.6, again showing the assignment of neighbours.

si,j+1 A

si−1,j si,j si+1,j B S D

si,j−1 C

(a) (b)

Figure 10.6: Mapping between sites on an Ising spin lattice (a) and nodes of a T2QC (b), showing the nearest neighbour assignments for a one dimensional lattice.

The circuit given for the 2D Ising model is not optimal and can in fact be realised with considerably fewer and simpler gates using a ‘full-adder’ type conﬁguration. It is presented in this form as it is more apparent which sections of the circuit operate for ∆E = 0J, 4J and 8J, making the operation of the circuit more transpar- ± ± 10.5. STREAMING AND PARALLELISATION 141

Ai-1 Si-1 Bi-1 Ai Si Bi Ai+1 Si+1 Bi+1 P An P An P An Node i-1 Node i Node i+1

Figure 10.7: Each node is initialised with the on-site spin Si and copies of the neighbouring on-site spins Si 1 and Si+1 − ent. If this circuit were to be realised experimentally, it could be broken down into the appropriate two qubit operations which suited the particular implementation [BBC+95]. This can result in considerable simpliﬁcation, depending on the details of the experimental system.

10.5 Streaming and parallelisation

The streaming process for the 1D Ising chain is illustrated in Fig. 10.7. Each node is initialised using the value for the on-site spin Si and copies of its nearest neighbours

Ai = Si 1 and Bi = Si+1. The superposition qubit (P ) and the ancilla qubit (An) − for each node are re-initialised every iteration with the same value, as discussed in section 10.3 and 10.4. The streaming rule for the 2D Ising lattice is similar to the 1D case. As discussed earlier, all nodes cannot be updated simultaneously otherwise the lattice will oscillate and not reach a stable state, resulting in the ‘feedback catastrophe’. A checkerboard update scheme can be used instead, as illustrated in Fig. 10.8. Each ‘black’ node is initialised with its spin value Si,j and its four nearest neighbours A = Si,j+1,

B = Si 1,j, C = Si,j 1 and D = Si+1,j. The new value of S is calculated and then − − the same process is followed for the ‘white’ nodes. Once this process is completed, every node in the lattice has been updated. As a result of the reversibility of quantum circuits and the fact that the control qubit is left unchanged after the application of a CNOT gate, the values of the neighbouring spins A-D are preserved after the computation. This means that no additional storage space is required to store the spins that are not being modiﬁed at 142 Chapter 10. SIMULATING THE ISING MODEL ON A T2QC

Si,j+1

Si 1,j Si,j Si+1,j −

Si,j 1 −

Figure 10.8: The checkerboard update scheme with the spin value Si,j and its four nearest neighbours A = Si,j+1, B = Si 1,j, C = Si,j 1 and D = Si+1,j illustrated − − each evolution (assuming error free operation). It also means that the value of an arbitrary neighbour (ie: A) can be used to initialise the spin S at the next time step. This results in only half as many nodes being required to simulate the lattice. This algorithm therefore requires N/2 computational nodes for N Ising spins. In this situation two applications of the quantum circuit are required to update every node in the lattice once. An alternative solution is to include a parity bit which controls whether the rest of the circuit functions or not. The lattice is then initialised so that the parity bits alternate across the lattice and they are inverted at each evolution.

10.6 Possible implementations

The algorithm, as discussed, is independent of implementation and would therefore be suited to a range of different quantum computing architectures. Of particular interest is that the final state of the spin after each evolution collapses to either a 1 or a 0 when measured. Existing diffusion based type-II algorithms require the probability amplitudes to be measured at each step, which requires either time or ensemble averaging. This cellular automaton based algorithm, on the other hand, does not require the measurement of the full ensemble of states and is therefore well suited to solid state or ion-trap architectures. One important consideration is that the accuracy with which the temperature of the system can be controlled is directly related to the accuracy with which the superposition state P can be created. The amplitude (γ = √P ) of this superposi- | i 4J/T 2J/T tion is given by γ2D = e− for the 2D case and γ1D = e− for 1D. The accuracy 10.7. ENSEMBLE STREAMING 143 with which this superposition can be created and controlled puts a lower bound on the temperature resolution the system can simulate. This is especially important for architectures where only a binary output is measured, as the superposition must be created using single qubit rotations. The accuracy in the temperature is therefore related to the accuracy of the rotation gate. By using standard uncertainty analysis, the maximum allowable error rate (δγ) can be calculated for a given temperature resolution using Eq. (10.2), dγ δγ = δT. (10.2) dT ¯ ¯ ¯ ¯ ¯ ¯ For example, in the 1D case, this gives ¯ ¯

2 2/T δγ = e− δT (10.3) 1D T 2 as the gate error corresponding to a temperature resolution of δT . Fig. 10.9 shows the maximum allowed gate error rate for a single qubit rotation (δγ) needed to achieve a temperature resolution (δT ) of 0.1, as a function of system temperature (T ). The curves δγ1D and δγ2D correspond to the maximum allowed gate error rate for the 1D and 2D systems respectively. For T < 0.5 the required gate accuracy is eﬀectively inﬁnite but in this region the simple Ising system displays trivial behaviour. For the region of interest (1.5

10.7 Ensemble streaming

The algorithm detailed so far is formally equivalent to the classical Metropolis algorithm for the Ising model. An interesting side note involves determining what would happen if the measurement step were to be replaced with an ensemble measurement. An ensemble measurement consists of measuring many copies of the system and determining the magnitude of the probability distribution for each qubit. This distribution can then be used to re-initialise the other spins, ready for the next it- 144 Chapter 10. SIMULATING THE ISING MODEL ON A T2QC

Figure 10.9: Maximum allowable error rate required for a temperature resolution of δT = 0.1, as a function of Ising lattice temperature. The critical temperature for the 2D lattice is plotted to illustrate the region of interest. As the temperature approaches zero, the required accuracy increases exponentially and therefore the allowable gate error decreases exponentially. In this regime, the lattice displays trivial behaviour which is independent of temperature and therefore insensitive to gate accuracy fluctuations. eration of the algorithm. This situation corresponds to either time averaging over many runs or ensemble averaging, as is the case with NMR based implementations, and will be referred to as ‘ensemble streaming’. This is especially interesting as the only type-II quantum computer in existence at the time of writing is based on the NMR architecture and therefore uses ensemble streaming [PCYC02] [PCCY03]. The result of this process is that the spins no longer take on definite integer values but can have continuous values depending on their neighbours and the temperature dependant function. The magnetisation as a function of temperature is shown for this ensemble streaming (Ensemble T2QC) for the 2D Ising model (the dot-dash line in Fig. 10.10). By allowing the full ensemble of states to propagate through the lattice, the model now bears some similarity to a mean-field style approximation. As each node can vary continuously from 0 to 1, the deviation from the mean spin value quickly decays to zero for any one spin site. After a few iterations, each node on the lattice takes on the same expectation value, resulting in a smooth curve even for a very 10.7. ENSEMBLE STREAMING 145 small lattice of four spins. As a result, the curve for any number of spins is found to be the same as for the four spin case, the smallest lattice which can form an infinite lattice with trivial periodic boundary conditions. In addition the ensemble version converges much faster than the classical Metropolis algorithm (less than 5 iterations per temperature point). The curve (Fig. 10.10) corresponds to the system being heated from the ground state up to a state where every configuration is equally likely, which is equivalent to the randomised state. If the system was then cooled again, it tends to stay completely randomised and therefore the zero magnetisation state is the stable solution all the way back to zero temperature. This is a consequence of allowing the spins to be continuous (an ensemble of states) rather than taking discrete values and is a non-physical solution.

For comparison, the solution for the ensemble T2QC is plotted in Fig. 10.10 with the analytic solution due to Onsager [Ons44] for an infinite lattice, a classical Metropolis MC run using a 1000 1000 point lattice and the classical mean field × solution for an infinite lattice [Cha87]. The classical Metropolis MC run was performed using a temperature step size of 0.01 with a minimum of 20 and a maximum of 104 iterations per temperature point. This includes 10002 spin flips per iteration with each spin flip requiring the generation of a random number, the calculation of the change in energy and the evaluation of an exponential weighting function. By comparison, for the standard T2QC algorithm with one-shot measurement all these steps are performed through the evolution of the circuit in Fig. 10.5. This occurs for each spin in the lattice simultaneously which means that the circuit need only be applied once per iteration. The number of iterations required for convergence using the one-shot measurement algorithm is identical to the classical algorithm.

The ensemble measurement algorithm underestimates the critical temperature, as can be seen in Fig. 10.10, giving T 2.109, compared to the analytic solution c ≈ of Tc = 2.269 [Ons44]. The form of the ensemble solution is a polynomial approximation rather than the accepted power law behaviour. This is due to the fact that as the true Ising lattice approaches the critical temperature, large clusters of spins form in random locations in the lattice, resulting in non-zero magnetisation. The ensemble measurement algorithm, on the other hand, eﬀectively averages over all 146 Chapter 10. SIMULATING THE ISING MODEL ON A T2QC

Figure 10.10: Magnetisation as a function of temperature for the 2D Ising model. The ‘Exact’ solution is the analytic solution for an infinite lattice, ‘Mean field’ is the mean field approximation for an infinite lattice, ‘MC’ is a classical Metropolis monte-carlo run using a 1000 1000 lattice and ‘Ensemble T2QC’ is a simulated run of the Ising algorithm for a× T2QC using ensemble streaming on 4 nodes possible configurations. As the clusters occur in random locations their effects cancel out, resulting in a zero average magnetisation. This discrepancy accounts for the difference in critical temperature and gives some insight into the regime where clustering is an important effect. It is expected that a large cluster generalisation of this algorithm would improve the critical behaviour near Tc as the forming of clusters would be more accurately modelled.

10.8 Scalability of a type-II quantum computer for simulating the Ising model

The theoretical exponential scaling of a type-II quantum computer is yet to be proven, as discussed in section 2.5.2. This is largely due to the lack of a suitable algorithm which displays both an exponential speedup in the number of qubits and the parallelisation inherent in classical distributed problems. The implementation of the Metropolis algorithm discussed in this chapter does not demonstrate a speedup over conventional (classical) information processing but it does provide an example 10.9. CHAPTER SUMMARY 147 of a more efficient use of resources, in this case (qu)bits. It has been suggested that the lack of viable algorithms mean that type-II quantum computers do not present a viable pathway to scalable quantum computing. As the algorithm presented in this chapter does not demonstrate a significant advantage when compared to classical computing, this would support that opinion. However, it does provide an interesting framework in which to investigate the quantum/classical crossover and one could speculate that this may lead to a greater understanding of quantum effects in information processing. This, in turn, may lead to novel applications which use large numbers of few qubit quantum information processors to solve computational problems of interest which do not require globally coherent behaviour.

10.9 Chapter summary

In this chapter we have investigated the use of type-II quantum computers to simulate classical and quantum cellular automata rules. As an example we formulated the Metropolis Monte-Carlo algorithm as a probabilistic cellular automata rule and demonstrated how this could be used to simulate the Ising model on a type-II quantum computer. The algorithm requires only N/2 nodes to simulate N spins and requires a maximum of 5 qubits per node for the 1D case and 12 qubits per node for the 2D case, though this is an upper bound for arbitrary interactions. This compares well with other classical implementations which require many more classical bits for random number generation. It also suggests that other cellular automata models may be similarly reformulated to run on a type-II quantum computer. This extends the range of applications for a type-II quantum computer beyond those based on the lattice Boltzmann hydrodynamic equations, such as diﬀusion. Additionally, we studied the eﬀect of running this algorithm on a type-II quantum computer which uses ensemble streaming as this corresponds to the present state of the art in experimental systems. 148 Chapter 10. SIMULATING THE ISING MODEL ON A T2QC Chapter 11

CONCLUSIONS

The first part of this thesis was primarily concerned with methods for characterising a few state quantum system, which is, in general, unknown. This is particularly relevant given the recent work on using controllable quantum systems to perform quantum information processing. While the Hamiltonian governing a controllable quantum systems is often known, in solid state systems it is inescapably fabrication dependent. This means that methods to efficiently calibrate a device once it has been fabricated are crucial to obtain the high precision demanded for QIP applications. In chapter 3 we presented a novel characterisation technique for the two-state system as this is the prototypical controllable quantum system. We showed that mapping the time evolution of a suitable observable and then analysing this data in the Fourier domain provides estimates for the components of the Hamiltonian, up to an unobservable phase. We then demonstrated how this procedure can be extended to measure the relative phase between different Hamiltonians. This procedure can therefore be used to ‘boot-strap’ an unknown two-state system to obtain all the information necessary to use this system as a qubit. In chapter 4 we investigated the effect of decoherence on a two-state system using the Lindblad model and demonstrated how to include the effects of this model in the characterisation procedure. We solved the density matrix equations analytically in the Fourier domain and obtained a general solution for a system undergoing relaxation, absorption and dephasing. To the best of our knowledge, this solution has not appeared anywhere else in the literature. We extended our characterisation procedure in chapter 5 by including uncertainty estimates and therefore providing a direct method to estimate the precision with which we can measure the Hamiltonian parameters. Using conventional uncertainty analysis, we obtained error propagation equations to estimate the uncer-

149 150 Chapter 11. CONCLUSIONS tainty in the Hamiltonian estimates, based on the spectral noise. This method was simulated to compare the uncertainty estimates from one application of the characterisation process with the statistical spread of the data over many applications of the procedure, ﬁnding very good agreement, with the discrepancies being attributed to simpliﬁcations made in the error propagation analysis. Using these techniques, the Hamiltonian of an unknown two-state system can be measured with uncertainty bounds that can be linked directly to expected gate error rates for quantum computing.

Chapter 6 provided an experimental demonstration of the Hamiltonian characterisation technique using the magnetic sublevels of the nitrogen-vacancy defect centre in diamond. This centre provides an excellent test system as it has been suggested as a promising qubit and has been investigated extensively using both spectroscopy and tomography. We can therefore directly compare the information obtained via Hamiltonian characterisation with other forms of system identiﬁcation. The results obtained showed good agreement between the experimentally measured eﬀective Hamiltonian and that expected based on theoretical grounds. Even with relatively poor data, the characterisation process was able to produce reliable estimates for the system parameters as well as uncertainties for those estimates.

The generalisation of the Hamiltonian characterisation technique to higher dimensions was addressed in chapter 7. As an example system, the Heisenberg interaction between eﬀective spin states was investigated as this is common interaction exploited to create two qubit gates in the solid-state. The coupling parameters of this Hamiltonian can be measured by mapping the evolution of the system from a set of input states. We also investigated mapping the entanglement generated as a function of time and found that this also allows the determination of the Hamil- tonian parameters. A simpliﬁed uncertainty analysis was used to determine how the parameter precision scales with increasing numbers of measurements. This was used to determine the minimum number of measurements required to characterise a two-qubit gate for use in scalable quantum computing using concatenated quantum error correction.

We will now consider possible extensions to the Hamiltonian characterisation 151 process and topics for further investigation. The higher dimensional case considered in this thesis (the Heisenberg Hamiltonian) is only a very special case. This method has been further generalised in reference [DCH06] to include Hamiltonians which are amenable to canonical decomposition, though this does not include all possible two-qubit interactions. A more general method for two qubit interactions would be advantageous to account for situations where the structure of the Hamiltonian is not known beforehand or when the assumptions made in chapter 7 regarding single qubit terms are not justiﬁed. Other higher dimensional extensions include the detection and characterisation of weakly coupled levels which take the system out of the two- state subspace or auxiliary levels required for two qubit interactions, such as those used to mediate interactions in ion trap systems.

The statistical analysis performed in this thesis consists of a pragmatic approach, using standard error propagation theory. As is common with this method, it often overestimates the uncertainty for a given set of data [PR04]. The application of more sophisticated techniques such as Bayesian estimation or maximum likelihood estimation to Hamiltonian characterisation should provide more efficient and accurate methods to estimate the uncertainties. A comparison of the efficiency of various system identification methods would also provide a quantitative analysis of which method is most suited to constructing quantum gates in any given architecture.

The experimental results presented in chapter 6 using the NV centre in diamond represent the first demonstration of these techniques in the laboratory. Repeating these experiments at lower temperatures and with a range of different magnetic and laser field configurations would provide increasingly high accuracy confirmation of the theory including the ability to directly measure the decoherence channels. The use of single-shot readout would also allow meaningful estimates of the effective measurement error rate, which directly affects the use of this system for QIP. Applying this characterisation technique to other solid-state systems, such as superconducting flux or charge qubits, would also demonstrate the technique’s versatility.

The later chapters of this thesis involve the use of quantum systems for (quantum) information processing directly. The use of quantum-dot cellular automata for classical information processing provides an alternative to the traditional transistor 152 Chapter 11. CONCLUSIONS based logic. While this field is still in its infancy it does show promise, if not to replace conventional chip design, at least for niche applications requiring low power consumption or high density for few bit processors. Type-II quantum computers, on the other hand, provide an interesting application of few qubit coherent systems using some level of quantum information processing to solve large scale simulation tasks. The concept of a QDCA cell based on buried donor atoms in a semiconductor matrix was presented in chapter 8. We calculated the energy levels and switching times for incoherent operation using a hydrogenic approximation. The switching speeds were found to be an order of magnitude slower than equivalent systems using mesoscopic quantum dots, while the state localisation was found to be good at temperatures of less than 1K. The operation of a buried donor QDCA was found to be theoretically possible, though the technical requirements are still quite challenging. The use of coherent switching of a QDCA cell was investigated in chapter 9. This could allow the switching of a buried donor QDCA cell on time scales of order picoseconds, though this depends on the dephasing time of the system which in turn depends strongly on materials fabrication issues. Coherent adiabatic switching allows high fidelity transfer with minimal requirements in terms of pulse shaping and timing and is applicable to a range of coherent systems, not just those based on buried donors. The scaling behaviour of a binary wire being switched using adiabatic switching was investigated and found to be approximately linear, though this depends strongly on the model used. This work could be extended by investigating other models for the relaxation process and comparing these to the well known equations for coupled quantum dots which describe relaxation via tunnel junctions. The link between these models is unclear as the equations used to describe single donors are fundamentally different to those used for mesoscopic dots, though theoretically one should constitute the limiting case of the other. A clearer picture of how these are linked would aid research into devices which approach the single atom level. Coherent adiabatic switching in solid-state systems show much promise, both as a transport mechanism and as a method of changing the state of a coherent system. 153

If this is to be successfully applied to the QDCA computing paradigm, the link between irreversible and reversible computation still needs to be investigated. Most QDCA architectures which have been proposed include a clocking sequence to both limit the effect of noise and provide signal gain. It is not clear if there is a coherent QDCA equivalent of this clocking procedure and therefore this or other methods for signal gain need to be investigated. In chapter 10 we investigated the use of a type-II quantum computer for simulating the Ising model using the Metropolis Monte-Carlo method. This provides an example of an algorithm for modelling a physical system using a probabilistic cellular automaton. In a T2QC we can use quantum superposition to perform the random number generation required for the Metropolis algorithm without requiring the large number of bits used to generate random numbers on a conventional computer. This algorithm constitutes a novel use of a T2QC as the existing algorithms are all based on the lattice-gas model for hydrodynamics. A T2QC provides a convenient model with which to study the classical/quantum correspondence in computing. Given a cellular automata model which shows both clear quantum and classical behaviour in the appropriate limits, a T2QC could conceivably be used to study this, both theoretically and experimentally. The development of other algorithms for this architecture would also provide benefits, though given the difficulty of finding new algorithms for quantum information processing in general, this is not a trivial problem. The integration of quantum mechanical concepts into the field of information processing continues to provides surprising results, with important repercussions for both the fundamental and applied branches of the sciences. The goal of constructing a quantum information processor is pushing the limits of current experimental techniques but there is still much we can learn along the way. The ability to both understand and manipulate the world around us is what drives science and technology forward and hence fabrication and control of few state quantum systems is an important area of study, both now and in the future. 154 Chapter 11. CONCLUSIONS Appendix A

QUANTUM CIRCUIT FORMALISM

This appendix provides a brief overview of the quantum circuit formalism [Deu89, Yao93] in order to deﬁne the notation and concepts used throughout this thesis. For a more complete discussion, see chapters 1 and 4 of Nielsen and Chuang [NC00].

The circuit model of quantum computation provides a direct link to conventional Boolean logic, with computation being achieved through the application of gates, the order and selection of which deﬁnes the algorithm or ‘program’. In moving from classical to quantum information processing, the fundamental unit is no longer the bit (0 or 1) but the qubit which can be in states 0 or 1 or any superposition of | i | i these states. The general state of a qubit can be expressed as

φ = α 0 + β 1 (A.1) | i | i | i where α 2 + β 2 = 1 (A.2) | | | | according to the Born interpretation of the wavefunction. The state of the system, assuming no interactions between the qubit, is then given by the tensor product of each qubit’s wavefunction, ie. φ = φ φ . . . φ . | sysi | 1i ⊗ | 2i ⊗ | ni

The concept of logic gates in classical computation is replaced with quantum gates or unitary operators. Application of these operators allows the state of the system to be changed, creating any arbitrary superposition. This ability to create a superposition and then to use this to create maximally entangled states, such as

155 156 Appendix A. QUANTUM CIRCUIT FORMALISM the Bell states

00 + 11 B00 = | i | i, | i √2 01 + 10 B01 = | i | i, | i √2 00 11 B10 = | i − | i, | i √2 01 10 B11 = | i − | i, (A.3) | i √2 lies at the heart of quantum information theory and quantum computing.

Fig. A.1 illustrates an example of a quantum circuit diagram, where the horizontal lines represent qubits and time runs from left to right. Single qubit unitary operations act on these qubits and are typically represented by boxes, whereas multiple qubit gates act on several qubits at once and are indicated by vertical lines.

initial single multiqubit state qubit gates gates q1 | i q2 Z | i q3 X Z qubits | i q4 Z | i time -

Figure A.1: Example of a quantum circuit showing example single qubit unitaries and multiqubit gates acting on the initial state q1 q2 q3 q4 . The gate ordering in time is indicated by horizontal gate position,| i ⊗ | withi ⊗computation | i ⊗ | i proceeding from left to right.

While, in general, computation can be realised with a range of diﬀerent unitary and non-unitary operations, it is common to consider a discrete set of unitary gates. The most common set of single qubit gates are deﬁned as

0 1 1 0 1 1 1 X = , Z = , H = , (A.4) 1 0 0 1 √2 1 1 − −       157

1 0 1 0 S = , T = , (A.5) 0 i 0 eiπ/4     where the action of a gate U on the state φ can be written as ψ0 = U ψ . The | i | i | i X-gate is known as a bit-ﬂip or NOT gate, similarly the Z-gate is a phase ﬂip while the Hadamard gate (H) transforms between the computational and symmetric/antisymmetric bases, ie.

1 1 1 α 1 α + β H φ = = . (A.6) | i √2 1 1 β √2 α β − −       The controlled-NOT or CNOT gate is illustrated in Fig. A.2 and is accompanied by the corresponding truth-table. The basic operation of the CNOT gate is to invert the target bit ( b ) if the control bit ( a ) is equal to one. The convention is to use a | i | i ﬁlled circle to represent the control bit while inversion of the target bit is indicated by the ‘crossed-circle’.

a b a b a a 0 0 ⊕0 | i | i 0 1 1 b a b 1 0 1 | i | ⊕ i 1 1 0

Figure A.2: Notation for a CNOT gate and the truth-table showing classical operation

The CNOT gate is an example of a maximally entangling gate, other examples are the CPHASE gate and the √SWAP gate. The appropriate operators for these gates are

1000 100 0 0100 010 0  U = , U = , (A.7) CNOT   CPHASE   0001 001 0          0010 0 0 0 1    −      158 Appendix A. QUANTUM CIRCUIT FORMALISM and 10 00 1 0 1 i 1+ i 0 U = − (A.8) √SWAP 2   0 1+ i 1 i 0  −    00 01     respectively. Maximally entangling gates such as these can be used to convert a separable state into a maximally entangled state, for example

1000 1 1 0 + 1 1 0100 0 1 0 UCNOT | i | i 0 = = = B00 . (A.9) √2 ⊗ | i √2     √2   | i µ ¶ 0001 1 0             0010 0 1             The CNOT gate can be extended to include two or more input states, an example of which is the CCNOT or Toﬀoli gate, shown in Fig. A.3. As expected, the Toﬀoli gate inverts the target bit if both control bits are equal to 1.

a b c c ab a a 0 0 0 ⊕0 | i | i 0 0 1 1 b b 0 1 0 0 | i | i 0 1 1 1 c c ab 1 0 0 0 | i | ⊕ i 1 0 1 1 1 1 0 1 1 1 1 0

Figure A.3: Notation for a Toﬀoli gate and the truth-table showing classical operation

Other gates of interest are the inversion or NOT gate (Fig. A.4(a)) and the swap gate (Fig. A.4(b)) which just swaps the input states a and b . The symbol | i | i in Fig. A.4(c) is used to indicate a destructive measurement, where that qubit is measured and therefore all information about the quantum superposition is lost. If the dependance on the control bit of any of these gates is reversed, this is indicated by a open circle as shown in Fig. A.5. The fact that the control bits for both the CNOT and Toﬀoli gates have not changed after the operation is a result 159

a b a a¯ | i | i | i | i b a | i | i (a) (b) (c)

Figure A.4: (a) Alternative notation for a NOT gate. (b) Notation for a swap gate and (c) a destructive measurement. of the reversibility of these gates.

a a a a | i | i | i | i b a¯ b b a¯ b | i | ⊕ i | i | ⊕ i

Figure A.5: Notation for a CNOT gate where the target bit is inverted if the control bit is equal to 0

The set of gates including those shown in Eq A.4 and the CNOT gate constitutes a universal set. This means that any arbitrary computation may be expressed in terms of this ﬁnite set of gates [BMP+99]. There are other universal gate sets, including examples in which the entangling gate is a Toﬀoli gate [BBC+95]. It turns out that almost any two-qubit gate is universal when combined with arbitrary single qubit unitary gates [DBE95, Llo95]. In practice, the gate set which is used to construct an algorithm is chosen based on the physical architecture in question. 160 Appendix A. QUANTUM CIRCUIT FORMALISM BIBLIOGRAPHY

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Minerva Access is the Institutional Repository of The University of Melbourne

Author/s: Cole, Jared H.

Title: Controllable few state quantum systems for information processing

Date: 2006-10

Citation: Cole, J. H. (2006). Controllable few state quantum systems for information processing , PhD thesis, School of Physics, University of Melbourne.

Publication Status: Unpublished

Persistent Link: http://hdl.handle.net/11343/39199

File Description: Controllable few state quantum systems for information processing

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