Realistic Read-Out and Control for Si:P Based Quantum Computers

Realistic read-out and control for Si:P based quantum computers

Matthew James Testolin BSc(Hons)

Submitted in total fulﬁlment of the requirements for the degree of

Doctor of Philosophy

School of Physics The University of Melbourne Australia

August, 2008

Abstract

This thesis identifies problems with the current operation proposals for Si:P based solid-state quantum computing architectures and outlines realistic alternatives as an effective fix. The focus is qubit read-out and robust two-qubit control of the exchange interaction in the presence of both systematic and environmental errors.

We develop a theoretical model of the doubly occupied D− read-out state found in Si:P based nuclear spin architectures. We test our theory by using it to determine the binding energy of the D− state, comparing to known results. Our model can be used in detailed calculations of the adiabatic read-out protocol proposed for these devices. Regarding this protocol, preliminary calculations suggest the small binding energy of the doubly occupied read-out state will result in a state dwell-time less than that required for measurement using a single electron transistor (SET). We propose and analyse an alternative approach to single-spin read-out using optically induced spin to charge transduction, showing that the top gate biases required for qubit selection are significantly less than those demanded by the adiabatic scheme, thereby in- + creasing the D D− lifetime. Implications for singlet-triplet discrimination for electron spin qubits are also discussed. On the topic of robust two-qubit control, we demonstrate how by using two-qubit composite rotations a high fidelity controlled-NOT (CNOT) gate can be constructed, even when the strength of the interaction between qubits is not accurately known. We focus on the exchange interaction oscillation in silicon based solid-state architectures with a Heisenberg Hamiltonian. This method easily applies to a general two-qubit Hamiltonian. We show how the robust CNOT gate can achieve a very high fidelity when a single application of the composite rotations is combined with a modest level of Hamiltonian characterisation. Operating the robust CNOT gate in a suitably characterised system means concatenation of the composite pulse is unnecessary, hence reducing operation time, ensuring the gate operates below the threshold required

i for fault-tolerant quantum computation. We finish by outlining how the effects of charge noise on a pair of spins coupled via the exchange interaction can be calculated by modelling the charge fluctuations as a random telegraph noise (RTN) process using probability density functions. We develop analytic expressions for the time-dependent superoperator of a pair of spins as a function of fluctuation amplitude and rate. We show that the theory can be extended to include multiple fluctuators, in particular, spectral distributions of fluctuators. These superoperators can be included in time-dependent analyses of the state of spin systems designed for spintronics or quantum information processing to determine the decohering effects of exchange fluctuations. We discuss the implications of the charge noise produced from a single fluctuator on the operation fidelity of our robust CNOT gate, comparing the performance to an uncorrected CNOT gate.

ii Declaration

This is to certify that

(i) the thesis comprises only my original work towards the PhD except where indicated in the Introduction,

(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 tables, maps, bibliographies, appendices and footnotes.

I authorise 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.

Signature

Date

iii

Acknowledgements

This thesis and the work it entailed would not have been possible without the support of a number of people who I now wish to acknowledge. Firstly, I would like to thank Lloyd Hollenberg for his dedicated supervision. His insight, guid- ance and patience have been invaluable throughout my PhD candidature and honours year and his door always open. Most of the early work in this thesis would not have been possible without the assistance of Cameron Wellard. I particularly value Cam’s practical approach and much of his advice has remained with me throughout this PhD. Both he and Andrew Greentree never seemed to tire of my numerous questions and concerns and I thank them both for that. I would also like to acknowledge Charles Hill and Jared Cole for their collaboration on parts of this research. Working closely with peers has been one of the most enjoyable of all the PhD experiences. There are many others from the School of Physics that have had either direct or indirect inﬂuence on both myself and this PhD thesis, however I would like to highlight the contributions of a few. The members of the DMAP group as well as my roommates (from 402 and 606) in particular Gajendran Kandasamy, Jared Cole, Jason Doukas, John McIntosh, Joo Chew Ang and Vince Conrad. I’m also thankful for the constant source of useful distraction provided by those on the sixth ﬂoor. I reserve a very special thanks for Kristian McDonald. Without close friends outside the School of Physics, completing this thesis would have been an onerous task. I’d like to thank all those who helped me to escape from the pressures of a PhD, especially Cameron Grech, Matt Lynch, Anthony Lewis, James Copes, Andrew Allan, Adrian Polizzi, James Hartigan, Gerard McMahon, Wojtek Ruszkowski, Jane Leigh, Sally Malone and Nadia Polizzi. Thanks for your friendship, advice, the taunts... Finally, I would like to thank my family for their loving support. I’m not sure how you have put up with me and continued to encourage me after all these years of study but I am extremely grateful for it. Thank you mum, dad and Sarah. v

List of Publications

Throughout the course of this project a number of the key results presented in this thesis have appeared in publication. These publications are listed here for reference, along with all conference presentations, which have been given based on the work in this thesis.

REFEREED PUBLICATIONS

M. J. Testolin, A. D. Greentree, C. J. Wellard and L. C. L. Hollenberg, • “Optically induced spin-to-charge transduction in donor-spin readout”, Physical Review B 72, 195325 (2005).

M. J. Testolin, C. D. Hill, C. J. Wellard and L. C. L. Hollenberg, “Ro- • bust controlled-NOT gate in the presence of large fabrication-induced variations of the exchange interaction strength”, Physical Review A 76, 012302 (2007).

REFEREED CONFERENCE PROCEEDINGS

M. J. Testolin, L. C. L. Hollenberg, A. D. Greentree and C. J. Wellard, • “Single-spin detection and read-out for the solid-state quantum computer via resonant techniques”, Proceedings of SPIE International Society of Optical Engineers 5650, 516-526 (2005).

CONFERENCE ABSTRACTS

M. J. Testolin, L. C. L. Hollenberg, C. J. Wellard and A. D. Greentree, • “Single-spin optical readout scheme for solid-state quantum computers”, SPIE International Symposium. Smart Materials, Nano-, and Micro- Smart Systems, Sydney, Australia, December 12-15, 2004.

vii M. J. Testolin, L. C. L. Hollenberg, C. J. Wellard and A. D. Green- • tree, “Optical readout of single-spins for solid-state quantum computing”, Australian Institute of Physics - 16th National Congress, Canberra, Australia, January 30 - February 4, 2005.

M. J. Testolin, C. D. Hill, C. J. Wellard, A. D. Greentree and L. C. L. Hol- • lenberg, “Robust CNOT gates to correct for variations in donor based exchange coupling”, Sir Mark Oliphant International Frontiers of Science and Technology Conference on Quantum Nanoscience, Noosa, Queens- land, January 22-26, 2006.

M. J. Testolin, C. D. Hill, C. J. Wellard, and L. C. L. Hollenberg, “Imple- • menting a robust CNOT gate to correct for fabrication induced variations in donor based exchange coupling”, Australian Institute of Physics - 17th National Congress, Brisbane, Australia, December 3-8, 2006.

viii Contents

Abstract...... i Declaration ...... iii Acknowledgements ...... v ListofPublications...... vii TableofContents...... ix ListofFigures...... xi ListofTables ...... xix

1 Introduction 1 1.1 Structureofthethesis ...... 2

2 Background 5 2.1 Anewparadigmforcomputation ...... 5 2.2 Thequantumcircuitmodel ...... 7 2.3 Devicearchitectures ...... 12 2.3.1 TheKanearchitecture ...... 12 2.3.2 The global electron spin architecture ...... 14

3 The read-out state for Si:P based architectures 17 3.1 Kaneadiabaticread-outprotocol ...... 18 3.2 Modelling the Si:P qubit ...... 24 3.2.1 Thesiliconlattice...... 25 3.2.2 Impurity donors and eﬀective mass theory ...... 28 3.2.3 Binding energy of a phosphorus donor in silicon . . . . . 33

3.3 Determining the binding energy of the D− read-outstate . . . . 35 0 3.4 Improving the modelling of the D and D− states ...... 38

ix 3.5 Summary ...... 39

4 Optically induced charge transfer for donor spin read-out 41 4.1 Gatedresonantspintransfer ...... 43 4.2 ResonantFIRlasertransfer ...... 49 4.3 Chargetransferﬁdelity ...... 53 4.4 Singlet-triplet read-out for electron spin qubits ...... 55 4.5 Summary ...... 57

5 A CNOT gate immune to large fabrication induced variations 59 5.1 Constructing robust gates using composite rotations ...... 61 5.2 Correcting for an unknown exchange interaction strength.... 66 5.2.1 Gatecount ...... 69 5.2.2 Gatetime ...... 70 5.3 The role of two-qubit Hamiltonian characterisation ...... 71 5.4 Summary ...... 77

6 Modelling eﬀects of charge noise on the exchange interaction 79 6.1 Thenoisemodel...... 80 6.2 Calculating the probability density function ...... 84 6.2.1 Approximating Ω (ξ, T )...... 85 6.3 Using the PDF to determine Q (t)...... 87 6.4 Multipleﬂuctuators...... 90 6.5 Usingthesuperoperators...... 92 6.5.1 Removingthecrossterms ...... 94 6.6 Gate operations in the presence of charge noise ...... 95 6.6.1 The perfectly characterised system ...... 97 6.6.2 The imperfectly characterised system ...... 101 6.7 Summary ...... 103

7 Conclusions 105

Bibliography 109

x List of Figures

2.1 The Bloch sphere is useful for visualising the state of a single qubit, with points on the surface of the sphere, deﬁned by the vector ψ , representing the qubit state...... 8 | i

2.2 An example of a quantum circuit containing three qubits. Boxes show the single qubit operations, whereas multi-qubit gates are usually represented by vertical lines connecting two or more qubits. Projective measurements are represented by the ﬁnal element to act on the third qubit. Time runs from left to right. 9

2.3 Circuit representation and truth table for the CNOT gate. ... 10

2.4 Circuit representations for (a) the controlled-U operation and (b) a generalised controlled operation which acts on n target qubits, conditional on the state of m controlqubits...... 11

2.5 Schematic of the Kane proposal for a phosphorus in silicon quantum computer. The metallic surface gates are fabricated on a high quality oxide layer. A gates control single qubit operations and J gates generate qubit-qubit interactions...... 13

3.1 Kane’s read-out proposal involves the adiabatic, spin-dependent charge transfer of a single electron from the qubit being mea-

sured to a nearby reference qubit in order to form the D− state. 19

xi 3.2 Eigenspectrum for the electronic states of a two-qubit system under exchange and Zeeman Hamiltonians as a function of the exchange coupling strength J (constant B). The Zeeman terms resolve the degenerate triplet states, lowering the triplet for |↓↓i J <µBB/2. When J>µBB/2, the singlet state is lowered in energy with respect to the triplet state ...... 21 |↓↓i

3.3 Evolution of the eight lowest electron-nuclear energy levels as a function of J for the two-qubit system. From highest in energy to lowest (on the left) they are: a 11 , a a , a s , | ei| i | ei| ni | ei| ni a 00 , 11 , s , a , 00 . Adiabatically in- | ei| i |↓↓i| i |↓↓i| ni |↓↓i| ni |↓↓i| i creasing the exchange coupling past J = µBB/2 results in anti- crossing behaviour for those electron-nuclear energy levels connected via the hyperfine interaction. Specifically, 00 and |↓↓i| i a evolve to a a and a 11 respectively. All other |↓↓i| ni | ei| ni | ei| i states remain unaffected by the hyperfine interaction...... 23

3.4 Band structure of silicon plotted along the [100] direction, including both the valence (red) and conduction (blue) bands. The conduction band minimum, of which there are six degenerate minima for silicon, occurs at k 0.85 2π/a. The donor | | ≈ × electron wave function is well localised around them in recipro- calspace...... 28

3.5 Constant energy ellipsoids at the six conduction band minima of silicon. These minima are located at the six equivalent [100]

directions. The mass anisotropy m∗ = m∗ , results in the no- k 6 ⊥ ticeable elongation of the surface in the direction parallel to the axisonwhichthesurfacelies...... 31

3.6 Comparison of the isotropic and anisotropic eﬀective mass wave functions. Both wave functions are plotted along the [100] direction by holding two variables constant...... 35

xii ad 4.1 The dc ﬁeld (Fdc ) required to adiabatically transfer an electron + between two donors is larger than the ﬁeld (Fdc∗ ) the D D−

system can safely sustain for a D− dwell time long enough for SETdetection...... 44 4.2 Schematic of the device for the gated resonant spin-dependent chargetransferofasingleelectron...... 45 4.3 Energy level diagram for the three-state system as a function

of dc electric ﬁeld strength Fdc, for a donor separation of R = 30 nm. For even small values of F , the state RR , is well dc | i decoupled from the other two states, therefore allowing us to treat the system as an eﬀective two-state system...... 47 4.4 Ground and excited state populations as a function of time showing spin-dependent charge transfer. In the presence of a

small dc electric field, Fdc, the Rabi solution closely matches the numerical solution obtained from the three-state calculation. The inset shows a close-up indicating the very good agreement between the two solutions. Noticeable are the high frequency oscillations due to a small coupling to the off-resonant state RR . 48 | i 4.5 Charge transfer probability as a function of FIR laser detuning for a range of intensities, showing the characteristic sinc function dependence. The narrowing of the central peak with decreasing laser intensity serves as a means by which to avoid unwanted singledonortransitions...... 49 0 0 + 4.6 Scaling of the energy gap for the D D D D− ( LR LL ) → | i→| i transition as a function of Fdc. Simulation results are shown for donor separations of R = 20 nm and R = 30 nm against the allowed single donor energy levels (relative to the donor ground state)...... 51 4.7 State probabilities for the resonant FIR laser transfer between donors. Simulation results are shown for donor separations of (a) R = 20 nm and (b) R = 30 nm. The reduction in the magnitude of the high frequency oscillations with increased donor separation, due to a smaller off-resonant coupling, is noticeable. 52

xiii 2 4.8 Fidelity of transfer for an FIR laser intensity of 0.37 Wcm− and donor separation of R = 20 nm. The improved ﬁdelity is noticeable with increased detuning. Above we show results for

Fdc values of (a) 8.9 kV/m and (b) 22.2 kV/m. Note that the scalesonthey-axesarediﬀerent...... 53

4.9 Increasing the donor separation to R = 30 nm reduces the off- resonant dipole matrix element, resulting in improved fidelity (as compared to a donor separation of R = 20 nm, seen in Fig. 4.8). Transfer fidelity is shown for an FIR laser intensity 2 of 0.37 Wcm− ...... 54

4.10 Simulation results for a donor separation of R = 30 nm showing fast, high-ﬁdelity transfer. Transfer on these time scales requires higher powered lasers which may cause heating and unwanted singledonortransitions...... 55

4.11 Electron spin energy levels for the spatially separated and single site states. Using an FIR laser to perform read-out when the states and are degenerate results in a singlet-triplet |↓↑i |↑↓i read-out protocol. This may be useful in cluster state quantum computation...... 56

5.1 Circuit diagram for a CNOT gate constructed from an Ising π interaction, where H is a Hadamard gate, Z π = exp i σZ − 2 4 and U π = exp i π σ σ ...... 62 I 2 − 4 Z ⊗ Z 5.2 Procedural ﬂowchart for constructing a robust CNOT gate using composite rotations, and concatenating to higher implementa- tionlevels...... 63

5.3 CNOT error, (1 ), as a function of the fractional error in our −F knowledge of the coupling strength, ∆, for various implementation levels. These composite rotations provide improvement over an uncorrected implementation for ∆ ( 1, 1)...... 65 ∈ − xiv 5.4 Exchange couplings in an unbiased, J (V = 0), system for donors at fcc lattice sites misplaced by a distance δ in all directions from the target separation of 20.6 nm (in the [100] direction). The exchange coupling strengths are given as a fraction of the target coupling strength, J 0.1 µeV...... 67 0 ≈

5.5 CNOT ﬁdelity as a function of donor separation in the [100] direction for various implementation levels. The resulting ﬁ- delities are determined based on a target donor separation of 20.6 nm. Note that interpolating curves between lattice sites indicate donor separation scenarios for a given implementation, and vertical dotted lines guide the eye between implementations. 68

5.6 Exchange interaction strength as a function of donor separation along the [100] direction, showing a large variation in the coupling strength with donor misplacement (dots indicate actual site separations). For an uncharacterised system the coupling is

set to the fabrication target J0, with the actual placement giving coupling J. The resulting CNOT error, (1 ), for a one −F site deviation (∆ 0.49) from the target separation can be 0 ≈ − seen on the inset plot as a function of implementation level. In

the characterised system the coupling is set to Jc. The CNOT

error for a system characterised to the 10% level (δJc/Jc =0.1), taking J 0.9J (∆ = 0.1), is shown as a function of imple- c ≈ c mentation level inset also. Note that all curves are included purelytoguidetheeye...... 74

5.7 CNOT error, (1 ), as a function of the total number of char- −F acterisation measurements required to achieve a given ﬁdelity for various implementation levels. The results demonstrate the usefulness of combining composite rotations with Hamiltonian characterisation when constructing a robust CNOT gate. Thresh- 4 old reference line at 10− errorrateisshown...... 75

xv 5.8 CNOT error, (1 ), as a function of the total gate time for an −F unbiased, J(V = 0), system (T2 = 60 ms assumed). Results are shown for a range of separations in the [100] direction, larger than the targeted 20.6 nm separation. We consider various CNOT gate constructions, namely an uncorrected CNOT, one constructed from both a single and two applications of composite rotations and ﬁnally a CNOT constructed using composite rotations in conjunction with characterisation to the 10% level (δJ /J = 0.1) taking J 0.9J. Only for this ﬁnal method c c c ≈ have more than two sites been included as for other methods 4 results will clearly be worse. Threshold reference lines at 10− errorratesareshown...... 76

6.1 Comparison of the analytical (exact and approximate) and exact numerical solutions of Q(nu) (t). Plotted is the non-zero matrix (nu) element of Q (t), denoted QNU, as a function of time for a range of ﬂuctuator rates, which span each of the three limiting regimes. The results show very good agreement between all three solutions, with the analytic approximation deviating only slightly when the ﬂuctuator rate is on the time scale of T . ... 90

6.2 Multiple instances of the superoperator Q (t) separated by a non-commuting Hadamard operation. Attempting to use the superoperators to determine the eﬀects of charge noise in a process like this can lead to the introduction of errors in the slow ﬂuctuatorregime...... 92

xvi 6.3 CNOT gate fidelity as a function of λ for the uncorrected and robust gates. The analytic simulations are represented by solid lines, whilst exact numerical simulations are represented by points. The robust gate clearly outperforms the uncorrected gate when λ is small (λT 1). When fluctuations occur gate ≪ on the time scale of the CNOT gate (λT 1), the robust gate ≈ gate fidelity drops, as the composite rotations do not effectively correct when the coupling strength varies. In the large λ limit (λT 1), the averaging effect on J causes a revival in the gate ≫ ZZ fidelity of the robust gate, however it is still outperformed by the uncorrected gate (see inset for a close-up of the CNOT gate error for large λ). Very good agreement is achieved between the exact numerical and exact analytic results. The approximate analytic result breaks down when the fluctuations occur on the timescaleoftheCNOTgate...... 98

6.4 CNOT gate ﬁdelity as a function of λ for the uncorrected and robust gates. The analytic simulations are represented by solid lines, whilst exact numerical simulations are represented by points. The approximate analytic solution has been improved dramatically by removing the higher order cross terms...... 99

6.5 CNOT gate ﬁdelity as a function of α in the small λ limit (λT 1), for the uncorrected and robust gates. The ap- gate ≪ proximate analytic solution used to determine the robust gate ﬁdelity clearly breaks down for large α, deviating from the exact numerical result. In the case of the uncorrected gate, the analytic superoperator solution achieves perfect agreement with expected results, as for a single application of the two-qubit interaction the superoperator solution is exact...... 100

6.6 CNOT gate ﬁdelity as a function of α in the large λ limit (λT 1), for the uncorrected and robust gates. No devi- gate ≫ ation from the expected results is observed with either analytic approach...... 101

xvii 6.7 CNOT gate ﬁdelity as a function of λ for the uncorrected and robust gates when a small amount of uncertainty exists in the

coupling strength (JZZ′ =0.99JZZ). The results show that the robust gate now clearly outperforms the uncorrected gate in both the small and large λ limits. This suggests that although the composite rotations are designed to correct for static errors, in a noisy system which is not perfectly characterised, using the composite rotations to construct robust gates may be advantageous...... 102

xviii List of Tables

2.1 A comparison of the times required to perform X and CNOT gate operations in both the Kane nuclear spin [1] and global electron spin [2] architectures...... 15

5.1 CNOT gate times for various pulse implementation levels in the electron spin solid-state quantum computing architecture..... 70

xix

Chapter 1

Introduction

Quantum physics was one of the great theoretical breakthroughs of the last century. The predictions and implications of quantum mechanics have been well tested over the years and it has driven many new scientiﬁc advances, due to a greater insight into the behaviour of matter at the atomic and sub-atomic scales. Of an equally impressive nature were the technological advances, in particular in modern computing, that occurred over the second half of the last century. In fact, the rapid increase in classical computation power (characterised by Moore’s law, which says that the number of transistors on a chip approximately doubles every 18 months), has no doubt in part been possible as a result of the insight provided by quantum mechanics, as device minia- turisation and the fundamental building block of the processor, the transistor, reach the atomic scale. Nevertheless, the operation of conventional computers to date relies on essentially classical theory, despite the physical size of these devices rapidly approaching the quantum regime. Building a computer that can harness this strange quantum mechanical behaviour lies on the frontier of nano-science. Quantum computing promises to be a powerful new application of a theory that has been around since the early 1900s, opening a new realm of possibilities, including access to a complexity class of problems previously intractable, and eﬃcient simulation of quantum systems without the exponential increase in resources required by their classical counterpart. 2 Introduction

The task of designing and fabricating a working quantum computer is formidable. Theoretical modelling of aspects of this process can provide feed- back about many of the design and fabrication requirements. Furthermore, it can highlight the challenges that lie ahead during the operation of such a device. In this thesis we are concerned with investigating solid-state Si:P based architectures, identifying some of the problems and weaknesses of these architectures and formulating novel solutions. We begin by investigating the proposed read-out protocol for the Kane nuclear spin quantum computer [1] and build a theoretical description of the read-out state. We propose an alternate qubit measurement scheme, one which preserves the fragile read-out state. Following this, we turn our attention to implementing robust two-qubit gate operations in the presence of large fabrication induced variations of the solid-state environment. Finally, we develop a theoretical description of the environmental charge noise, which may contribute to decoherence phenomena in solid-state devices. We include this theory in simulations of gate operations to determine the eﬀects of a realistically noisy environment on the ﬁdelity of two-qubit operations.

1.1 Structure of the thesis

The material presented in this thesis is original except for the review included in chapter 2 and the necessary background review material presented at the beginning of each chapter. Some of the work has appeared in refereed journal articles and where this is the case, it has been outlined below. In chapter 2 we present a brief overview of the background material in order to contextualise the work in this thesis. We provide an introduction to quantum information theory and the quantum circuit formalism, as well as a description of the solid-state quantum computing architectures discussed in this thesis. Chapter 3 focuses on developing a theoretical framework for the read-out state of the Si:P based Kane nuclear spin architecture, beginning with a review of the proposed adiabatic read-out protocol. The motivation for this is the fragility of the state due to its weak binding energy. We introduce the effective 1.1 Structure of the thesis 3 mass theory to model the phosphorus donors in silicon, beginning with the qubit state and finishing with the read-out state, for which we calculate the binding energy,comparing to experiment. The agreement with experiment can be used as a measure of the accuracy of the theory. In chapter 4 an alternative to the Kane adiabatic read-out protocol is presented. This resonant optical scheme is analogous to the gated resonant scheme of Hollenberg et al. [3], which is also outlined for completeness. We show how these schemes are superior to the original Kane proposal due to the greatly reduced field bias required, resulting in a sufficiently large dwell time of the read-out state, enabling detection by a measurement device. Charge transfer fidelities are examined for a range of donor separation scenarios and optical field intensities. A singlet-triplet read-out protocol for electron spin qubits using the same method is also discussed. The work in this chapter is published in Ref. [4]. In chapter 5 we demonstrate how to create a robust controlled-NOT (CNOT) gate using composite rotations, even when large fabrication induced variations mean the exchange interaction strength between qubits is not well known. Us- ing simulated exchange data we provide an analysis of gate time and count as a function of the pulse concatenation level. We motivate the importance of including a two-qubit Hamiltonian characterisation procedure with the composite rotations in order to minimise the concatenation level of the pulse, whilst still achieving a high fidelity operation. Combining these procedures in unison allows for construction of a robust CNOT gate which operates below current fault-tolerant error thresholds. The published version of this work appears in Ref. [5]. An analytical model describing charge fluctuators is introduced in chapter 6. From a probability density function describing the fluctuator statistics we develop expressions for the superoperator of a pair of spins coupled via the exchange interaction as a function of time, fluctuation amplitude and rate. We derive an approximate expression for the superoperator by examining appropriate limits of the probability density function and compare this to the general expression. The theory is extended to multiple fluctuators, including spectral distributions of fluctuators. Finally, we study the effects of a single 4 Introduction charge fluctuator on the operation fidelity of a two-qubit gate. Specifically, we examine the viability of constructing a CNOT operation from composite rotations as in chapter 5, determining in which fluctuator regimes, if any, the robust gate outperforms an uncorrected CNOT implementation. Chapter 2

Background

The fundamental concepts of quantum information theory and quantum computation necessary to the work in this thesis are presented in this chapter. Firstly, in section 2.1 a review of the early developments in the ﬁeld of quantum information processing (QIP) is presented. The fundamental model of QIP is the quantum circuit formalism and in section 2.2 an overview of this model is included. For a comprehensive review of the material presented in sections 2.1 and 2.2 the reader is directed to Ref. [6]. Finally, in section 2.3 the speciﬁc device architectures modelled in this thesis are introduced for completeness. An outline of the developments for each individual topic is included in the relevant chapter beginning.

2.1 A new paradigm for computation

The relentless push to increase the processing power of classical computers will inevitably lead to a paradigm shift in computing. Improvements in the current technology continue to drive down the physical size of components on integrated circuits, a trend which is remarkably well predicted by Moore’s law. In 1965, Gordon Moore, co-founder of Intel, predicted that the number of components on an integrated circuit would double roughly every year for constant cost [7], a trend that remained accurate until 1975. In 1975, Moore revised his original prediction to a doubling of the component density every 6 Background

18 months and shortly after, this trend became known as Moore’s law [8]. Extrapolation of this law into the 21st century led to the realisation that the transistor size would be of the order the atomic scale by the second decade. At this point, quantum effects are no longer negligible. During the 1970s and 1980s some of the first work on quantum information processing (QIP) began to emerge. In 1982, Richard Feynman suggested that computers built on the principles of quantum mechanics may help to avoid the difficulties of efficiently simulating quantum systems on classical computers [9]. Shortly after, in 1985, David Deutsch, motivated by the desire to find a model of computation which could efficiently simulate all other models, defined the quantum analogue of the Turing machine [10]. This computational device was based on the principles of quantum mechanics just as Feynman had suggested. Not until the 1990s, when the first quantum algorithms demonstrated the capability of quantum computers to solve certain problems more efficiently than classical computers, did the field of study attract significant attention. In 1992, Deutsch and Jozsa derived an algorithm that could determine whether a function was a constant binary or a balanced function quicker than any classical counterpart [11]. This was followed shortly after by Peter Shor’s remarkable quantum prime factoring algorithm [12], which has profound implications on the security of public key cryptosystems such as RSA encryption. The security of such cryptosystems is based closely on the inability of classical computers to factor large numbers, a problem efficiently soluble using Shor’s algorithm. Grover’s contribution in 1995 [13], an algorithm for searching an unstructured search space quadratically faster than classical algorithms, provided further proof of the power of quantum computers. One of the final developments to cement the experimental viability of using quantum states for information processing was the introduction of quantum error correction (QEC) by Robert Calderbank, Peter Shor and Andrew Steane in 1995-96 [14, 15, 16]. Without QEC, the quantum states would be subject to environmental noise ultimately causing the states to decohere, potentially rendering them useless for computation. Much progress has been made towards a full working quantum computer in a very small space of time, yet much remains to be achieved. We now introduce 2.2 The quantum circuit model 7 the fundamental model for QIP, the quantum circuit model.

2.2 The quantum circuit model

The quantum bit or qubit is the quantum analogue of the of the classical bit and is the fundamental constituent of QIP. However, whilst the bit must be either of the states 0 or 1, the qubit state may be either 0 or 1 or any | i | i linear superposition of these two states. The state of an arbitrary qubit may be expressed as ψ = α 0 + β 1 , (2.1) | i | i | i where α and β are in general complex coeﬃcients, constrained by normalisation requirements to satisfy α 2 + β 2 =1, (2.2) | | | | and the computational basis states 0 , 1 , form an orthonormal basis for {| i | i} the complex vector space. A single qubit may be visualised geometrically by re-expressing Eq. 2.1 as

θ θ ψ = cos 0 + eiϕ sin 1 , (2.3) | i 2| i 2| i where θ and ϕ deﬁne a point on the surface of a sphere known as the Bloch sphere. In this representation the qubit state ψ , is interpreted as any point | i on the surface of the Bloch sphere (see Fig. 2.1). An operation on the qubit may be visualised as a rotation about an axis deﬁned by the Hamiltonian, which traces out a path on the surface of the Bloch sphere. For example, an arbitrary single qubit rotation about the z-axis is achieved with a Hamiltonian comprised of the σZ Pauli matrix. The Pauli matrices

0 1 0 i 1 0 σX = , σY = − , σZ = , (2.4) 1 0 ! i 0 ! 0 1 ! − are extremely important single qubit operators, as they can be used to deﬁne 8 Background

0 | i z ψ | i θ

y x ϕ

1 | i Figure 2.1: The Bloch sphere is useful for visualising the state of a single qubit, with points on the surface of the sphere, deﬁned by the vector ψ , representing the qubit state. | i rotations about each of the x, y and z axes,

θ θ θ cos i sin i σX 2 2 − 2 Rx (θ) = e = θ − θ , (2.5) i sin cos ! − 2 2 θ θ θ cos sin i σY 2 2 − 2 Ry (θ) = e = θ − θ , (2.6) sin 2 cos 2 ! i θ θ e 2 0 i σZ − R (θ) = e− 2 = . (2.7) z i θ 0 e 2 !

It is common to write the Pauli matrices as X, Y and Z respectively when referring to them as quantum gates, as in Eq. 2.12. In a system comprised of many qubits, assuming no interactions have occurred between qubits, the state of the system is just the product state, Ψ = ψ ψ . . . ψ , where is the tensor product. Interactions be- | i | 1i⊗| 2i ⊗ | ni ⊗ tween qubits generate entanglement and entangled states, which are crucial to QIP. The well known Bell states B , are an example of maximally entangled | iji 2.2 The quantum circuit model 9 states, i.e., they cannot be separated out into a product state as above,

00 + 11 B00 = | i | i, (2.8) | i √2 01 + 10 B01 = | i | i, (2.9) | i √2 00 11 B10 = | i−| i, (2.10) | i √2 01 10 B11 = | i−| i. (2.11) | i √2

A quantum circuit is a sequence of single and multi-qubit gate operations arranged to achieve an algorithm for QIP. An example of a quantum circuit is shown in Fig 2.2. The circuit operates from left to right, with horizontal lines representing the passage of time for a qubit. Single qubit gates operate along a single horizontal line, whereas multi-qubit gates act on two or more qubits and are usually represented by vertical lines connecting qubits. Projective measurements are represented by the ﬁnal element to act on the third qubit in Fig. 2.2.

|ψ1i Z |ψ1′ i

|ψ2i Z X |ψ2′ i

qubits |ψ3i X

time Figure 2.2: An example of a quantum circuit containing three qubits. Boxes show the single qubit operations, whereas multi- qubit gates are usually represented by vertical lines connecting two or more qubits. Projective measurements are represented by the ﬁnal element to act on the third qubit. Time runs from left to right.

Unlike classical gate operations, which form the basis of Boolean logic, 10 Background quantum gate operations must be unitary in order to preserve the normalisation condition (see Eq. 2.2). This means quantum gates are necessarily reversible. An N qubit gate is represented by a 2N 2N matrix U, which × satisfies the unitarity condition U †U = I, where U † is the adjoint of U and I is the 2N 2N identity matrix. × The most common single qubit gates are the X-gate or bit flip/NOT gate (X), Y -gate (Y ), Z-gate or phase flip (Z), Hadamard gate (H), phase gate (S) and π/8 gate or T -gate (T ). Matrix representations for these gates are

0 1 0 i 1 0 X = ,Y = − , Z = , 1 0 ! i 0 ! 0 1 ! −

1 1 1 1 0 1 0 H = , S = , T = . (2.12) √2 1 1 ! 0 i ! 0 eiπ/4 ! − It is worth noting that H = (X + Z) /√2 and S = T 2. The application of any gate U, transforms the general state ψ , to a new state ψ′ , such that | i | i ψ′ = U ψ . For the X-gate the result is | i | i 0 1 α β ψ′ = X ψ = = . (2.13) | i | i 1 0 ! β ! α !

Multi-qubit gates generate interactions between qubits leading to entanglement, an essential property of QIP. The CNOT gate is the prototypical multi-qubit gate. The function of the CNOT gate is to ﬂip the state of the target qubit b if the control qubit a is in the state 1 . The circuit diagram | i | i | i and truth table for the CNOT gate appear in Fig. 2.3. The operator for the

|ai |ai a b a ⊕ b 0 0 0 0 1 1 1 0 1 |bi |a ⊕ bi 1 1 0

Figure 2.3: Circuit representation and truth table for the CNOT gate. 2.2 The quantum circuit model 11

CNOT gate is 1000  0100  UCNOT = . (2.14)  0001     0010      In general, a controlled-U operation [see Fig. 2.4(a) for the circuit representation] will perform the arbitrary single qubit operator U on the target qubit provided the control qubit is set to one. In total generality, there exists an extension of the controlled-U operation to a gate consisting of m control and n target qubits. The unitary operation Un, acting on the n target qubits is applied if and only if all m control qubits are set to one. The circuit for this gate operation appears in Fig. 2.4(b).

. . m controls . .

U . . n targets . Un .

(a) (b)

Figure 2.4: Circuit representations for (a) the controlled-U operation and (b) a generalised controlled operation which acts on n target qubits, conditional on the state of m control qubits.

Any unitary operation acting on N qubits can be approximated to arbitrary accuracy using a discrete set of gates [6]. This set of gates is said to be universal for quantum computation. The standard universal set of gates is the H, S, T and CNOT gates [17], however this is not the only possibility [18, 19, 20]. If arbitrary single qubit unitaries are allowed then any unitary operation on N qubits can be implemented exactly [6] and almost any two-qubit gate may be used [21, 22]. The final necessary element is measurement. Measurement most commonly refers to projective or destructive measurement, whereby the state of the qubit is projected onto one of the computational basis states, leaving a single bit of 12 Background classical information. For example, measuring the first qubit of the Bell state B would yield either 0 or 1 , each with 50% probability, collapsing the | 00i | i | i final state to 00 or 11 respectively. | i | i In the following section we provide an overview of the quantum computer architectures relevant to the work in this thesis.

2.3 Device architectures

Many quantum systems meet the criteria [23] for fabricating scalable quantum computers. The myriad of quantum computing proposals includes linear optical systems [24, 25], ion traps [26], nuclear magnetic resonance (NMR) on molecular qubits [27, 28], superconducting implementations [29, 30], spin qubits in silicon-germanium heterostructures [31], quantum dot designs [32, 33], cavity quantum electrodynamics (QED) devices [34, 35] and buried donor systems, such as the Kane nuclear spin proposal of phosphorus donors in silicon [1]. In this thesis we focus on the Kane nuclear spin proposal and related electron spin buried donor devices [2, 36], however much of the work is in principle applicable to a wider range of systems.

2.3.1 The Kane architecture

The Kane proposal for a quantum computer consists of an array of nuclear spins located on phosphorus donors in silicon (see Fig. 2.5). The phosphorus impurities 31P, possess nuclear spin I =1/2, whereas the isotopically pure 28Si lattice has nuclear spin I = 0. The orientation of the nuclear spin with respect to an externally applied magnetic field B, defines the qubit state. Control of the qubits is achieved by biasing the metallic surface gates that change the local electrostatic environment. Operations are performed with the electrons fully spin polarised by the uniform field B = 2 T and low op- e e 12 e erating temperature T = 100 mK, such that n /n 10− , where ni is the ↑ ↓ ≈ number density of donor electrons with spin i projection along the B-field axis. Broadly speaking, the A gates, which are located directly above qubits, control the single qubit operations by altering the hyperfine interaction between the 2.3 Device architectures 13

A gates J gates

SiO2

e− e−

31P+ 31P+

Figure 2.5: Schematic of the Kane proposal for a phosphorus in silicon quantum computer. The metallic surface gates are fabricated on a high quality oxide layer. A gates control single qubit operations and J gates generate qubit-qubit interactions. donor nuclear and electron spins, which in turn changes the nuclear magnetic resonance frequency of the qubit. Appropriately biasing the A gate brings the qubit in and out of resonance with a global ac magnetic field Bac, allowing arbitrary rotations to be performed on the nuclear spins. In the Kane proposal, two-qubit interactions are mediated by the electrons, which couple via the exchange interaction. The J gates, which are located cen- trally between donors, facilitate this, as the exchange interaction is dependent on the wave function overlap of the electrons. Coupling qubits in this way means evolution of the system can occur on a faster time scale than would be possible by directly coupling nuclear spins, a process normally governed via the weak dipole-dipole coupling. By considering also the hyperfine interaction and Zeeman splitting (due to the uniform magnetic field B) the full two-qubit electron-nuclear state space can be mapped. For a more detailed analysis of gate operations in the Kane quantum computer, the reader is directed to Refs. [1, 37]. Qubit measurement is a complex process owing to the difficulties in directly measuring the state of a single nuclear spin. In section 3.1 we present a detailed review of the read-out process. Kane’s ingenious solution (and the equivalent scheme of Loss and DiVincenzo as proposed in Ref. [32] for quantum dots) 14 Background was to convert the problem of single spin detection into one of measuring the formation of a spin-dependent charge signature using a highly sensitive electrometer. The presence or absence of this charge signature, where the qubit donor electron has tunnelled to an auxiliary donor forming a state known as the D−, infers the state of the qubit. The Kane architecture possesses a number of strengths, the most notable being the advantage afforded by the long decoherence times of nuclear spins, due to their isolation from the environment. Another significant advantage is the ability to leverage off the semiconductor industry in order to up-scale the device, with silicon fabrication techniques already well developed. This final point makes silicon an attractive candidate for other quantum computing proposals. One of these is the global electron spin proposal of Hill et al. [2], which is complimentary to the original Kane proposal. We describe this scheme in the following section.

2.3.2 The global electron spin architecture

Encoding the qubits on the polarisation of electron spins rather than nuclear spins is a natural extension of the Kane design, with many QIP proposals for electron spin qubits existing in a variety of systems [31, 32, 33, 38]. The larger magnetic moment of electrons means that an electron spin encoding would result in faster gate operations and simpler read-out. Donor electron spins also exhibit long decoherence times, an essential QIP requirement, with recent experiments estimating the dephasing time T2, of phosphorus donors in silicon to be greater than 60 ms at 7 K [39]. The global electron spin proposal [2] exploits the Kane design and features of the Si:P system to great advantage. A 2-D architecture, which operates on the principles of the global electron spin device was also recently proposed [36] and promises considerable scalability beneﬁts over the linear nearest neighbour device. The idea of the global electron spin quantum computer is to take full advantage of the fast electron evolution time scales by tuning a global ac magnetic

field Bac, with the electron resonance of unbiased qubits (A = A0). Local A gate control brings a target qubit out of resonance, causing it to rotate faster 2.3 Device architectures 15 about an off-resonant axis. The result is a relative rotation between the target and spectator qubits when all qubits are brought back into resonance with Bac. This technique allows for the realisation of single qubit operations once a final correction step accounts for any unwanted spectator evolution. Two-qubit interactions are achieved directly via the exchange coupling using J gate control. As first shown in Ref. [2], typical gate times for the global electron spin device are considerably faster than for the nuclear spin device. In Table 2.1 a comparison of the times taken to perform X and CNOT operations in these devices is provided to illustrate this.

Table 2.1: A comparison of the times required to perform X and CNOT gate operations in both the Kane nuclear spin [1] and global electron spin [2] architectures.

qubit TX TCNOT n-spin 6 µs 16 µs e-spin 30 ns 148 ns

Read-out of electron spins is easier than nuclear spins owing to the electrons larger magnetic moment. One possible technique for measurement is magnetic resonance force microscopy, which has recently exhibited the sensitivity to detect a single electron spin [40]. Resonant spin-to-charge transduction [3, 4] and ancilla assisted methods [41] are also viable alternatives.

Chapter 3

The read-out state for Si:P based architectures

The isolation of electron and nuclear spins from their environment makes them attractive qubit candidates [1, 2, 31, 32, 33, 38, 42, 43]. Conversely, and as a consequence, this means the detection of a single spin is notoriously diﬃcult, with only a few instances being reported experimentally [40, 44, 45]. Often the task of detecting a single spin is cleverly transformed into the easier problem of measuring charge, via spin-to-charge transduction [3, 4, 32, 41, 46, 47, 48]. In the Kane nuclear spin quantum computer [1] measurement of the qubit state relies on detection of a charge signature, or read-out state, known as the

D−, a doubly occupied phosphorus donor. The D− state is only very weakly bound, with a binding energy of 1.7 meV [49, 50], making it susceptible to ionisation by the large top gate bias ﬁelds controlling the device. A scenario such as this would be disastrous for the read-out process, which requires a D− dwell time longer than the time taken to measure by the measurement device. In the case of the Kane quantum computer, this measurement device is the single electron transistor (SET). For small induced charge levels (∆q < 0.05qe), single-shot read-out with a radio-frequency single electron transistor (rf-SET) operating near the quantum limit is of the order T 1 µs [51]. A theoretical SET ≈ description of the D− read-out state is therefore crucial for modelling aspects of the experimental design of the read-out process. Being able to model this 18 The read-out state for Si:P based architectures

system will help determine important information such as the D− dwell time and the maximum tolerable bias field of the D− state. In this chapter we take preliminary steps towards doing this by developing a theoretical description of the D− read-out state. Using this, we can calculate the binding energy of the D− state in order to verify the validity of our model. This theoretical description can then be used for more sophisticated modelling of the read-out process in the Kane architecture. This chapter is organised as follows, in section 3.1 we present a review of the Kane read-out scheme originally outlined in Refs. [1, 37]. In section 3.2 we outline the theory required to develop a description of the D− read-out state, by firstly describing how the Si:P qubit, D0, may be modelled. This is then extended to the difficult task of determining the binding energy of the D− state in section 3.3. Finally, 0 in section 3.4, future improvements to the modelling of the D and D− states are proposed.

3.1 Kane adiabatic read-out protocol

In order for quantum computers to operate without decoherence they must be completely isolated from their surrounding environment, (in actual fact, error correcting codes allow quantum computers to still function with a small amount of decoherence). This requirement makes the task of controlling and measuring qubits within a quantum computer onerous, particularly the measurement of single nuclear spins, which only couple weakly to their surrounding environment. In the Kane nuclear spin quantum computer, adiabatic read-out of the state of the nuclear spin qubit is conditional on the detection of the D− charge signature, a spin singlet state formed when the transition 0 0 + D D D D− causes two electrons to occupy the same site. The transition is → controlled so as to be dependent on the state of the nuclear spin in question.

Detection of the D− state is facilitated via measurement using a highly sensitive electrometer such as an SET. This transition has been observed in optical studies of bulk-doped silicon [52]. The doubly occupied donor is known to be bound by 1.7 meV as compared to the neutral donor, D0, which has a binding energy of 45.5 meV [53]. We now outline how this transition can be exploited 3.1 Kane adiabatic read-out protocol 19 to determine the state of the qubit following Ref. [37]. The system we wish to study comprises an unknown qubit (qubit 1), whose state is to be determined and a reference qubit (qubit 2), whose state is independent of the read-out process and need not be known (see Fig. 3.1). The

Read-out SET Control gates (A, J, A)

SiO2 Si

Reference qubit Unknown qubit dc ﬁeld Figure 3.1: Kane’s read-out proposal involves the adiabatic, spin- dependent charge transfer of a single electron from the qubit being measured to a nearby reference qubit in order to form the D− state. reference qubit is situated near the SET and is the target site for the D− state. The Kane Hamiltonian (ignoring Bac terms, which are not relevant to the read-out process) for this two-donor system is comprised of a contribution from three separate Hamiltonian terms, a Zeeman Hamiltonian HB, hyper-

ﬁne Hamiltonian HA and an exchange Hamiltonian HJ , such that the total Hamiltonian is

H = HA + HB + HJ . (3.1)

To describe the read-out process the evolution of the electron-nuclear energy levels must be mapped from computation (J<µBB/2) to read-out (J>µBB/2). We begin by calculating the energy levels relevant to this process, starting with the eigenspectrum of the pure exchange Hamiltonian,

H = Jσ1e σ2e, (3.2) J · = 4JS S , (3.3) 1e · 2e = 2J(S2 S2 S2 ), (3.4) − 1e − 2e 20 The read-out state for Si:P based architectures where J is the exchange coupling, the σie are the Pauli matrices for the ith electron, S is the total spin operator of both electrons and the Sie are the spin operators of the ith electron. The eigenvectors of this Hamiltonian are the singlet, 1 χsinglet = [ ] , (3.5) | i √2 |↑↓i − |↓↑i and triplet,

|↓↓i χ = 1 [ + ] , (3.6) | tripleti  √2 |↑↓i |↓↑i  |↑↑i spin states with eigenvalues 3J and J respectively. By turning on a static B − ﬁeld, the Zeeman terms cause the splitting of those electronic and nuclear states whose total spin state z-component is non-zero. This breaks the degeneracy of the spin triplet. The Hamiltonian for this is

H = µ Bσ1e g µ Bσ1n + µ Bσ2e g µ Bσ2n, (3.7) B B z − n n z B z − n n z where µB is the Bohr magneton, µn is the nuclear magneton, gn is the nuclear ie th in g-factor, σz is the Pauli matrix for the i electron spin, σz is the Pauli th matrix for the i nuclear spin and we have implicitly assumed ge = 2 for the electron. In Fig. 3.2, we show the electronic states as a function of the exchange coupling. In the presence of a magnetic ﬁeld, the triplet state , is shifted |↓↓i below the singlet state. Increasing J lowers the singlet state below , when |↓↓i the singlet-triplet splitting is equal to the Zeeman splitting (4J = 2µBB). This is important because in the Kane quantum computer all computations are performed with the electron spins fully polarised (kT J<µ B/2) but ≪ B measurement occurs when the singlet state is lowest in energy (J>µBB/2). Splitting of the nuclear spin energy eigenstates occurs on a much smaller energy scale since µ µ . The result is each electronic level being split into four n ≪ B closely spaced electron-nuclear energy levels, meaning sixteen states in total.

The interaction between electron and nuclear spins can now be included by 3.1 Kane adiabatic read-out protocol 21

5 singlet 4 triplet |↑↑i 3 2 1 [|↑↓i + |↓↑i] √2 B 1 B 0 E/µ |↓↓i -1

-2 1 [|↑↓i − |↓↑i] √2 µB B -3 J = 2 -4 0 0.2 0.4 0.6 0.8 1 1.2

J/µBB

Figure 3.2: Eigenspectrum for the electronic states of a two-qubit system under exchange and Zeeman Hamiltonians as a function of the exchange coupling strength J (constant B). The Zee- man terms resolve the degenerate triplet states, lowering the |↓↓i triplet for J <µBB/2. When J >µBB/2, the singlet state is lowered in energy with respect to the triplet state . |↓↓i

considering the hyperﬁne interaction

H = A σ1e σ1n + A σ2e σ2n, (3.8) A 1 · 2 ·

in where A1 and A2 are the hyperfine interaction energies, σ are the Pauli matrices of the ith nuclear spin and σie is defined as before. The hyperfine coupling between nuclear and electron spins is controllable via the top gate voltages, which effect A1 and A2. Because the contribution from this process is small, the relevant section of the Hamiltonian can be treated as a perturbation. The effect of the perturbation on the energy levels relevant to the read-out process can be calculated to second order in A, allowing the evolution of the electron-nuclear energy levels to be approximated analytically as a function of 22 The read-out state for Si:P based architectures

J. For A1 = A2 = A, the relevant perturbed energy levels are

(2) E 11 = 2µBB +2gnµnB + J +2A, (3.9) |↓↓i| i − 2 (2) 2A E sn = 2µBB + J , (3.10) |↓↓i| i − − µBB + gnµnB 2 (2) 2A E an = 2µBB + J , (3.11) |↓↓i| i − − µBB + gnµnB 2J (2) − E 00 = 2µBB 2gnµnB + J 2A |↓↓i| i − − − 2A2 2A2 , (3.12) −µ B + g µ B − µ B + g µ B 2J B n n B n n − where we have adopted the following abbreviations

1 se = [ + ] , (3.13) | i √2 |↑↓i |↓↑i 1 ae = [ ] , (3.14) | i √2 |↑↓i − |↓↑i 1 sn = [ 10 + 01 ] , (3.15) | i √2 | i | i 1 an = [ 10 01 ] . (3.16) | i √2 | i−| i

We now turn our attention to explaining what happens when J is adiabatically increased past the level crossing at J = µBB/2 initiating qubit read-out. The ability to track each of the four triplet states through to read-out is crucial |↓↓i in determining the qubit state. From Fig. 3.2, when J <µ B/2 the electrons are spin polarised in . B |↓↓i With A1 > A2 and J = 0, the lowest four eigenstates are (from highest in energy to lowest): 11 , 10 , 01 , 00 . As ∆A = A A |↓↓i| i |↓↓i| i |↓↓i| i |↓↓i| i 1 − 2 is adiabatically reduced to zero and J adiabatically increased from zero but remaining below J = µ B/2, nuclear states 10 and 01 gradually become B | i | i degenerate, evolving into s and a respectively. Full numerical simulation | ni | ni of the system reveals that as J is increased further, past J = µBB/2, two of the four electron-nuclear states described by Eqs. 3.9-3.12 exhibit anti-crossing behaviour with two of the singlet states, indicating a connection via the hyperfine perturbation. More specifically, 00 and a adiabatically evolve |↓↓i| i |↓↓i| ni 3.1 Kane adiabatic read-out protocol 23 to a a and a 11 respectively. Both 11 and s are unaffected | ei| ni | ei| i |↓↓i| i |↓↓i| ni by the hyperfine interaction and cross the singlet states. The results of the simulation appear in Fig. 3.3. They show the eigenspectrum of the eight lowest electron-nuclear energy levels as a function of J. In order to follow the

-1.49

-1.495 B

B -1.5 E/µ -1.505

-1.51

0.497 0.498 0.499 0.5 0.501 0.502 0.503

J/µBB

Figure 3.3: Evolution of the eight lowest electron-nuclear energy levels as a function of J for the two-qubit system. From highest in energy to lowest (on the left) they are: ae 11 , ae an , a s , a 00 , 11 , s , a , | i|00 .i Adiabat-| i| i | ei| ni | ei| i |↓↓i| i |↓↓i| ni |↓↓i| ni |↓↓i| i ically increasing the exchange coupling past J = µBB/2 results in anti-crossing behaviour for those electron-nuclear energy levels connected via the hyperfine interaction. Specifically, 00 |↓↓i| i and an evolve to ae an and ae 11 respectively. All other states|↓↓i| remaini unaffected| i| by thei hyperfine| i| i interaction. evolution of these eigenstates from computation to read-out we summarise the relevant results

µB B µBB J=0, A1>A2 0 2 , A1=A2 11 11 11 , |↓↓i| i −→ |↓↓i| i −→ |↓↓i| i 10 s s , (3.17) |↓↓i| i −→ |↓↓i| ni −→ |↓↓i| ni 01 a a 11 , |↓↓i| i −→ |↓↓i| ni −→ | ei| i 00 00 a a . |↓↓i| i −→ |↓↓i| i −→ | ei| ni 24 The read-out state for Si:P based architectures

For the states which have adiabatically evolved into the spin singlet state 0 0 + a , appropriately biasing the A gates will allow the transition D D D D− | ei → to occur, forming the D− beneath the SET. Formation of the D− state is conditional on the system being in a spin singlet, which is in turn conditional on the nuclear spin of the first qubit being in the state 0 . If after an adiabatic | i increase in J the system remains in the spin triplet , then no charge transfer |↓↓i will occur and it can be inferred that the nuclear spin state of the first qubit must be 1 . Interestingly this process does not require any prior knowledge | i about the state of the reference qubit, (qubit 2). We now turn our attention to the modelling of the Kane Si:P qubit, D0. Building a theoretical description of this state is the first step towards modelling of the D− read-out state. We begin with a review of the Bloch theory used to describe the periodic silicon lattice, and then move to the effective mass theory, which can be used to model impurity donors in this silicon lattice. We outline the assumptions that have been made in order to accomplish the difficult task of modelling the Si:P qubit.

3.2 Modelling the Si:P qubit

Group-V donors in silicon are often referred to as a solid-state analogue of the hydrogen atom. Phosphorus atoms are similar in size to silicon given their proximity in the periodic table. This property means phosphorus is a stable substitutional donor and to a good approximation four of its five valence electrons are able to covalently bond with the neighbouring silicon atoms. The fifth electron is weakly bound by the excess charge of the phosphorus nucleus, with an experimentally measured binding energy 45.5 meV below the conduction band minimum of silicon. Shallow donor states created by impurities like phosphorus in the silicon crystal have been well studied using effective mass theory (see Ref. [54] for a detailed review of effective mass theory). Modelling these shallow donor states is non-trivial due to the complexities of the silicon lattice, which has six degenerate conduction band minima. Our task is to ultimately model the phosphorus donors in the silicon host using effective mass theory. 3.2 Modelling the Si:P qubit 25

Often a simple initial approach to modelling the phosphorus dopants in silicon involves using scaled hydrogenic orbitals given their resemblance to hydrogenic atoms. Lattice effects are incorporated by treating the silicon host as a uniform dielectric and giving the electron an isotropic effective mass. The Hamiltonian includes a Coulombic potential term for the donor scaled by the silicon dielectric constant ε = 11.9, and the electron mass is given by the effective mass m∗ = 0.2me. The scaled Bohr radius and bound state energies given by the hydrogenic solutions are

ε a∗ = aB = 31.5 A˚, (3.18) m∗ m∗ E∗ = E = 19.2 meV. (3.19) n ε2 n

The binding energy predicted by this simple approach is not consistent with the experimental value of 45.5 meV. A more sophisticated approach which accounts for the structure of the silicon lattice and the anisotropic nature of the eﬀective mass can account for some of this discrepancy. We now go beyond this simple approach using eﬀective mass theory and develop a wave function which takes into account the silicon lattice structure.

3.2.1 The silicon lattice

To develop a sophisticated understanding of the phosphorus impurity we require an understanding of the silicon host. The silicon atom (Z = 14), has 10 core and 4 valence electrons. The silicon lattice is of face-centered-cubic (fcc) structure, with each silicon atom in the fcc lattice bound to four others via the four valence shell electrons. The core electrons are more tightly bound to the nuclei than the valence electrons and as such, along with the nuclei, they may be treated as part of the core potential of the silicon atom. This greatly sim- plifies the problem of describing the electronic structure of the silicon lattice, which in general would require solving the many-body Hamiltonian for all the electrons. For a macroscopic object of N atoms this is still a formidable task as there are 4N valence electrons in the lattice. Further simplification is possible by using the mean-field approximation, which amounts to assuming that 26 The read-out state for Si:P based architectures each electron experiences an average potential due to the other electrons. The many-body Hamiltonian reduces to a single particle mean-field Hamiltonian for a valence electron in the silicon lattice

~2 2 H0 = + VSi (r) , (3.20) −2me ∇ where VSi (r) is the crystal potential for silicon. The silicon potential is periodic in the fcc lattice and is invariant under translations by any fcc lattice vector R, such that VSi (r)= VSi (r + R). The eigenstates of H0 are the Bloch states [55]

ik r φk (r)= uk (r) e · , (3.21) where uk (r)= uk (r + R) also exhibits the periodicity of the lattice. The uk (r) can be expanded in terms of plane waves in the reciprocal lattice vectors G,

iG r uk (r)= Ak (G) e · , (3.22) G X where the Ak (G) are coefficients to be determined. The reciprocal lattice vectors provide a description of the crystal lattice in momentum space. Due to the extended naure of the lattice it is convenient to restrict the problem to a unique region of momentum space - the first Brillouin zone. In this representation, known as the reduced zone representation any vector k which falls outside the first Brillouin zone can be translated back inside using an appropriate reciprocal lattice vector. When we restrict k to be within the first Brillouin zone in this fashion, there will be a number of solutions φk (r) to the eigenvalue equation

H0φnk (r)= Enkφnk (r) , (3.23) each corresponding to a particular k. It is convention to label these solutions by a subscript n as above, in the order of increasing energy Enk, to distinguish the solutions. This is the origin of the band structure. The Bloch states may now be expressed i(k+G) r φnk (r)= Ank (G) e · . (3.24) G X 3.2 Modelling the Si:P qubit 27

In order to determine the coefficients Ank (G) and silicon band structure we substitute the Bloch states as defined above into Schrödinger’s equation with Hamiltonian as in Eq. 3.20 and simplify

~2 2 i(k+G) r (k + G) Ank (G) e · − 2me G X i(k+G) r i(k+G) r + VSi (r) Ank (G) e · = Enk Ank (G) e · . (3.25) G G X X The silicon potential can be transformed into reciprocal space and the expres- i(k+G ) r sion simpliﬁed by multiplying by e− ′ · and integrating over r

2 ~ 2 (k + G′) Ank (G′)+ Ank (G) VSi (G G′)= EnkAnk (G′) . (3.26) −2me G − X The potential can be described by adopting the method of pseudopoten- tials. The basic idea of the pseudopotential method is to ignore the rapidly varying behaviour of the wave function at the nucleus, instead describing the behaviour of the valence electrons at a distance away from the nucleus. For silicon, which consists of two interlaced fcc lattices displaced from one another by a vector 2s = (a/4)(1, 1, 1), (where a is the silicon lattice constant, taken to be 5.43 A)˚ the pseudopotential can be written as

V (G G′)= V (G G′) cos [(G G′) s] , (3.27) Si − s − − · where the Vs are symmetric pseudopotential form factors which give the contribution to the silicon potential from one of the silicon atoms. The potential decreases like V (q) 1/q2 for large q, so only contributions from vectors Si ∼ with a small magnitude need be included. For a full description of the pseudopotential method the reader is directed to Ref. [56]. The band structure of silicon can now be determined by solving the Schr¨odinger equation in Eq. 3.26 including the pseudopotential form in Eq. 3.27. The calculated band structure of silicon along the [100] crystallographic direction for the lowest eight energy levels can be seen in Fig. 3.4. The ﬁgure shows both valence (red) and conduction (blue) bands. One of silicons six degenerate con- 28 The read-out state for Si:P based architectures

(eV) 10 E 5

-5 0 0.2 0.4 0.6 0.8 1 k (2π/a) | | Figure 3.4: Band structure of silicon plotted along the [100] direction, including both the valence (red) and conduction (blue) bands. The conduction band minimum, of which there are six degenerate minima for silicon, occurs at k 0.85 2π/a. The donor electron wave function is well localised| | ≈ around× them in reciprocal space. duction band minima (located along the [100], [010], [001] axes) can be seen at k 0.85 2π/a. In the following section we show how in eﬀective mass the- | | ≈ × ory the donor electron wave function is expressed as a localised wave function in reciprocal space around these conduction band minima.

3.2.2 Impurity donors and eﬀective mass theory

Phosphorus donors can to a good first approximation be treated as hydrogenic with four of the five valence electrons able to bond covalently with the surrounding silicon. At the beginning of section 3.2 we demonstrated that this approach does not accurately reproduce the binding energy of the extra electron. We now introduce the theory of shallow donor states, originally developed by Kohn and Luttinger in the 1950s [57, 58, 59, 60]. Known as effective mass theory, the theory gives a much more accurate account of the energies of impurities introduced to the silicon lattice. 3.2 Modelling the Si:P qubit 29

Kohn and Luttinger showed that the wave function for a donor in silicon could be expressed 6

ψ (r)= αjFj (r) φkj (r) , (3.28) j=1 X where the sum is over equivalent conduction band minima j, the αj are coef- ﬁcients determined by the point group symmetry of the phosphorus donor in silicon, and the Fj (r) are anisotropic envelope functions. The process of determining the wave function for a phosphorus donor in silicon involves solving the Schr¨odinger equation for the donor state

Hψ (r) = [H0 + U (r)] ψ (r)= Eψ (r) . (3.29)

The impurity potential U (r) is to good approximation Coulombic, scaled by the silicon dielectric constant ε, to account for the silicon lattice such that U (r)= e2/εr. To arrive at the form of the wave function given by Eq. 3.28 − requires a number of steps and carefully applied approximations, which we now outline. We begin by expanding the solution to the Schr¨odinger equation ψ (r), in terms of the eigenstates of the unperturbed Hamiltonian, H0

ψ (r)= Fnkφnk (r) . (3.30) k Xn,

Substitution into the Schr¨odinger equation (Eq. 3.29), multiplication by φ∗ k (r) n′ ′ and integration over all space yields

k k k k k EFn = En Fn + φn∗k (r) U (r) φn′ ′ (r) Fn′ ′ dr, (3.31) k Z nX′, ′ where we have swapped the primed and unprimed quantities. This expression can be simpliﬁed further by expanding out the Bloch states using Eq. 3.24 and taking the Fourier transform of the potential such that

k k k k k EFn = En Fn + An∗k (G) An′ ′ (G′) U (k k′ + G G′) Fn′ ′ k G G − − n′, ′ , ′ X X (3.32) 30 The read-out state for Si:P based architectures

The previous expression is exact and to proceed further we are required to make some simplifying approximations. It is usual in eﬀective mass theory to consider only one energy band, or in the case of degenerate bands, all bands which are degenerate. This restricts the applicability of the resulting equations to particular forms of the potential U (r), however our choice remains valid. From now on the band index n is therefore dropped.

Silicon has six degenerate conduction band minima located at the points kj in momentum space, where

2π k ( 0.85, 0, 0) , (0, 0.85, 0) , (0, 0, 0.85) , (3.33) { j} ≈ a { ± ± ± } and the crystallographic directions of the conduction band minima are along [100], [010] and [001]. The donor impurity wave function is well localised around these degenerate conduction band minima so we write

6 Fk = α Fk (k k ) , (3.34) j j − j j=1 X such that at each minimum kj, there is a well localised Fkj . Provided the impurity donor potential possesses a high degree of symmetry (i.e., radial) the

αj coeﬃcients may be determined from the point group symmetry of the donor.

For an impurity in silicon this is the Td point group symmetry. If however the local donor environment is inhomogenous, such as if an external potential is present, the αj will in general depend on this external potential [46]. In the vicinity of the conduction band minima the conduction band energies can be approximated by expanding them to second order in (k k ) − j ~2 2 k2 k j j Ek E0 + k + ⊥ , (3.35) ≈ 2 m∗ m∗ ! k ⊥

where k j is the component of (k kj) parallel to the axis the conduction band k − minima lies along and k j are the components of (k kj) perpendicular to this ⊥ − axis. The masses m∗ and m∗ are the eﬀective electron masses corresponding k ⊥ to these directions and are non-isotropic, which is apparent from viewing the 3.2 Modelling the Si:P qubit 31 energy surfaces at these minima (see Fig. 3.5). In unstrained silicon each of the six conduction band minima are equivalent, meaning the masses m∗ and k m∗ are not valley dependent. The potential in the region where Fkj is well ⊥

k 0.85 2π/a | |≈ ×

Figure 3.5: Constant energy ellipsoids at the six conduction band minima of silicon. These minima are located at the six equivalent [100] directions. The mass anisotropy m∗ = m∗ , results in the k 6 ⊥ noticeable elongation of the surface in the direction parallel to the axis on which the surface lies. localised can be well approximated by ignoring all terms (G G′) = 0, since − 6

U (k k′ + G G′) U (k k′) , (3.36) | − − |≪| − | which follows from the Fourier transform of the potential, U (q) 1/q2. ∼ Next, the plane wave expansion coeﬃcients of the Bloch functions are

k k treated as approximately independent of k by setting A (G) = A ′ (G), an assumption that is found to be well satisfied in the vicinity of the conduction band minima. Finally, the sum over k′ ranging across the first Brillouin zone 32 The read-out state for Si:P based architectures is replaced by an integral over all of k-space. Along with the normalisation 2 of the plane wave coefficients, Ak (G) = 1, these assumptions result in G | | the simplification of the expressionP in Eq. 3.32 to

~2 k2 k2 j j k ⊥ E′ αjFkj (k kj)= + αjFkj (k kj) − 2 m∗ m∗ − j ! j X k ⊥ X

+ U (k k′) αjFk k′ k′ dk′, (3.37) − j′ − j Z j X where E′ = E E . The localisation of the Fk around minima k means we can − 0 j j neglect intervalley coupling and decouple the above result into six independent one-valley eﬀective mass equations

~2 2 k2 k j j k ⊥ E′Fkj (k kj)= + Fkj (k kj) − 2 m∗ m∗ ! − k ⊥

+ U (k k′) Fk k′ k′ dk′. (3.38) − j′ − j Z This equation is isomorphic to a non-isotropic Schr¨odinger equation in momentum space and is in general diﬃcult to solve. Kohn and Luttinger [59] and independently Kittel, Mitchell and Lampert [61, 62], determined an approx-

i(k kj ) r imate solution for the F (r), where F (r) = e − · Fk (k k ) dk and j j j − j has the functional form (corresponding to a minimumR along the [001] axis for example)

1 2 2 2 Fj z (r)= exp (x + y /a )+ z /a . (3.39) ± 6πa2 a − ⊥ k ⊥ k h q i p The six independent solutions unavoidably lead to a six-fold degenerate ground state contradicting experiment, which showed that the ground state of the donors is actually split into three levels, a singlet, doublet and triplet [63]. In earlier work Kohn postulated that by including intervalley terms, which go beyond the one-valley eﬀective mass theory, the levels could be split [60]. This degeneracy can be explained by considering the Td point group symmetry of the phosphorus donor in silicon. The six degenerate solutions may be used 3.2 Modelling the Si:P qubit 33

to form a basis for a representation of the group Td. This representation can be reduced into the irreducible representations A1, E and T1, which are the singlet, doublet and triplet levels found by experiment. The coeﬃcients αj corresponding to the irreducible representations of the point group symmetry

Td, are 1 (1, 1, 1, 1, 1, 1) A √6 1 1 (1, 1, 1, 1, 0, 0)  2 − − E  1  (1, 1, 0, 0, 1, 1) )  2 − − αj =  1 . (3.40)  (1, 1, 0, 0, 0, 0)  √2 − 1 (0, 0, 1, 1, 0, 0) T √2  1  −   1 (0, 0, 0, 0, 1, 1)   √2  −   To ﬁrst order in k the total wave function is then given by

ψ (r)= αjFj (r) φkj (r) , (3.41) j=1 X as in Eq. 3.28, where for the ground state energy of the donor impurity in silicon the αj should be chosen to correspond to the irreducible representation

A1. Using the standard one-valley eﬀective mass theory we now calculate the binding energy of a D0 phosphorus donor in silicon. This is a preliminary step towards calculating the binding energy of the two electron D− state.

3.2.3 Binding energy of a phosphorus donor in silicon

In order to determine the binding energy of the two electron D− state, we must ﬁrstly calculate the binding energy of the neutral phosphorus donor in silicon (D0 state). For two electrons bound to the same donor centre the ground state corresponds to when both electrons occupy the A1 symmetry state, with spins anti-aligned in the spin singlet, [ ] /√2. This is analogous to |↑↓i − |↓↑i calculating the ground state energy of the helium atom in free space and is in general a diﬃcult problem to solve as it involves adding the contribution from both electrons as well as the interaction between electrons. In order to simplify 34 The read-out state for Si:P based architectures

the problem of calculating the D− binding energy we treat the energy surfaces at the conduction band minima as spherical, which amounts to ignoring the mass anisotropy by setting m∗ = m∗ . Although this will not provide the best k ⊥ result for the binding energy of the D− state, this approach should greatly simplify the complexity of the problem for a ﬁrst attempt at calculation. In fact it has been noted that including the eﬀect of electron correlations in the two electron wave function has a much greater impact than the mass anisotropy does on the resulting energy [64].

The binding energy of a phosphorus donor in silicon is now calculated using the eﬀective mass formalism, ignoring the mass anisotropy. The position of the phosphorus donor is actually displaced from the fcc lattice by the vector s = (a/8)(1, 1, 1). The electron wave function of the D0 state, derived from the anisotropic form in Eq. 3.28, has the form

6 i(kj +G) r ψ (r s) ψ (r′)= F (r′) Ak (G) e · , (3.42) − ≡ j j=1 G X X where F (r′)= 0 exp ( r′/a∗) is the isotropic envelope function, r′ = r s ND − | − | and the normalisation 0 absorbs the α , which are symmetric for the A ND j 1 ground state. The parameter a∗ is the effective Bohr radius defined previously in Eq. 3.18. In Fig. 3.6 we compare wave functions with isotropic and anisotropic envelopes along the [100] crystallographic direction. The results for the anisotropic case were calculated by Wellard et al. in Ref. [65]. The wave functions show fairly good agreement, which seems to validate the choice of ignoring mass anisotropy in the first instance.

The Hamiltonian for the donor impurity electron is

~2 2 2 e H = + VSi (r) . (3.43) −2me ∇ − εr′

Expressions for the energy expectation value of Eq. 3.43 were determined analytically and evaluated numerically. The energy of the D0 state is found to be 34.7 meV, which is in closer agreement with the experimental value of 45.5 meV than the scaled hydrogenic solution (see Eq. 3.19). At the conclu- 3.3 Determining the binding energy of the D− read-out state 35

[100]

isotropic anisotropic ) r ( ψ

-15 -10 -5 0 5 10 15 r (nm)

Figure 3.6: Comparison of the isotropic and anisotropic eﬀective mass wave functions. Both wave functions are plotted along the [100] direction by holding two variables constant. sion of the following section we discuss how improvements to this calculated value may be obtained, hence closing the gap between theory and experiment. However, we now turn our attention to calculating the binding energy of the doubly occupied D− state.

3.3 Determining the binding energy of the D− read-out state

The importance of the D− state for read-out in the Kane Si:P quantum computer has already been outlined in section 3.1. The detection (or absence of detection) of the D− charge signature by the SET can be used to determine the state of the nuclear spin qubit. Paramount to the formation of the D− 0 0 + is the charge transfer process D D D D−. Experimental realisation would → involve applying a large gate bias across the A gates in order to eﬀect the transition provided the system was in the spin singlet as outlined in section 3.1. 36 The read-out state for Si:P based architectures

Due to the low binding energy of the state, which is experimentally known to be 1.7 meV, until fabrication of the two-qubit array is possible, it is unknown whether or not the presence of the bias ﬁeld will cause the extra electron to be lost to the conduction band forever. It is therefore of great importance to know how the D− state behaves under changes in gate potentials. An important initial step towards treating this problem is to be able to accurately model the state, including calculating characteristics such as the binding energy. This requires the use of a sophisticated wave function, as the task of modelling the two electron state is known to be notoriously diﬃcult.

The Kane read-out protocol requires that the D− is a spin singlet state. This implies that the spatial part of the wave function must be symmetric in order to satisfy Fermi-Dirac statistics, which demand a totally antisymmetric state overall. The obvious extension to the single electron isotropic envelope wave function used in section 3.2.3 is the Chandrasekhar wave function, which was originally used for the H− problem and is known to give a good account of D− centres [64]

αr1 βr2 βr1 αr2 F (r , r )= e− − + e− − (1 + λr ) . (3.44) D− 1 2 ND− 12 th The α, β and λ are evaluated variationally, the ri give the position of the i electron relative to the phosphorus donor, r12 is the electron correlation term and is the normalisation of the D− state. Including the Bloch structure ND− and the oﬀset of the phosphorus donor from the fcc lattice as before, the full spatial part of the D− wave function is

i(k +G ) r i(k +G ) r r r r r k G k G i 1 1 j 2 2 ψ ( 1′ , 2′ )= FD− ( 1′ , 2′ ) A i ( 1) A j ( 2) e · e · i, j G1, G2 X X i(ki+G1) r2 i(kj +G2) r1 + e · e · . (3.45) Earlier attempts to calculate the binding energy of the D− state using a much 3.3 Determining the binding energy of the D− read-out state 37 simpler trial function

ξ(r +r ) i(ki+G1) r1 i(kj +G2) r2 ψ (r′ , r′ )= e− 1′ 2′ Ak (G ) Ak (G ) e · e · 1 2 N i 1 j 2 i, j G1, G2 X X i(ki+G1) r2 i(kj +G2) r1 + e · e · , (3.46) gave poor results, in particular, an unbound D− state [66]. From this analysis it was concluded that the complexity of the envelope wave function should be increased to account for electron-electron interactions and also to include independent variational parameters for both electrons as in Eq. 3.44.

To determine the variational parameters α, β and λ, the Chandrasekhar variational wave function (Eq. 3.44) is minimised with respect to the isotropic eﬀective mass Hamiltonian

~2 ~2 2 2 2 2 2 e e e Heﬀ = 1 2 + . (3.47) −2m∗ ∇ − 2m∗ ∇ − εr1′ − εr2′ εr12

The results for these parameters agree with those of Ref. [64], α = 1.075,

β = 0.478 and λ = 0.312. It is now possible to determine the D− binding energy by calculating the energy expectation value of the Hamiltonian

~2 ~2 2 2 2 2 2 e e e H = 1 2 + VSi (r1)+ VSi (r2) + , (3.48) −2me ∇ − 2me ∇ − εr1′ − εr2′ εr12 using the full D− wave function given by Eq. 3.45. All integrations except those containing electron-electron interactions are performed analytically, however due to the sheer number of results and their size they have not been included here. The remaining integrations are performed numerically. The large number of summations and the oscillatory nature of the Bloch structure means numerical integration of the interaction terms is diﬃcult. To solve this problem an interpolation program was developed to store the Bloch structure in memory and hence greatly reduce computation time.

The calculated binding energy of the D− state is found to be 0.9 meV, which considering the difficult nature of the calculation and the approximations applied to simplify this first attempt, is in reasonable agreement with the 38 The read-out state for Si:P based architectures experimentally determined value of 1.7 meV. Considerable improvements are 0 needed in the modelling of the D and D− states though and this should be achieved by relaxing the approximations. With such a large difference between the calculated (34.7 meV) and experimental (45.5 meV) energies for even the D0 centre in silicon, it is difficult to conclude whether the calculated binding energy agrees by design, or coincidence. Nevertheless these preliminary results are encouraging. In the following section we briefly outline how future calculations can be improved.

0 3.4 Improving the modelling of the D and D− states

0 There are two obvious avenues for improvement of the D and D− calculations. These are, inclusion of the mass anisotropy, which is known to exist in the reciprocal space of the electron and the inclusion of valley-orbit coupling terms to describe the effects of intervalley coupling. Both of these have been neglected to date in order to simplify the already difficult nature of these calculations. We have already noted that we expect the inclusion of mass anisotropy to have only a small effect, nevertheless a proper treatment of the non-spherical nature of the energy surfaces at the conduction band minima of silicon requires mass anisotropy. Inclusion of the mass anisotropy terms would require using 0 an envelope of the form Eq. 3.39 for the D case and in the case of the D−, for a minimum corresponding to the [001] axis for example, an envelope of the form

2 2 2 2 2 2 an an α (x1+y1 ) α z1 β (x2+y2 ) β z2 F (r1, r2)= e− ⊥ − k − ⊥ − k D− ND− 2 2 2 2 2 2 h β (x1+y1) β z1 α (x2+y2 ) α z2 + e− ⊥ − k − ⊥ − k (1 + λr12) , (3.49) i would be required (note that the oﬀset of the phosphorus donor from the fcc lattice by s, has not yet been included in these envelope functions). The Hamiltonians would also have to reﬂect the mass anisotropy by including the transverse and longitudinal masses m∗ and m∗ just as in Eq. 3.38. ⊥ k 3.5 Summary 39

The valley-orbit coupling affects the distribution of the wave function across the different conduction band minima (valleys). This coupling term accounts for interactions between the electrons in equivalent valleys, “intravalley” (e.g., z and z axes) and non-equivalent valleys, “intervalley”( e.g., x and z axes). − Inclusion of the valley-orbit coupling lifts the six-fold degeneracy which remains if the one-valley effective mass theory is used. Baldereschi was the first to be show this using an effective mass type calculation in 1970 [67]. The valley orbit coupling can have a considerable effect on the resulting energy, particularly for the true ground state, or A1 level, which is well separated from the E and T1 levels and not well represented by the one-valley effective mass theory [54]. It should be noted that the Coulombic impurity potential itself is only an approximation to the true impurity potential, as it does not account for the core phosphorus electrons. Improvement to the modelling of this potential via central cell corrections is possible, although the result is unlikely to be as profound as corrections arising from the inclusion of mass anisotropy and valley-orbit coupling terms.

3.5 Summary

In this chapter we have given an introduction to the Kane nuclear spin read- out protocol. The D− state is crucial to the read-out process as detection (or absence of detection) of this charge signature determines the qubit state. The doubly occupied D− state is only weakly bound to the phosphorus donor, with experiment putting this value at 1.7 meV. This poses real problems for the survival of the state during the read-out protocol due to the large bias fields 0 0 + required to effect the transfer D D D D−. → We have taken a first step towards the realistic modelling of this state using effective mass theory, which improves considerably upon the use of scaled hydrogenic orbitals by including the lattice structure of silicon via the Bloch structure. We used this theory to determine that the binding energy of the

D− centre is 0.9 meV, a result which is in reasonable agreement with experiment. Improvements to this value are expected if the anisotropic nature of the conduction band minima of silicon is accounted for. Including corrections 40 The read-out state for Si:P based architectures beyond the one-valley effective mass theory such as the valley-orbit coupling are also expected to provide further corrections. In particular, the valley-orbit coupling lifts the six-fold degeneracy present in the one-valley effective mass calculation, bringing theory into agreement with experimental observation. In the following chapter, we turn our attention to modelling of the charge transfer process required for the Kane read-out protocol. We propose that replacing the large dc bias field with an ac field in resonance with the transition 0 0 + D D D D− results in considerably lower bias field strengths. This means → charge transfer can occur without the risk of ionising the important D− state. Chapter 4

Optically induced charge transfer for donor spin read-out

Solid-state quantum computer (SSQC) architectures are of particular interest for the development of a working quantum computer, as any such architecture could leverage the power of the semiconductor industry for scalability. The Kane architecture [1] is one contender for an SSQC. Here the qubits are phosphorus donors in isotopically pure 28Si with I = 0. The logical state of the qubit is encoded on the nuclear spin of the phosphorus donor which has nuclear spin I = 1/2. The advantage of encoding the qubit in this way is that these Si:P systems are known to exhibit long relaxation times [68, 69], meaning the nuclear spin is highly robust to decoherence. On the other hand, weak coupling to the environment (and hence a measurement device) renders measurement of the spin qubit extremely difficult. All operations are dependent on electron mediated interactions with the nucleus via the hyperfine interaction. Donor electron spin based proposals for an SSQC [2, 31, 36, 38] are also of interest. Electron spin qubits may offer enhanced simplicity for qubit control, read-out and gate operation speed (for exchange based proposals) over their nuclear spin counterpart. Recent measurements [39] of the electron spin coherence time, T2, for phosphorus donors in Si, give T2 > 60 ms at 7 K. Despite the electron spin coherence time being shorter than the coherence time for nuclear spins, relatively faster gate operations mean that of order 106 operations are 42 Optically induced charge transfer for donor spin read-out possible within the coherence time [2]. Measurement and intialisation are essential requirements of quantum computation. Experimental detection of a single electron spin in solid-state systems has only recently been reported. Detections of a single electron spin have now been made in a quantum dot system formed in the two-dimensional electron gas (2DEG) of a GaAs/AlGaAs heterostructure [44], via magnetic resonance force microscopy [40] in SiO2 and optically in nitrogen-vacancy (NV) defect centres in diamond [45]. Proposals exist for single spin read-out within a number of different qubit systems, ranging from spin-to-charge transduction techniques involving: adiabatic transfer [1], spin valves [32], gated resonant transfer [3], asymmetric confining potentials [46] and spin-dependent charge fluctuations [47]. Other novel methods include ancilla assisted read-out [41, 48] and optical read-out [70, 71]. In all spin-to-charge transduction processes, measuring the state of the nuclear spin qubit is turned into the task of measuring a spin-dependent electron charge transfer event: for example in Kane, the 0 0 + process whereby a two neutral donor system, D D becomes D D−. Indi- rect measurement of the spin state of the qubit in this way is possible due to the relative ease of coupling to a charge measurement device, e.g. an SET or quantum point contact (QPC).

The resultant doubly occupied state, D−, is very shallow, with a binding energy of 1.7 meV [49, 50] and hence may be easily ionised. Read-out via ∼ the adiabatic Kane protocol requires electric fields which may be too large to preserve the D− state long enough for detection by the SET. In particular, the maximum dc field strength tolerated for a “safe” D− dwell time of T 10 µs D− ≈ has been estimated to be an order of magnitude smaller than the field required for adiabatic charge transfer (see section 4.1) [3]. In this chapter, we describe a means by which to perform the resonant spin- dependent charge transfer proposed in Ref. [3] utilising a far infrared (FIR) 0 0 + laser at resonance with the transition D D D D−. This FIR laser induced → resonant transfer is related to that implied by Larionov et al. [72] in generating qubit gates in a D− based quantum computer proposal. In section 4.1 we review the work presented in Ref. [3] on gated resonant spin transfer and expand on some of the results by treating the three-state system as an effective 4.1 Gated resonant spin transfer 43 two-state system using the Rabi solution. An optically driven version of the gated resonant transfer using an FIR laser is outlined in section 4.2. This has the advantage of separating the terahertz source and gating circuitry from the device, hence reducing high speed on-chip switching. The fidelity of the charge transfer process is determined in section 4.3 as a function of donor separation, detuning from the off-resonant state and laser intensity. In section 4.4 we discuss how in an electron spin architecture, optically driven spin-to-charge transduction would be a means by which to perform singlet-triplet read-out, which is sufficient for cluster state quantum computation [73]. The material in this chapter has been published in Ref. [4].

4.1 Gated resonant spin transfer

The short lifetime of the D− state in the presence of a dc electric field motivates the proposal for gated resonant spin transfer [3]. The adiabatic charge transfer 0 0 + proposed by Kane relies on a slowly varying dc field to effect the D D D D− → transition. The shallow phosphorus donors are 45.5 meV below the conduction band edge and the doubly occupied D− state is bound by only 1.7 meV. The problem with the existing adiabatic charge transfer scheme is that application of the static dc field is likely to ionise the D− state to the conduction band. By using additional suitably placed gates it may be possible to protect the system from ionisation during the read-out process, however, this would require the fabrication and control of complex arrays of gate structures. The measurement time for small induced charge levels (∆q < 0.05qe) using single-shot read- out with an rf-SET operating near the quantum limit is of the order T SET ≈ 1 µs [51]. This means that for read-out to be successful, the survival of the

D− state must be longer than T 1 µs. SET ≈ In order to quantify this, the maximum dc field strength Fdc∗ for a “safe” ad D− dwell time of T 10 µs was calculated in Ref. [3]. The dc field (F ) D− ≈ dc required in order to adiabatically transfer the charge between the two donors was also calculated as in the earlier work of Fang et al. [74] and found to be much greater than Fdc∗ for all cases of donor separation tested. Specifically, for ad a donor separation of R = 30 nm, F /F ∗ 11. Essentially this means that dc dc ≈ 44 Optically induced charge transfer for donor spin read-out the read-out proposal based on Kane adiabatic charge transfer is problematic as the D− state is not sufficiently long-lived for high-fidelity SET detection (see Fig. 4.1).

F ad >F T − < T dc dc∗ ⇒ D SET D0 D− D0

ad Figure 4.1: The dc ﬁeld (Fdc ) required to adiabatically transfer an electron between two donors is larger than the ﬁeld (Fdc∗ ) the + D D− system can safely sustain for a D− dwell time long enough for SET detection.

Gated resonant spin transfer was proposed in Ref. [3] as an alternative to the adiabatic charge transfer scheme discussed in the previous chapter. The idea behind gated resonant spin transfer is to replace the adiabatic dc electric ad field (F ) with a small dc electric field F F ∗ for qubit selection and an dc dc ≪ dc ac electric field with amplitude F F ∗ at resonance with the energy gap ac ≪ dc 0 ∆E(Fdc) of the two states, D and D−. A schematic of the device can be seen in Fig. 4.2. 0 0 + We begin by studying the dynamics of the D D D D−transition driven → by gate fields only. The Hamiltonian for the system is

(t)= + (t)+ (t), (4.1) H H0 Hdc Hac where

~2 ~2 2 2 2 2 2 2 2 qe qe qe qe qe 0 = 1 2 k k k k + k (4.2) H −2m∗ ∇ − 2m∗ ∇ − r1 − r2 − r1′ − r2′ r12 and the dc and ac terms (applied along the donor separation axis, deﬁned to 4.1 Gated resonant spin transfer 45

Read-out SET Control gates (A, J, A)

SiO2 Si

SET donor - |Li Qubit donor - |Ri ac ﬁeld Figure 4.2: Schematic of the device for the gated resonant spin- dependent charge transfer of a single electron. be the x-direction) are given by

(t) = q (x + x R)F (t), (4.3) Hdc e 1 2 − dc (t) = q (x + x R)F (t) sin ωt. (4.4) Hac e 1 2 − ac

is the ungated two-donor Hamiltonian in the effective mass approximation H0 relevant for the Si system (m∗ = 0.2m , ε = 11.9ε ). Here, r′ = r R , e 0 i | i − | where the ri give the electron positions relative to a phosphorus donor at the origin and R =(R, 0, 0), specifies the double donor separation. The Coulombic constant relevant for the Si system is k = 1/4πε. The Fdc(t) and Fac(t) are square-pulses with time-dependence to signify that the turn-on times of these pulses will in general differ from each other. Energies are scaled to the D0 centre ground state energy in Si (45.5 meV). In this work we assume that the electric fields used to generate gated resonant spin transfer are uniform as described by (t) and (t). A more Hdc Hac complete analysis of the problem for specific gate structures would account for the non-uniformity of these fields, the effects due to mirror charges in the gates and the presence of charge traps at the Si-SiO2 interface. The inclusion of a non-uniform field would alter the details of time scale and bias required for charge transfer, yet should not be too different from the analysis carried out here. Including these effects presents an opportunity for future work. 46 Optically induced charge transfer for donor spin read-out

At an operating temperature of 100 mK the electrons will only occupy the 1s orbitals. The starting state describes the D0D0 system at B = 0 with wave function

r1 r′ r′ r2 ψLR = NLR(e− − 2 + e− 1− ). (4.5)

In this notation, L and R refer to the position of the electrons with respect to the left and right donors of a two donor system. As discussed in chapter 3, + the D D− system is well described by the Chandrasekhar wave function

αr1 βr2 βr1 αr2 ψLL = NLL(e− − + e− − )(1 + λr12), (4.6)

αr′ βr′ βr′ αr′ ψRR = NRR(e− 1− 2 + e− 1− 2 )(1 + λr12), (4.7) however, we note a more complete treatment would include lattice effects as in chapter 3. The α, β and λ are evaluated variationally [64]. All total wave functions (ΨLL,ΨLR,ΨRR) are correctly anti-symmetrised when the spin component is considered, χ = χ = [ ]/√2. This spin singlet state is as |↓↑i − |↑↓i required for the charge transfer stage of read-out. To effect a transition from LR to the arbitrarily chosen doubly occupied | i LL state, the gated fields described by (t) and (t) are pulsed and the | i Hdc Hac transition from LR LL is studied numerically. | i→| i For the parameters considered, we find the lowest two states are effectively decoupled from the third state (see Fig. 4.3), and we can therefore treat the system as an effective two-state system. The Rabi solution [75] to this problem gives the excited state population PLL (with ground state population PLR = 1 P ) − LL V 2 V 2 t P = | | sin2 ∆2 + | | , (4.8) LL ~2∆2 + V 2 ~2 2 | | r ! where the dipole matrix element is

V = q LL (x + x R) LR F . (4.9) eh | 1 2 − | i ac

∆= ω ω is the detuning of the ac ﬁeld (with frequency ω) with respect to the − 0 transition (with frequency ω0) and R is the donor separation. For the case of resonant excitation, i.e., ω = ω0, the populations given by Eq. 4.8 as a function 4.1 Gated resonant spin transfer 47

-40 |LLi |RRi

-60

-80

|LRi |LRi |LRi -100 Energy (meV)

-120 |RRi |LLi

-140 -2000 -1000 0 1000 2000

Fdc (kV/m)

Figure 4.3: Energy level diagram for the three-state system as a function of dc electric ﬁeld strength Fdc, for a donor separation of R = 30 nm. For even small values of Fdc, the state RR , is well decoupled from the other two states, therefore allowin| g usi to treat the system as an eﬀective two-state system.

of time are plotted in Fig. 4.4 (for Fdc = 22.2 kV/m, Fac = 44.5 kV/m), and match well with the numerical solutions obtained from the three-state calculation. Complete transfer is achieved by applying a π-pulse, and the time for this is π~ 1 π~ M t = = = , (4.10) π V F q LL (x + x R) LR F | | ac eh | 1 2 − | i ac where π~ M = . (4.11) q LL (x + x R) LR eh | 1 2 − | i The transition time is inversely proportional to the ﬁeld strength Fac. To a very good approximation, the time given by the analytic Rabi solution is equivalent to the numerical solution which results from the resonant transfer calculations. Fig. 4.4 shows this and an inset close-up of a selected region of the transition. The numerical simulation includes all three states, and a small oﬀ-resonant coupling to the third state, which is responsible for the oscillations 48 Optically induced charge transfer for donor spin read-out

1 numerical solution Rabi solution

0.8 PLR

0.6 0.6 Fdc = 22.2 kV/m 0.5 Fac = 44.5 kV/m 0.4

Probability 0.4 45 50 55 0.2 PLL

0 20 40 60 80 100 t (ps)

Figure 4.4: Ground and excited state populations as a function of time showing spin-dependent charge transfer. In the presence of a small dc electric ﬁeld, Fdc, the Rabi solution closely matches the numerical solution obtained from the three-state calculation. The inset shows a close-up indicating the very good agreement between the two solutions. Noticeable are the high frequency oscillations due to a small coupling to the oﬀ-resonant state RR . | i

visible in Fig. 4.4. We comment on the ﬁdelity of this transfer as a result of these oscillations in section 4.3.

0 0 + We also examined the D D D D− transition probability as a function of → detuning, ∆, to observe the response to varying ac ﬁeld strength. The results are given in Fig. 4.5, showing the characteristic sinc function dependence and narrowing of the central peak with decreasing ﬁeld strength. This well known result serves as a means by which to avoid single donor level transitions.

Spin-to-charge transduction by gated resonant transfer is a promising technique since the dc selection ﬁeld is very low compared to the critical ﬁeld sustainable by the doubly occupied state before electron loss occurs. 4.2 Resonant FIR laser transfer 49

1 2 0.5 Wcm− 2 4.0 Wcm− 2 0.8 16.0 Wcm−

0.6

0.4 Probability

0.2

0 -60 -40 -20 0 20 40 60 ∆ (GHz)

Figure 4.5: Charge transfer probability as a function of FIR laser detuning for a range of intensities, showing the characteristic sinc function dependence. The narrowing of the central peak with decreasing laser intensity serves as a means by which to avoid unwanted single donor transitions.

4.2 Resonant FIR laser transfer

An optically driven version of gated resonant transfer is preferable, given that the separation of the terahertz source and gating circuitry from the rest of the chip, reduces noise from high speed on-chip switching and aids transmission of the signal to the device. An FIR laser operating at wavelengths of 34 µm ∼ could provide the required radiation ﬁeld. This is on the outer limits of current technology however various candidates exist, including methanol lasers and their deuterated derivatives CD3OH [76], and possibly synchrotrons and free- electron lasers. Promising FIR technology also utilises the Si:P system as the 0 0 + active medium for lasing [77]. The observation of the D D D D− transition → in optical studies of bulk-doped silicon [52] suggests that this transition may be observed resonantly in this optical implementation. To analyse the optical version, we rewrite the Hamiltonian of Eq. 4.1

(t)= + (t)+ (t), (4.12) H H0 Hdc Hopt 50 Optically induced charge transfer for donor spin read-out where

(t) = q (x + x R) F (t) , (4.13) Hdc e 1 2 − dc iq ~ (t) = e (A ∇ + A ∇ ) , (4.14) Hopt − m · 1 · 2 i(k r ωt) i(k r ωt) A (r, t) = A0 (ω) ǫˆ e · − + e− · − . (4.15) Using the dipole approximation, which is valid for the wavelength of the FIR ﬁeld required here (since k R 1), the Hamiltonian matrix elements for the · ≪ perturbation reduce to equivalent form

4q2ω2 A (ω) 2 cos2 ωt Ψ ǫˆ (r + r ) Ψ 2 = e 0| 0 | |h LL| · 1 2 | LRi| q2F 2 sin2 ωt Ψ (x + x R) Ψ 2, (4.16) e ac |h LL| 1 2 − | LRi| assuming linear polarisation, ǫˆ = (1, 0, 0). This yields the following relation- ship for the amplitudes of the vector potential and ac ﬁeld at resonance,

2 2 2 Fac M A0(ω0) = 2 = 2 2 , (4.17) | | 4ω0 4ω0tπ and the previous analysis can be applied. Thus, resonant transfer can in principle be achieved via FIR laser excitation. To do this, the frequency of the laser should be set to the energy gap between the LR and LL states. | i | i We simulate the transition using the hydrogenic wave functions described in Eqs. 4.5-4.7.

It is essential that the small dc oﬀset Fdc, which serves the purpose of qubit selection, is smaller than the critical dc ﬁeld strength, Fdc∗ as outlined in section 4.1. Staying below this critical value will ensure that ionisation of the

D− state to the conduction band does not occur. We show the energy levels as a function of the dc ﬁeld strength, Fdc, for a donor separation of R = 30 nm in Fig. 4.3. It is also important to examine the scaling of the energy gap between states LR and LL as a function of the dc oﬀset F . We do this for donor | i | i dc separations of R = 20 nm and R = 30 nm. Fig. 4.6 shows the results against the relevant single donor levels (relative to the ground state). This will ensure 4.2 Resonant FIR laser transfer 51

1s → 2p (39.1) F ∗ (R = 30 nm) ± 38 dc R = 20 nm R = 30 nm 36 Fdc∗ (R = 20 nm) (meV) 1s → 2p0 (34.1) E 34 ∆

32 0 0 + D D → D D−

30 0 50 100 150 200 250 300

Fdc (kV/m)

0 0 + Figure 4.6: Scaling of the energy gap for the D D D D− ( LR LL ) transition as a function of F . Simulation→ results | i→| i dc are shown for donor separations of R = 20 nm and R = 30 nm against the allowed single donor energy levels (relative to the donor ground state).

that Fdc may be chosen to avoid exciting these single donor transitions. We also note that there will be no linear Stark eﬀect for the relevant single donor levels [60, 78] and hence these energy levels will remain unperturbed to ﬁrst order. The results are shown in Fig. 4.6 for the dipole allowed transitions,

1s 2p0 and 1s 2p . Spectroscopic observations in bulk doped silicon [52] → → ± show that the widths of the 1s 2p transitions are considerably less than → 0 0 + 1 meV, and can be neglected compared to the D D D D− power broadened → transition width. We note that keeping the laser intensities low will also reduce the probability of causing oﬀ-resonant transitions as explained in section 4.1.

The qubit selection ﬁeld, F , is chosen to be below F ∗ ( 130 kV/m for dc dc ∼ R = 30 nm) and to avoid single donor transitions. The time scale of the FIR induced transfer is controlled directly by the laser intensity. The required laser intensity is

2 1 2 2 1 M I(ω0)= ε0cω0 A0(ω0) = ε0c 2 . (4.18) 2 | | 8 tπ 52 Optically induced charge transfer for donor spin read-out

For a charge transfer time of order nanoseconds the required laser power is of order a few milliwatts. This is in the regime of fast transfer given that T 1 µs [51]. At the same time the required laser wavelength can be varied SET ≈ by altering the strength of the local dc field, Fdc, allowing some flexibility in the requirement for a 34 µm FIR laser. Restrictions on the value of Fdc (as discussed earlier) are required in order to avoid coupling to single donor transitions or causing electron loss. For donor separations less than R = 30 nm there is larger coupling to the off-resonant state, RR , due to a larger dipole matrix element (see Fig. 4.7(a)). | i An example of the low transfer fidelity of such a transition is seen in Fig. 4.8.

1 (a) 1 (b)

0.8 0.8 PLR PLR

0.6 0.6 2 2 0.01 Wcm− 0.37 Wcm− 0.4 0.4 Probability Probability

PLL PLL 0.2 0.2

0 0

0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 t (ns) t (ns)

Figure 4.7: State probabilities for the resonant FIR laser transfer between donors. Simulation results are shown for donor separations of (a) R = 20 nm and (b) R = 30 nm. The reduction in the magnitude of the high frequency oscillations with increased donor separation, due to a smaller oﬀ-resonant coupling, is noticeable.

Increasing the detuning from this off-resonant state will in principle improve the fidelity, however the local dc field must remain below the critical dc field strength, Fdc∗ , which limits the process. This suggests that donor separations less than 30 nm will not be practical. In Fig. 4.7 we give examples of FIR transfer for separations of R = 20 nm and R = 30 nm. The reduction in the off-resonant coupling with increased donor separation is prominent. 4.3 Charge transfer fidelity 53

4.3 Charge transfer ﬁdelity

Fidelity of charge transfer is dependent both upon donor separation and detuning from the off-resonant state. For a given separation, fidelity may be improved by increasing the detuning which is achieved by increasing the local dc field, Fdc (see Fig. 4.8). This process is of course limited by the critical dc

ﬁeld strength, Fdc∗ and the eﬀect is small.

0 -0.1 (a) -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 ) -0.8 LL P -0.9 − 0 (1

10 -0.2 (b) log -0.4 -0.6 -0.8 -1 -1.2 -1.4 0 50 100 150 200 250 300 t (ps)

Figure 4.8: Fidelity of transfer for an FIR laser intensity of 2 0.37 Wcm− and donor separation of R = 20 nm. The improved ﬁdelity is noticeable with increased detuning. Above we show results for Fdc values of (a) 8.9 kV/m and (b) 22.2 kV/m. Note that the scales on the y-axes are diﬀerent.

Increasing the donor separation in turn reduces the dipole matrix elements to unwanted states which results in smaller oﬀ-resonant oscillations, thereby improving ﬁdelity (see Figs. 4.8 and 4.9).

Maximising ﬁdelity should be achieved by careful selection of Fdc as well 54 Optically induced charge transfer for donor spin read-out

-1

-1.5

) -2 LL P

− -2.5 (1 10

log -3

F = 22.2 kV/m -3.5 dc

-4 1 1.04 1.08 1.12 1.16 1.2 t (ns)

Figure 4.9: Increasing the donor separation to R = 30 nm reduces the off-resonant dipole matrix element, resulting in improved fidelity (as compared to a donor separation of R = 20 nm, seen in Fig. 4.8). Transfer fidelity is shown for an FIR laser intensity of 2 0.37 Wcm− .

as an understanding of the timing window for FIR laser pulsing. Low powered lasers are preferential to avoid heating, minimise unwanted single donor transitions, and increase the timing window over which high-ﬁdelity transfer may occur. Faster transfer is possible, however the timing window over which the FIR laser must be pulsed to achieve high-ﬁdelity transfer is narrow (see Fig 4.10). Such high speed switching is possible using laser activated semiconductor switches [79, 80].

Within the approximations used in this preliminary analysis we ﬁnd greater than 99% transfer ﬁdelity with ample scope for improvement. Read-out need 4 not operate at the 10− error threshold demanded of logic gates, provided that logic gates can be operated at this threshold or better [81]. 4.4 Singlet-triplet read-out for electron spin qubits 55

-0.5

-1

-1.5 ) LL

P -2 −

(1 -2.5 10 log -3 2 65.72 Wcm− -3.5

-4 60 65 70 75 80 85 90 95 100 t (ps)

Figure 4.10: Simulation results for a donor separation of R = 30 nm showing fast, high-ﬁdelity transfer. Transfer on these time scales requires higher powered lasers which may cause heating and unwanted single donor transitions.

4.4 Singlet-triplet read-out for electron spin qubits

Single spin read-out fails for the Si:P electron spin SSQC when the 1s energy levels are split by an externally applied magnetic field, B, as in Fig 4.11. The convention used here labels the state of each site, such that s represents the | ·i doubly occupied spin singlet state formed on the left donor. Read-out fails in this paradigm given that the states , are degenerate and have equal |↓↑i |↑↓i 0 0 + dipole matrix elements for the D D D D− transition. This means the → |↓↑i state cannot be preferentially selected without first lifting the degeneracy. Spectrally resolving the and would provide a physically interesting |↓↑i |↑↓i system to study direct charge transfer and re-initialisation: an inhomogeneous magnetic field provides one such mechanism for this. Abe et al. [82] proposed a SSQC architecture that relies on a dysprosium (Dy) micromagnet to generate a gradient magnetic field in order to selectively access different nuclear spin qubits using a resonant field. These Dy micromagnets can generate field gra- 56 Optically induced charge transfer for donor spin read-out

s | ·i

ωFIR

|↑↑i |↓↑i |↑↓i |↓↓i

Figure 4.11: Electron spin energy levels for the spatially separated and single site states. Using an FIR laser to perform read-out when the states and are degenerate results in a singlet- |↓↑i |↑↓i triplet read-out protocol. This may be useful in cluster state quantum computation.

2 dients of order 20 T/µm [83], resulting in a state separation of 7 10− meV × for R = 30 nm. This yields better than 99.99% transfer fidelity. If instead we apply a site selective hyperfine interaction, A, above one donor and not the other in the Si:P electron spin SSQC, the resultant state separation is only 4 2A 2.4 10− meV. This is not sufficiently resolved to perform single spin ≈ × read-out and re-initialisation and only 50% transfer fidelity is achieved. In a 2-D SSQC architecture with separated read-out and interaction zones, spin resolved read-out and initialisation may be possible. Such separation requires qubit transfer, and examples include the shuttling process of Skinner et al. [42] and adiabatic passage [36, 84]. By performing read-out away from the qubit interaction zone, a Dy micromagnet could be included to provide the required frequency resolution as described above. Having read-out off site means that the field gradient from the Dy micromagnet would in principle only affect read-out qubits. Thus qubits in the interaction zone would not need to be characterised to allow for the gradient. Alternatively, one may use localised magnetic fields (similar to those produced by the current-carrying wire array structures of Lidar et al. [85]). Performing read-out in the Si:P electron spin quantum computer with the states , degenerate (see Fig. 4.11) results in singlet-triplet read-out. |↓↑i |↑↓i 4.5 Summary 57

Such a scheme could be used for cluster state computation [73]. Tuning the FIR laser to the energy diﬀerence between the s and the degenerate , | ·i |↓↑i states results in charge transfer provided the spatially separated states are |↑↓i in an anti-symmetric (singlet) superposition. The absence of charge detection by the SET projects the electrons onto the spin triplet manifold after read-out.

4.5 Summary

Optically driven single spin read-out for the nuclear spin SSQC via use of an FIR laser has been investigated as an alternative to the adiabatic transfer method of Kane [1]. The advantage of this method is that the ﬁelds required are much weaker than the ﬁeld strength which would cause ionisation of the

D− read-out state. High-ﬁdelity transfer was shown to be possible on pico to nanosecond time scales which is fast compared to the time required for high- ﬁdelity single-shot measurement by an rf-SET. We explain how singlet-triplet read-out can be performed in an electron spin paradigm and suggest that it may be used for measurement in cluster state quantum computation. Spectral resolution of the degenerate states in the electron spin SSQC architecture allows direct single spin read-out and re-initialisation. We note that the methods developed are in principle adaptable to any buried donor system. In the following chapter we turn our attention to robust control of two- qubit gate operations. We demonstrate how to optimally construct a robust CNOT gate using composite rotations from the Heisenberg interaction, even when uncertainty exists in the coupling strength between qubits.

Chapter 5

A CNOT gate immune to large fabrication induced variations

The ability to correct errors arising from the construction or operation of any quantum computing architecture is essential for a successful implementation. Without the ability to correct the random and/or systematic errors that arise throughout operation, the implementation of large scale quantum algorithms is hopelessly undermined. In a realistic device the threshold for fault-tolerant 4 quantum computation is likely to be well below 10− , placing severe constraints on the tolerable magnitude of errors due to decoherence or lack of precision in quantum control. The work in this chapter focuses on minimising a particular type of systematic error, namely, uncertainty in the coupling strength of two- qubit devices as a result of imperfect fabrication, which causes systematic under- or over-rotation control errors. We use recently developed two-qubit composite control sequences to correct for this uncertainty in the strength of the electron spin exchange interaction in Si:P based architectures [1, 36]. The exchange interaction is also common to other spin qubit systems [31, 32, 33]. Our results also apply more generally and could be used to correct this type of systematic error in a range of solid-state systems. The strength of the exchange interaction coupling between donors in silicon based solid-state architectures is known to be highly sensitive to donor placement. The cause of this is the inter-valley interference between the six 60 A CNOT gate immune to large fabrication induced variations degenerate conduction band minima of silicon, resulting in oscillations of the exchange coupling strength [65, 86, 87, 88]. Exact positioning of donors to better than 2-3 sites is difficult [89] and therefore we expect significant uncertainty in the un-biased strength of the coupling between donors. The uncertainty in our knowledge of the coupling, leads to error in gate operation, however, we will show that systematic errors of this kind are correctable to high precision using composite rotations. Recently two-qubit composite rotations have been considered for systems with uncertainty in their coupling strength [90, 91]. Experimental applications already exist in a variety of quantum systems demonstrating the usefulness of composite rotations for ensuring robust operations [92, 93, 94, 95, 96, 97]. In section 5.1, we follow the method for creating a robust CNOT gate developed in Ref. [91] and quantitatively study the performance of the robust CNOT gate using simulated exchange oscillation data in section 5.2. We specifically consider the global Si:P electron spin control case where the interaction is of Heisenberg type and gate times are in the (10-100 ns) regime. This O technique is readily generalisable to any two-qubit Hamiltonian, and for a full treatment, the reader is directed to Ref. [91]. Misplacement of donors by only one lattice site can lead to large variations in the exchange coupling strength, even in Si:P systems with voltage bias applied to top gates [87], meaning a single application of the composite rotations may not be enough to guarantee a high fidelity CNOT gate. Concatenating the pulse by feeding it back into itself can help to achieve correction to a higher level, however, performing multiple concatenations costs a large increase in time. In certain cases using composite rotations alone will not improve the fidelity of the operation above an uncorrected CNOT gate, as the composite rotations are designed to work within a specific uncertainty range. In section 5.3 we show that in unison with Hamiltonian characterisation [98, 99, 100, 101, 102], the process of experimentally determining a Hamil- tonian, a single application of the composite rotations guarantees a high fidelity CNOT operation with an error rate below the fault-tolerant error threshold. Operating the CNOT gate this way helps remove the need for concatenation, and strikes a balance between fully characterising the system and using com- 5.1 Constructing robust gates using composite rotations 61 posite rotations to construct robust operations. The material in this chapter has been published in Ref. [5].

5.1 Constructing robust gates using composite rotations

Composite rotations have been widely used in NMR experiments to correct for pulse length errors and off-resonance effects [103, 104]. In the case of pulse length errors, a deviation of the field strength from its nominal value leads to systematic under- or over-rotations. Although originally designed for applications involving single spin quantum systems, composite rotations may be extended to two-spin operations. In the context of quantum computation, only a certain class of composite rotations, sometimes referred to as fully compensating pulses are applicable, as they work on any initial state. Using these fully compensating pulses, the application of composite rotations for constructing robust two-qubit gates against pulse length error has already been found for an Ising Hamiltonian [90], and a general two-qubit Hamiltonian [91].

In Ref. [91], it was noted that for a general two-qubit Hamiltonian expanded in the Pauli basis, H = J σ σ , (5.1) ij i ⊗ j i,j= I,X,Y,Z {X } any interaction term can be eﬀectively extracted using a technique called term isolation [105]. The isolation of a given term will in general not be exact but can be made arbitrarily accurate. This result is particularly useful and can be used to isolate the Ising coupling term, JZZ , such that we can construct a CNOT gate from this interaction as in Fig. 5.1. In the case of the Heisenberg interaction with isotropic couplings,

H = J(σ σ + σ σ + σ σ ), (5.2) H X ⊗ X Y ⊗ Y Z ⊗ Z 62 A CNOT gate immune to large fabrication induced variations

Z π − 2

π UI ` 2 ´

HHZ π − 2

Figure 5.1: Circuit diagram for a CNOT gate constructed from an π Ising interaction, where H is a Hadamard gate, Z π = exp i σZ − 2 4 and U π = exp i π σ σ . I 2 − 4 Z ⊗ Z

the isolation of the JZZ term is exact,

exp ( iJ tσ σ )= (Z I) exp ( iH t) − ZZ Z ⊗ Z − π ⊗ − H (Z I) exp ( iH t) , (5.3) × π ⊗ − H where for single qubit gates Za is a rotation about the σZ axis by an angle a, and similarly for other operators, JZZ =2J, and the global phase factor is included.

We now consider constructing a robust CNOT gate using composite rotations, whereby we replace the interaction term with one created using composite rotations. Doing this compensates for any uncertainty in our knowledge of the exchange interaction coupling strength, J. In Fig. 5.2 the entire process of constructing a robust CNOT gate from composite rotations is demonstrated schematically.

In an ideal system with a perfectly characterised coupling strength, the evolution operator generated by the Ising interaction is

θ θ U (θ) = exp i σ σ . (5.4) 0 ≡ ideal − 2 Z ⊗ Z Here, θ is a two-qubit rotation by an angle θ about the σ σ axis. In 0 Z ⊗ Z general, θa is a two-qubit rotation by an angle θ around an axis tilted from the 5.1 Constructing robust gates using composite rotations 63

WITHOUT CONCATENATION WITH CONCATENATION

Start with YESDesired NO concatenation level achieved? H = J(σX σX + σY σY + σZ σZ ) H ⊗ ⊗ ⊗

Isolate Ising component Correct σZ σZ operations to higher level ⊗ (i+1) (i) (i) (i) (i) (i) exp ( iJZZ tσZ σZ )= (Zπ I) exp ( iHHt) − ⊗ − ⊗ − θ0 = (θ/2)0 ∗ πφ ∗2π3φ∗πφ ∗ (θ/2)0 ∗ (Zπ I) exp ( iH t) × ⊗ − H for i = 1, 2,...

Re-isolate Ising component Construct composite rotations (i) (i) (i) Nr θ ∗ = (Xπ Xπ) (I Zπ)[θ/(2Nr)] (I Zπ)[θ/(2Nr)] 0 h ⊗ ⊗ 0 ⊗ 0 i (1) (0) (0) (0) (0) (0) for i = 1, 2,... θ0 = (θ/2)0 πφ 2π3φ πφ (θ/2)0 where Nr is the level of re-isolation

Construct robust CNOT gate

CNOT=(I H)(Z π Z π ) π (i)(I H) ⊗ − 2 ⊗ − 2 2 0 ⊗ for i = 1, 2,...

Figure 5.2: Procedural ﬂowchart for constructing a robust CNOT gate using composite rotations, and concatenating to higher implementation levels.

σ σ axis towards the σ σ axis by an angle a, Z ⊗ Z Z ⊗ X θ θ = exp i (σ σ cos a + σ σ sin a) . (5.5) a − 2 Z ⊗ Z Z ⊗ X This two-qubit rotation is achievable via,

θa =(I Ya) θ0 (I Y a) . (5.6) ⊗ ⊗ −

We make the assumption that all single qubit unitaries are error free, but note that single qubit operations may also be made robust using existing techniques developed in the context of NMR.

In reality a fractional error, ∆, in the two-qubit operation will be present due to the uncertainty in our knowledge of the actual coupling strength, JZZ,

J ∆= ZZ 1. (5.7) JP − 64 A CNOT gate immune to large fabrication induced variations

Here, JP is our prediction of the Ising coupling strength based on the targeted donor separation. Therefore the actual rotation performed will be

θ θ(0) U (θ) = exp i (1+∆)σ σ . (5.8) 0 ≡ − 2 Z ⊗ Z

(b) The superscript of θa in the above equation indicates the implementation level of the actual (non-ideal) rotation, with “(0)” being an uncorrected implementation and higher levels signifying subsequent corrections from composite rotations. The implementation level should not be confused with concatenation level, (e.g., 2nd implementation level is the 1st concatenation level). It has been previously noted that single qubit composite rotations can be extended to two-qubit composite rotations for use in quantum computation [90, 91] using fully compensating pulses. A class of these composite rotations known as BB1 [104, 106] is particularly useful for applications involving quantum (0) computation [107]. Replacing the pulse θ0 with the symmetrised BB1 class composite pulse (1) (0) (0) (0) (0) (0) θ0 =(θ/2)0 πφ 2π3φ πφ (θ/2)0 , (5.9) where φ = arccos( θ/4π), will result in a higher fidelity operation. The fidelity − of an operation is defined as

Tr U † (θ) Uideal (θ) = . (5.10) F Tr U † (θ) Uideal (θ) ideal h i We may re-isolate the Ising component JZZ again to arbitrary accuracy as in Fig. 5.2. The re-isolated Ising component can then be used to correct to even higher order by passing this pulse back into each of the constituents of Eq. 5.9, (see Fig. 5.2). In principle there is no limit to how often this concatenation can be done, however, the increase in gate time means that in practice this process will be limited by the decoherence time of the system in which the CNOT gate is being implemented. In Fig. 5.3 the performance of the uncorrected CNOT gate is compared to the robust gate for various implementation levels, as originally calculated in Ref. [91]. Notice each subsequent implementation level performs better over a larger range of the fractional error, ∆. For example, 5.1 Constructing robust gates using composite rotations 65

-0.5 uncorrected 1st Level nd -1 2 Level 3rd Level ) -1.5 − F -2 (1 10 -2.5 log

-3

-3.5

-4 -1 -0.5 0 0.5 1 ∆ (fractional error)

Figure 5.3: CNOT error, (1 ), as a function of the fractional error in our knowledge of the−F coupling strength, ∆, for various implementation levels. These composite rotations provide improvement over an uncorrected implementation for ∆ ( 1, 1). ∈ −

the ﬁdelity of the uncorrected CNOT gate is

π2∆2 1 , (5.11) F ≈ − 32 whilst after the ﬁrst level of the composite control sequence the ﬁdelity is

63π6∆6 1 . (5.12) F ≈ − 65536

We now apply the robust CNOT gate to the Si:P architecture with large fabrication induced variations (and hence uncertainty) in the exchange interaction strength. 66 A CNOT gate immune to large fabrication induced variations

5.2 Correcting for an unknown exchange interaction strength

Systematic errors arising from imperfections in the fabrication process are correctable. In Kane type architectures [1, 36] where phosphorus donors are implanted into an isotopically pure 28Si matrix, two fabrication processes are being pursued concurrently [108]. The top down approach uses ion beam implantation of phosphorus ions incident on the silicon substrate. Precise placement of phosphorus donors is limited in this approach due to scattering off the silicon atoms, in a process known as straggling. State of the art top down fabrication results in placement uncertainties of (10 nm) [109]. The bottom up O approach offers atomically precise fabrication using a phosphine gas. The gas is applied to a hydrogen terminated silicon substrate, where scanning tunneling microscopy has removed individual hydrogen atoms from the hydrogen mono- layer at the desired implantation sites. Once the phosphorus is integrated into the substrate, the mono-layer is removed and overgrown with silicon. Small deviations from target implantation by of (1 nm) (approx. 2-3 sites) can still O occur during the annealing process [89]. The exchange coupling J of the Heisenberg Hamiltonian (see Eq. 5.2), is highly sensitive to donor electron wave function overlap. This means that even small deviations from the targeted implantation sites can lead to large variations in the exchange coupling between donors [65, 86]. Calculated variations in the strength of J for small deviations in all directions from the targeted donor separation (in the [100] direction) in an unbiased, J (V = 0), system are shown in Fig. 5.4. This calculation was performed using the Heitler-London formalism, where the wave functions for the phosphorus donors in silicon were expressed in Kohn-Luttinger effective mass form, with Bloch states explicitly computed using the pseudopotential fit to the band structure. Details can be found in Ref. [65]. These Kohn-Luttinger calculations are representative only. It is now known through a more sophisticated band minima basis analysis including core correction and strain, that the extreme oscillations arising in the Kohn-Luttinger treatment are strongly tempered [110]. Small values of the coupling are also enhanced, with further improvements expected in the 5.2 Correcting for an unknown exchange interaction strength 67 biased, J (V = 0), system [65, 87]. Importantly, this type of systematic error 6 is correctable using the composite rotations described above. The case J =0 can never be corrected, however, as Fig. 5.4 shows there are no instances of J exactly zero.

10 )

0 1 J/J 0.1

0.01

Exchange coupling ( 0.001

0.0001 0 0.5 1 1.5 2 2.5 |δ| (nm)

Figure 5.4: Exchange couplings in an unbiased, J (V = 0), system for donors at fcc lattice sites misplaced by a distance δ in all directions from the target separation of 20.6 nm (in the [100] direction). The exchange coupling strengths are given as a fraction of the target coupling strength, J 0.1 µeV. 0 ≈ In an uncharacterised system we assume that the exchange interaction strength is J0 and will be determined by the target donor separation and bias on the control gates. Fabrication induced donor misplacement will cause the true exchange interaction strength, J, to be quite diﬀerent from J0. The fractional error in our knowledge of the coupling strength is

J ∆0 = 1. (5.13) J0 −

These composite rotations will only provide an improvement over an uncorrected implementation for ∆ < 1. For ∆ > 1 these composite rotations | 0| | 0| are actually outperformed by the uncorrected implementation, so if J > 2J0 then the composite rotations provide a less robust operation. We address 68 A CNOT gate immune to large fabrication induced variations this point in section 5.3, where we show how one may always remain within the correctable range of the composite rotations using a systematic two-qubit interaction characterisation procedure.

Implementing the gate based on the target coupling strength J0, the ﬁ- delity of the resulting CNOT operation will be determined by the size of the fractional error ∆0 in the actual coupling strength. As an example, in Fig. 5.5 we demonstrate the resulting CNOT ﬁdelity for a number of donor separations in the [100] direction when the target separation is 20.6 nm. The results show

Target site uncorrected 1 1st Level 2nd Level 0.95 3rd Level

0.9

F Lattice sites 0.85

0.8

0.75

0.7 20 22 24 26 28 30 32 Donor separation (nm)

Figure 5.5: CNOT fidelity as a function of donor separation in the [100] direction for various implementation levels. The resulting fidelities are determined based on a target donor separation of 20.6 nm. Note that interpolating curves between lattice sites indicate donor separation scenarios for a given implementation, and vertical dotted lines guide the eye between implementations. that using composite rotations improves the fidelity of operation for the CNOT gate. For example, if the actual separation is 21.7 nm, one application of the composite pulsing scheme improves the fidelity from 0.93 to 0.99, whilst ∼ ∼ a second application brings the fidelity above 0.9999. The successive improvements due to the various levels of pulse concatenation do however come at the expense of operation time. We examine this issue in the following sections. 5.2 Correcting for an unknown exchange interaction strength 69

5.2.1 Gate count

The robust CNOT gate outperforms the uncorrected CNOT gate given an error in the targeted coupling strength, J , for ∆ < 1. Each level of concatenation 0 | 0| provides further improvement, however the cost of this improvement is an exponential increase in the total number of gates required. An unavoidable consequence of this is an increase in the time required to perform these robust operations. To be of use for quantum computation we need to be able to perform many precise operations within the decoherence time of the system. We show how to minimise the time taken to perform a robust CNOT gate using Hamiltonian characterisation in section 5.3. Below, we consider the actual time costs of concatenated composite pulse correction. An uncorrected CNOT gate requires only 6 single qubit gates and 2 two- qubit gates. In comparison, a raw gate count for the number of single qubit gates required in constructing the robust Ising interaction for the CNOT gate yields

n1 = 16,

ni = 10Nr(ni 1 +2)+6, i =2, 3,... , (5.14) −

th where ni is the number of single qubit gates required for the i implementation level, and Nr, which we assume to be constant, quantiﬁes how much we re- isolate the Ising term for pulse concatenation. Constructing a robust CNOT gate requires an additional 4 single qubit gates, such that the total number of 1q single qubit gates required, ni , is

1q ni = ni +4, i =1, 2,... . (5.15)

The total number of two-qubit gates needed in the robust CNOT construction is 2q i i 1 ni = 10 Nr− , i =1, 2,... , (5.16) again assuming the same Nr for each level of concatenation. We may be able to reduce the total number of single qubit operations by compounding gates 70 A CNOT gate immune to large fabrication induced variations however this is not possible for the two-qubit operations. The viability of using multiple concatenation for constructing robust two-qubit gates lies in tenuous balance between the ability to perform the large number of operations required quickly, and adequate pulse timing control over the small two-qubit rotations which arise from the re-isolating of the Ising component. The strength of the exchange coupling of our system will determine whether these conditions can be satisﬁed.

5.2.2 Gate time

Each level of concatenation increases the time taken for the robust CNOT operation signiﬁcantly. In a working quantum computer this may be problematic as the decoherence time of the system sets an upper limit on how long operations may take. For phosphorus donors in Si the coherence time, T2, of donor electron spins has been measured to be T2 > 60 ms at 7 K [39]. We calculate the total time taken for the robust CNOT gate for various implementation levels based on gate times using global control methods [2]. The results for this appear in Table 5.1. As in Ref. [91], we assume that single qubit rotations by an angle π take 40 ns to perform as does the Hadamard gate. We also assume that two-qubit rotations by π/4 take approximately 2 ns if the coupling strength is given by J 0.1 µeV, taken from the calculated unbiased 0 ≈ exchange data [65]. Actual gate time will decrease under the application of a J gate bias [65, 87], however, we assume a worst case scenario here.

Table 5.1: CNOT gate times for various pulse implementation levels in the electron spin solid-state quantum computing architecture.

implementation gate times (ns) level single qubit two-qubit total 0 180 4 184 1 716 35 751 2 53257 2544 55801 5.3 The role of two-qubit Hamiltonian characterisation 71

As Table 5.1 demonstrates, operation time grows appreciably with concatenation. Furthermore, Fig 5.5 shows that the success of the robust CNOT gate is dependent on how accurately we can estimate the exchange coupling strength based on expectations of the fabrication process alone. In such an uncharacterised system we have shown that a sensible choice can be made based upon the target separation, yielding J0. Large variations in the exchange interaction strength due to donor misplacement, and the additional time cost for multiple concatenation means composite rotations alone can not always guarantee a feasible, robust CNOT gate. However, we will now show that composite pulses at the lowest level coupled with a systematic two-qubit interaction characterisation procedure allows for precise CNOT gate construction.

5.3 The role of two-qubit Hamiltonian characterisation

Using a combination of system indentification and composite rotations, we may always construct a high fidelity robust CNOT gate. Whilst many methods of system identification exist, we choose the procedure of Hamiltonian characterisation pioneered by Schirmer et al. [98] and subsequently extended by others [99, 100, 101, 102] because it provides direct knowledge of the Hamiltonian (which we require) in an efficient manner. This approach strikes a balance between the need for multiple concatenation and precision Hamiltonian characterisation, and may be particularly useful for systems whose Hamiltonian parameters require re-characterisation over time due to drift. Recent work shows how characterisation of a two-qubit Hamiltonian can be achieved via entanglement mapping of the squared concurrence relation [99, 100]. The identification of the Hamiltonian coefficients amounts to determining the oscillation frequency of this entanglement function for different input states. The only requirements are an accurately characterised Hadamard gate and measurement on both qubits. An important result from the work in 72 A CNOT gate immune to large fabrication induced variations

Ref. [99], is the fractional uncertainty in a frequency determination

δf 4 , (5.17) f ≥ Nt√Ne where, Nt is the number of discrete time points at which Ne projective measurements are made. An equivalent result can also be found in the earlier work of Huelga et al. in the context of Ramsey spectroscopy [111]. To accurately determine the frequency, the time over which the system is observed, tob, should be maximised, however this process is limited by the decoherence time of the system. An accurate frequency determination is still possible in the presence of decoherence by allowing tob to be relatively large and performing two measurements at Nt time points. The uncertainty in the frequency can then be reduced by evolving the system for a suitably long time before measuring at two final time points. This process is repeated Ne times to estimate the phase of the oscillation. The total number of required measurements is then N = 2(Nt + Ne). Characterising the system in this way results in the scaling of Eq. 5.17. To characterise the Heisenberg Hamiltonian with isotropic couplings requires determining the oscillation frequency of three different input states, meaning N = 6(Nt + Ne) total measurements are needed. The fractional uncertainty in the characterised exchange coupling, Jc, as a function of N for a given Nt is δJ δf 4√6 c . (5.18) Jc ≡ f ≥ N √N 6N t − t To illustrate the effect of composite rotations we consider a modest amount of characterisation by choosing Nt = 10. Increasing the number of time points results in higher precision characterisation.

In an uncharacterised system we assumed the coupling between donors, J0, to be determined by the target donor separation. Donor misplacement as a result of fabrication uncertainties lead to variations in the coupling strength,

J, from the target J0. We have seen how the robust CNOT gate for an uncharacterised system performs in Fig. 5.5. We now consider the performance of a robust CNOT gate in a characterised system. 5.3 The role of two-qubit Hamiltonian characterisation 73

Characterisation of the Hamiltonian can be performed to any level of precision at the expense of extra measurements, with the uncertainty given by Eq. 5.18. In a characterised system, the estimated coupling strength is set to the characterised coupling strength, J (with uncertainty bounds δJ ), rather c ± c than J0. The fractional error in this case is

J ∆c = 1, (5.19) Jc − where in general the characterised coupling strength, Jc, will be much closer to the true value of J, than the target value, J0 is to J. This guarantees that we remain well within the correctable bounds of composite rotations, hence ensuring a high fidelity operation with fewer levels of concatenation, independent of donor misplacement direction. Given that the total gate time increases so sharply with increased concatenation, operating with a single application of the composite rotations is preferential. For a one site deviation from the target separation, we show the resulting CNOT fidelity as a function of pulse implementation in a system characterised to the 10% level (δJc/Jc = 0.1) in Fig. 5.6. Characterisation to this level would require at least 156 measurements assuming the previous parameters. We take J 0.9J to be the characterised value of the exchange c ≈ coupling strength, as it corresponds to an extremal bound value. The results in Fig. 5.6 demonstrate that it is possible to construct a very high fidelity CNOT gate using one level of robust pulsing, provided a suitable amount of characterisation is first performed. The total number of characterisation measurements needed to achieve a given fidelity can also be determined as a function of the implementation level. These results appear in Fig. 5.7. In reality the fidelity may be substantially higher than the results of Fig. 5.7 indicate, as they provide a lower bound for the corresponding number of measurements. These results show the clear benefit in using a single level of composite rotations and characterisation to construct a robust CNOT gate. The improvements expected beyond this do not seem to warrant concatenation. Any quantum computation proposal requires that many operations be per- 74 A CNOT gate immune to large fabrication induced variations

0.2

Target separation (J0) 0.16 0

eV) -2 ) µ -4 −F 0.12 -6 (1

10 -8 Actual characterised separation (J) log -10 0.08 uncharacterised -12 δJc 0 1 2 Jc ≈ 0.9J Implementation level

Exchange strength ( 0.04

0 20 22 24 26 28 30 Donor separation (nm)

Figure 5.6: Exchange interaction strength as a function of donor separation along the [100] direction, showing a large variation in the coupling strength with donor misplacement (dots indicate actual site separations). For an uncharacterised system the coupling is set to the fabrication target J0, with the actual placement giving coupling J. The resulting CNOT error, (1 ), for a one site deviation (∆ 0.49) from the target separation−F can be seen 0 ≈ − on the inset plot as a function of implementation level. In the characterised system the coupling is set to Jc. The CNOT error for a system characterised to the 10% level (δJc/Jc =0.1), taking Jc 0.9J (∆c = 0.1), is shown as a function of implementation level≈ inset also. Note that all curves are included purely to guide the eye.

4 formed within the dephasing time, T2, of the system. The 10− level is widely assumed to be the fault-tolerant threshold for both environmentally induced and systematic errors [112], however more rigorous bounds [113] recently cal- 5 culated, suggest it could be closer to 10− . Figure 5.7 shows that it is possible to construct a CNOT gate to this precision level in the presence of signiﬁcant fabrication induced uncertainties, using either multiple concatenation of the composite rotations or a combination of the composite rotations and a modest level of characterisation. Assuming the system has been characterised to a modest level beforehand, 5.3 The role of two-qubit Hamiltonian characterisation 75

1 10−

2 10−

uncorrected 3 10− − F

1 4 error threshold ref. 10− 1st Level 5 10− 2nd Level

6 10− 101 102 103 104 105 106 Number of measurements (N)

Figure 5.7: CNOT error, (1 ), as a function of the total number of characterisation measurements−F required to achieve a given ﬁ- delity for various implementation levels. The results demonstrate the usefulness of combining composite rotations with Hamilto- nian characterisation when constructing a robust CNOT gate. 4 Threshold reference line at 10− error rate is shown. we now show that in order to remain below the threshold for environmentally induced errors also, the robust CNOT should be constructed using a single application of composite rotations and characterisation. In Fig. 5.8, these results are shown for a system with an unbiased J gate, J(V = 0), based on the 60 ms dephasing time in isotopically pure 28Si at 7 K [39], and for characterisation to the 10% level, again assuming the extremal bound value of J 0.9J. In c ≈ a biased system, the exchange coupling is stronger. Calculations suggest that for donors separated by 20 nm in the [100] direction, a 1 V bias applied ∼ to the control gates can strengthen the coupling by over two orders of magnitude [65, 87]. A robust CNOT gate comprising characterisation as described 7 above could therefore operate at close to the 10− level for environmentally induced errors. Performing additional measurements to characterise the system 7 to the 1% level would lower the systematic error level to well below 10− also, bringing it well within more rigorous threshold bounds [113], including those relevant to Si:P based architectures [114]. 76 A CNOT gate immune to large fabrication induced variations

100 Misplacement by: 1 uncorrected 1 site 10− 2 sites 6 sites 9 sites

2 ref. threshold dephasing 10−

3 st nd 10− 1 levelcomp. 2 level comp. − F 1 4 error threshold ref. 10−

5 1st level comp./10% char. 10−

6 10−

6 5 4 3 10− 10− 10− 10−

TCNOT/T2

Figure 5.8: CNOT error, (1 ), as a function of the total gate −F time for an unbiased, J(V = 0), system (T2 = 60 ms assumed). Results are shown for a range of separations in the [100] direction, larger than the targeted 20.6 nm separation. We consider various CNOT gate constructions, namely an uncorrected CNOT, one constructed from both a single and two applications of composite rotations and ﬁnally a CNOT constructed using composite rotations in conjunction with characterisation to the 10% level (δJc/Jc =0.1) taking Jc 0.9J. Only for this ﬁnal method have more than two sites been≈ included as for other methods results 4 will clearly be worse. Threshold reference lines at 10− error rates are shown.

For systems whose Hamiltonian parameters are not well known due to fabrication uncertainties, or may drift over time, this is an important result, suggesting that operating the CNOT gate in this way can guarantee that the error rate remains below the fault-tolerant error threshold. For the case of Si:P quantum computer architectures Fig. 5.8 suggests that this may be fabrication uncertainties within up to six sites of the target site, or 6.5 nm ∼ in the unbiased case, however in the J gate biased case this allowance may be much greater. The trade-oﬀ for operating in this manner is the need for periodic re-characterisation, however the cost of this should be minimal as the number of required measurements is small. 5.4 Summary 77

5.4 Summary

The performance of a robust CNOT gate constructed using two-qubit composite rotations has been examined. Multiple concatenation of the composite rotations results in a high fidelity CNOT gate provided the fractional uncertainty in J lies within the correctable range. Large variations in the exchange interaction coupling with donor separation means this is not always the case. Furthermore, multiple concatenation of composite rotations requires long overall gate times with respect to the decoherence time of the system and results in gate operation which exceeds the current error threshold required for fault-tolerant quantum computation. As an effective fix to this problem, we demonstrated how, in a system with large variations in the qubit coupling strength, a high fidelity CNOT gate which operates below this error threshold can be constructed from a single level of composite rotations in conjunction with Hamiltonian characterisation. In the following chapter we study another source of error that solid-state quantum computation will be subject to, environmental decoherence caused by charge noise. We develop analytic expressions for the time-dependent superoperator of a pair of coupled spins as a function of fluctuation amplitude and rate. These superoperators are used to determine the effectiveness of the robust CNOT gate discussed in this chapter against tackling this type of error.

Chapter 6

Modelling eﬀects of charge noise on the exchange interaction

We have previously noted that the isolation of electron and nuclear spins from their environment makes them attractive qubit candidates. This isolation means the coherence of spin states is relatively long compared to charge based qubits. Interactions between qubits such as the exchange coupling can introduce new decoherence channels. In the case of the exchange interaction the underlying electrostatic nature of the interaction makes the spin qubits more susceptible to environmental decoherence. Fluctuations in the local charge environment may therefore lead to gate errors and dephasing of these spin qubits. In this chapter we describe how the effect of charge noise on a pair of spins coupled via the exchange interaction can be calculated by modelling charge fluctuations as a random telegraph noise (RTN) process using probability density functions. We begin by introducing the superoperator formalism [115, 116, 117, 118, 119] as well as the model describing the noise process in section 6.1. In section 6.2 we develop analytic expressions for the probability density function of the RTN process and in section 6.3 use them to determine the corresponding analytic expressions for the time-dependent superoperator of a pair of spins as a function of fluctuation amplitude and rate. The extension to multiple fluctuators, in particular, spectral distributions of 80 Modelling effects of charge noise on the exchange interaction

fluctuators is considered in section 6.4. The superoperator formalism allows the effect of exchange fluctuations on spintronics, quantum control schemes and specifically QEC to be investigated. Before considering a specific example we outline how to use the superoperators in section 6.5. Finally, in section 6.6 we calculate the fidelity of a robust CNOT gate constructed from composite rotations in a realistically noisy environment and compare the performance to an uncorrected CNOT gate. The exchange interaction is of increasing importance in the study of controllable quantum mechanics using solid-state systems. As well as being funda- mentally important in many-body physics, it is this interaction which is often used to mediate spin flips or entanglement in spintronics and QIP [1, 31, 32, 33, 36, 120, 121]. For these reasons there has been considerable study recently on the origin and control of the exchange interaction [86, 110, 122, 123]. For applications involving the time varying control of the exchange interaction (such as QIP) the stability in time of this interaction is of crucial importance. As the origin of the exchange interaction is essentially the overlap of electron wave functions, the interaction strength is sensitive to the local charge environment. Recent work [124] has shown that the dependence of the exchange interaction is approximately linearly dependent on fluctuations in the local electric field.

6.1 The noise model

We begin by studying the exchange coupling Hamiltonian in the presence of a single charge fluctuator with the aim of understanding the fluctuator’s decohering effects. The Hamiltonian for the process is

H (t)= J (t) σ σ . (6.1) 1 · 2

The exchange coupling J (t) varies in time due to an RTN process, η (t) and we assume a net eﬀect of the form

J (t)= J0 + αη (t) , (6.2) 6.1 The noise model 81 where η (t) describes the ﬂuctuator. This RTN process couples with strength α (ultimately dependent on the distance between the coupled spins and the

ﬂuctuator) to the bare exchange term, J0. The time evolution of the system can then be described by the density matrix master equation

ρ˙ (t)= i [H (t) , ρ (t)] , (6.3) − where ρ (t) is the density matrix of the system. Additional terms can be added to this master equation to also model non-unitary evolution, such as decohering processes. As a matter of convenience we may re-express the system evolution in superoperator form. In superoperator form the density matrix is given a vector representation, denoted by ~ρ (t), by transforming the matrix into a single col- umn, one row at a time. A superoperator P contains all the evolution of the system (both unitary and non-unitary)

~ρ˙ (t)= P~ρ (t) . (6.4)

For purely Hamiltonian evolution the superoperator P can be written down in terms of H and the identity operator

P = i (H I) I HT , (6.5) − ⊗ − ⊗ where denotes the tensor product. The superoperator simpliﬁes to ⊗

P = iJ (t) σ , (6.6) − H for the Hamiltonian we consider. Here, σH is the Heisenberg interaction in superoperator form. If the Hamiltonian is constant in time, then the superoperator P is time independent and the density matrix at some time t is

Pt ~ρ (t) = e ~ρ0, (6.7) q (t) ~ρ , (6.8) ≡ 0 given an initial state ~ρ (t0) = ~ρ0. We show how this formalism is relevant to 82 Modelling eﬀects of charge noise on the exchange interaction our problem shortly.

The RTN process, η (t), is modelled as in Ref. [125]. The noise ﬂuctuates randomly between -1 and 1 with the frequency of the ﬂuctuations controlled by the correlation time 1/λ. Here, λ is the typical frequency of jump times, where the jump time instants are,

i 1 t = ln (p ) , (6.9) i −λ j j=1 X and the p are random numbers between 0 and 1, such that p (0, 1). The j j ∈ noise process η (t) is described as

Θ(t ti) η (t)=( 1)Pi − η (0) , (6.10) − where Θ (t) is the Heaviside step function, and η (t) can ﬂuctuate between η (0). We choose η (0) = 1 and control the coupling strength via α as in ± | | Eq. 6.2.

The density matrix evolution for our system can be found by numerically averaging over many such noise histories η (t) to obtain the correct system dynamics. For an initial state ρ0,

1 N ρ (t) = lim Ukρ0Uk†, (6.11) N →∞ N Xk=1 where the U are the evolution operators for trajectories η (t). Since the { k} k Hamiltonian (Eq. 6.1) commutes with itself at all times,

[H (ti) ,H (tj)]=0, (6.12) the U may be expressed as { k}

Uk (t, t0)= U (t ) U+ (t+) , (6.13) − − where t and t+ describe the total time the ﬂuctuator exists in the -1 and +1 − 6.1 The noise model 83 states respectively for a particular noise history, and

i(J0 α)σ1 σ2t U (t)= e− ± · . (6.14) ±

For each of the evolution operators U (t) the Hamiltonian is constant in time, ± so the result of Eq. 6.8 holds and Eq. 6.11 may be re-expressed in superoperator form 1 N ~ρ (t) = lim qk (t ) qk (t+) ~ρ0. (6.15) N − →∞ N Xk=1 The ensemble averaged superoperator, Q (t), is the average of all the individual trajectory superoperators qk (t),

1 N Q (t) = lim qk (t ) qk (t+) . (6.16) N − →∞ N Xk=1 This implies that Q (t) may be constructed by numerically averaging over many noise histories. The averaging is crucial in obtaining the correct system dynamics, as the RTN is a stochastic process and so there are many unique noise trajectories. Averaging over these noise trajectories results in non-unitary evolution despite the Hamiltonian (Eq. 6.1) being strictly unitary.

Rather than constructing Q (t) numerically, it can be constructed analytically by describing the stochastic RTN using an appropriate probability density function (PDF). By considering all unique qk (t) as a function of the average

ﬂuctuator state ξ = η (0) (t+ t ) /T , weighted by a PDF giving the occur- | | − − rence likelihood of the average ﬂuctuator state, and integrating this over all possible ξ, the resulting expression for Q (t) is

Q (t)= qξ (t)Ω(ξ, T ) dξ. (6.17) Zξ

Here, qξ (t) is the unique individual superoperator corresponding to a particular value of ξ and Ω(ξ, T ) is the PDF, which determines the probability that during the time interval T , the average ﬂuctuator state is ξ. In section 6.2 we show how to specify the PDF, so that we can use it to analytically determine 84 Modelling eﬀects of charge noise on the exchange interaction

Q (t) in section 6.3.

6.2 Calculating the probability density function

The statistical properties of an RTN process have been studied extensively in the context of reliability theory, alternating renewal processes and queueing theory [126, 127, 128, 129, 130, 131]. In our case, we are speciﬁcally interested in the probability of the RTN spending a certain fraction of the observation period in a particular state. The PDF for an RTN signal ﬂuctuating between the states 0 and +1 is given by [131]

λT τ p (τ, T )= λe− I 2λ τ (T τ) , (6.18) T τ 1 − r − h p i where I1 is the modified Bessel function of the first kind. This PDF assumes the initial state is +1, and that at least a single fluctuation occurs. Here, τ is used to describe the time spent in the state 0 and T is the duration of the process we are considering. The parameter λ characterises the fluctuator rate as before. Properly normalised the PDF is,

λ τ I1 2λ τ (T τ) p (τ, T )= − . (6.19) 2 T τ h sinhp2 λT i r − 2 We could equally describe a process which begins in the state 0, with T τ − describing the time spent in this state. Assuming at least a single ﬂuctuation occurs, the full PDF is obtained by averaging over both possible starting states

1 p′ (τ, T )= [p (τ, T )+ p (T τ, T )] . (6.20) 2 −

We may re-express this PDF in terms of the mean ﬂuctuator state ξ, where ξ [ 1, 1]. Taking care to preserve the normalisation, the PDF for an RTN ∈ − 6.2 Calculating the probability density function 85 process of duration T assuming at least one ﬂuctuation occurs is

T T Ω (ξ, T ) = p′ (ξ + 1) , T , (6.21) >0 2 2 2 λT I1 λT 1 ξ = − . (6.22) 4 1 ξ2psinh2 λT − 2 p The case where no ﬂuctuations occur must be treated separately. In this case we expect ξ to be either of 1. The properly normalised PDF for this ± case can be described using two delta functions,

1 Ω (ξ, T )= [δ (ξ 1) + δ (ξ + 1)] . (6.23) 0 2 −

The full, general PDF is constructed by appropriately weighting Ω0 (ξ, T ) and

Ω>0 (ξ, T ), with the ﬂuctuation probability given by the Poisson distribution

e λT (λT )k p (λT )= − , (6.24) k k! where k denotes the number of ﬂuctuations, such that

Ω(ξ, T )= p0 (λT ) Ω0 (ξ, T )+ p>0 (λT ) Ω>0 (ξ, T ) , (6.25) and p (λT )=1 p (λT ). After simpliﬁcation, the resulting PDF is >0 − 0

2 e λT λT I1 λT 1 ξ Ω(ξ, T )= − [δ (ξ 1) + δ (ξ +1)]+ − . (6.26) 2 − eλT 1 1p ξ2 − − p In the following section we examine the three limiting cases of the PDF and use these to construct an approximate PDF. The approximate PDF provides greater physical insight when working within these limits.

6.2.1 Approximating Ω(ξ,T )

Examining the two limiting cases of the PDF Ω>0 (ξ, T ), the fast and slow

ﬂuctuator limits and combining them with Ω0 (ξ, T ), which describes the zero 86 Modelling eﬀects of charge noise on the exchange interaction

fluctuation limit, leads to a simplified expression which approximates Ω (ξ, T ). We begin by considering the slow fluctuator limit λ 0 for the distribution → describing at least one fluctuation, Ω>0 (ξ, T ). This is the regime where no more than one fluctuation occurs. In this limit

1 x a I (x) , (6.27) a ∼ Γ(a + 1) 2 and sinh (x)= x + x3 . (6.28) O This reduces the PDF to

1 Ω (ξ, T ) , (6.29) >0 ≈ 2 Ω˜ (ξ, T ) . (6.30) ≡ 1

This uniform distribution implies that a ﬂuctuation is just as likely to occur at any time during the system evolution.

The limit λ represents a fast ﬂuctuator. In this regime →∞

1 x Ia (x) e , (6.31) ∼ √2πx and ex sinh (x) , (6.32) ≈ 2 which reduces the PDF to

2 λT λT ξ 3 2 Ω (ξ, T ) e− 2 1+ ξ , (6.33) >0 ≈ 2π 4 r 2 λT λT ξ 2 e− 2 + ξ . (6.34) ≈ r 2π O In this limit ξ will be small, therefore making the substitution µ =1/√λT , we 6.3 Using the PDF to determine Q (t) 87

ﬁnd the PDF to be Gaussian about the origin,

1 ξ2 µ2 Ω>0 (ξ, T ) e− 2 , (6.35) ≈ µ√2π Ω˜ (ξ, T ) . (6.36) ≡ >1 which we expect intuitively. We note that this approach is similar to that used by Happer and Tam when considering the Gaussian limit of rapid spin exchange in alkali vapors [132]. Weighting these two limiting cases and the PDF describing no ﬂuctuations using the Poisson distribution as before, allows us to construct an approximate PDF

Ω(ξ, T ) p (λT ) Ω (ξ, T )+ p (λT ) Ω˜ (ξ, T )+ p (λT ) Ω˜ (ξ, T ) , (6.37) ≈ 0 0 1 1 >1 >1 where p (λT )=1 p (λT ) p (λT ). This approximate Ω (ξ, T ) provides >1 − 0 − 1 nice analytic solutions for Q (t) in each of the three interesting ﬂuctuator regimes. While this is only an approximation to the exact solution (Eq. 6.26), it can provide more physical insight, as will become apparent later.

6.3 Using the PDF to determine Q (t)

The superoperator Q (t) can be derived analytically via Eq. 6.17 using the PDFs determined in the previous section. Of particular interest is the non- unitary part of the superoperator. Non unitary quantum processes are those which irreversibly leak information from the system into the surrounding environment, a phenomenon known as decoherence. This has huge implications for controllable quantum systems, in particular quantum computers, which require a high degree of coherence to operate with high ﬁdelity. Qubits subject to decoherence evolve from pure states into mixed states, hence introducing ignorance into our knowledge of the system. Protecting quantum information against decoherence is therefore imperative for the success of any quantum computing architecture. Quantum error correcting codes and fault-tolerant design play their role to this end, as do other factors, such as clever control 88 Modelling eﬀects of charge noise on the exchange interaction schemes to minimise unwanted external interactions with the environment.

The non-unitary superoperator can be found by expanding the superoperator into a unitary and non-unitary part, such that Q (t) = Q(u) (t) Q(nu) (t). The evolution in the absence of a ﬂuctuator is contained within the unitary part,

(u) iJ0σ t Q (t)= e− H , (6.38) whilst the eﬀect of the charge ﬂuctuator is contained within the non-unitary part Q(nu) (t). Note that these two parts can be factored out due to the commutation relation (Eq. 6.12). We now determine the non-unitary parts of the superoperator for Ω(ξ, T ) and its various approximations.

Beginning with the case where no ﬂuctuations occur and the PDF is given by Ω0 (ξ, T ), as in Eq. 6.23, we ﬁnd

(nu) σ Q0 (t) = cos(α Ht) , (6.39) where σH is the Heisenberg superoperator introduced earlier. When there is at least one fluctuation (see Eq. 6.22) the resulting form of the superoperator is 2 2 cos (ασHt) (λT ) cos(ασHt) (nu) − − Q>0 (t)= q 2 λT . (6.40) 2 sinh 2 Examining the limiting cases of the general PDF we find that for the slow fluctuator (see Eq. 6.30)

σ ˜ (nu) sin (α Ht) Q1 (t)= , (6.41) ασHt and in the fast ﬂuctuator limit (see Eq. 6.36)

σ 2 ˜ (nu) (αµ Ht) /2 Q>1 (t)= e− . (6.42)

It should be noted that this final superoperator corresponds exactly to that which would be obtained using the Lindbladian formalism [133, 134] if a decoherence operator of the form L = α σ σ was included. From this we √λ 1 · 2 6.3 Using the PDF to determine Q (t) 89 can deduce that the fast fluctuator limit is equivalent to purely Markovian decoherence due to interaction with the environment via an exchange like two-qubit decoherence channel. This is in contrast to conventional dephasing which is modelled using two independent σZ channels, one for each qubit. This distinction is particularly important as it implies that exchange fluctuations introduce correlated errors which can have important implications for fault-tolerant QEC [113]. Using the previous results, we can determine Q(nu) (t) for the full weighted PDF’s in both the approximate and exact cases. The exact PDF given in Eq. 6.26 yields

(nu) λT 2 Q (t)= e− cos(ασ t)+ H eλT 1 − cos (ασ t)2 (λT )2 cos(ασ t) , (6.43) × H − − H q whilst for the approximate PDF given in Eq. 6.37 we ﬁnd

σ (nu) λT λT sin (α Ht) Q (t) e− cos(ασHt)+ λT e− ≈ ασHt λT λT (αµσ t)2/2 + 1 e− λT e− e− H . (6.44) − − In general it is difficult to graphically compare these analytic forms of the superoperator to the numerical result. However, it is possible in this case, as the superoperator Q(nu) (t) is a sparse matrix with the same underlying structure of the σH superoperator which defines it. It follows from the definition of σH (see Eq. 6.6) that the only non-zero matrix elements of the Heisenberg superoperator are 2. Consequently, a comparison of the resulting non-zero matrix ± (nu) element of Q (t), denoted QNU, proves effective in determining the agreement between the analytic (exact and approximate) superoperators and exact numerical solution for the superoperator. The results, as a function of time for a range of fluctuator rates, shown in Fig. 6.1, reveal very good agreement between the exact analytic and numerical results for all rates λ. The approximate solution also matches closely, particularly in the slow and fast fluctuator limits. Slight deviations from the exact solution can be seen when the fluc- 90 Modelling effects of charge noise on the exchange interaction

λT = 100 0.8

0.6 numerical

NU exact Q 0.4 approximate

0.2 λT = 1 λT = 10

λT = 0.1 0 0 0.5 1 1.5 2 2.5

1 t `α− ´

Figure 6.1: Comparison of the analytical (exact and approximate) and exact numerical solutions of Q(nu) (t). Plotted is the non-zero (nu) matrix element of Q (t), denoted QNU, as a function of time for a range of fluctuator rates, which span each of the three limiting regimes. The results show very good agreement between all three solutions, with the analytic approximation deviating only slightly when the fluctuator rate is on the time scale of T . tuations occur on the time scale of the process we are considering (λT 1). ≈ In this regime the contribution from the uniform distribution Ω˜ 1 (ξ, T ) is at its maximum and approximately on par with contributions from the other two distributions. The deviation from the exact results do not come as a sur- prise, as the approximate PDF is constructed from contributions due to 0, 1 or many fluctuations. Adding contributions from 2, 3 and more fluctuations would reduce this discrepancy. The generalisation of the single fluctuator formalism to multiple fluctuators is now presented in the following section.

6.4 Multiple ﬂuctuators

Extending this formalism to multiple fluctuators is straight forward and provides a method for the treatment of many physically realistic scenarios. The 6.4 Multiple fluctuators 91 total ensemble averaged superoperator, Λ (t), for N fluctuators is just the product of all the individual ensemble averaged superoperators, Q (t), such that N (u) (nu) Λ (t)= Q (t) Qi (t) . (6.45) i=1 Y This result is useful for a finite number of fluctuators each with known strength and rate. However, in most instances only the spectral distribution in strength and rate will be known and therefore Eq. 6.45 offers no further insight. By (nu) considering all possible unique superoperators Q (αi,λi, t) weighted by their probability of occurrence p (where p [0, 1]), in a similar way to the method i i ∈ used to construct Q (t) in Eq. 6.17, we may re-express Eq. 6.45 as

M (u) (nu) Npi Λ (t)= Q (t) Q (αi,λi, t) , (6.46) i=1 Y where in general there are M possible ﬂuctuator types and N ﬂuctuators. We would like to interpret the pi as a spectral distribution function in αi and λi. By expressing Λ as a sum of logarithms

M (u) (nu) Λ (t)= Q (t) exp N pi ln Q (αi,λi, t) , (6.47) ( i=1 ) X and extending the deﬁnition of Λ to the continuum, we may replace the pi with a spectral distribution function S (α,λ) such that

Λ (t)= Q(u) (t) exp N S (α,λ) ln Q(nu) (α,λ,t) dαdλ , (6.48) Z ensuring that the spectral distribution function is properly normalised

S (α,λ) dαdλ =1. (6.49) Z

The eﬀects of a region of charge noise can now be modelled using either Eq. 6.45 or Eq. 6.48. The approach used will depend on exactly what information is known about the system. 92 Modelling eﬀects of charge noise on the exchange interaction

In the following section we outline how to implement the superoperators and provide a discussion of some important limitations of this approach.

6.5 Using the superoperators

The superoperators in sections 6.3 and 6.4 were constructed on the basis of the commutation relation for our Heisenberg Hamiltonian (see Eq. 6.12). The commutation relation meant we could express the total evolution operator as a product of two evolution operators, each describing the total time spent in one of the fluctuator states (see Eq. 6.13). The superoperators themselves also commute as a result. Without further approximation the superoperators can be used to model individual processes satisfied by the Hamiltonian in Eq. 6.12 (or a similar commuting Hamiltonian, such as the Ising interaction). In more complex superoperator applications, for example when multiple applications of the superoperators are separated by non-commuting operations, it may be necessary to make a further approximation. When modelling these more complex processes (see Fig. 6.2 for an example where two superoperators are separated by a Hadamard operation) a problem arises with the formalism when considering the slow fluctuator limit. Specifically, the superoperator terms which

Q (t1) Q (t2)

Figure 6.2: Multiple instances of the superoperator Q (t) separated by a non-commuting Hadamard operation. Attempting to use the superoperators to determine the effects of charge noise in a process like this can lead to the introduction of errors in the slow fluctuator regime. should describe no fluctuations at all, actually account for the possibility of a fluctuation occuring between superoperator applications. We refer to these terms as cross terms. As the fluctuation rate increases the Poissonian weighting 6.5 Using the superoperators 93 of these cross terms in the overall superoperator reduces, hence reducing the cross terms significance. We now consider a simple example which illustrates how these cross terms manifest themselves, before showing how an approximate solution can be constructed for the slow fluctuator regime by removing the cross terms.

When the superoperators do not commute as in the example shown in Fig. 6.2, two or more applications of the superoperators leads to the introduction of unphysical cross terms in the slow fluctuator limit. This becomes apparent when we consider the action of the superoperator describing no fluc- (nu) tuations Q0 (t), which may be expanded in terms of the superoperators, Q± (t), each describing one of the two fluctuator states ξ, in the no fluctua- 0 ± tor limit (nu) 1 + Q (t)= Q (t)+ Q− (t) . (6.50) 0 2 0 0 The cross terms from the product of two (or more) of these superoperators which sandwich non-commuting operations results in the description of a single (or multiple) fluctuation(s). For example, consider the system in Fig. 6.2. The Hadamard operation does not commute with the superoperators

Qtotal = Q (t2) HsoQ (t1) , (6.51)

where Hso is the Hadamard superoperator. Expanding out each of the superoperators using

(u) (nu) (nu) Q (t)= Q (t) p0 (λt) Q0 (t)+ p>0 (λt) Q>0 (t) , (6.52) h i and Eq. 6.50, with some rearranging we ﬁnd

p [λ (t + t )] Q = 0 1 2 Q(u) (t ) Q+ (t ) H Q+ (t ) total 4 2 0 2 so 0 1 + + + Q0 (t2) HsoQ0− (t1)+ Q0− (t2) HsoQ0 (t1) (u) + Q0− (t2) HsoQ0− (t1) Q (t1)+ ... , (6.53) where we have only shown the terms which should describe no ﬂuctuations. 94 Modelling eﬀects of charge noise on the exchange interaction

This entire expression should represent the total superoperator describing no fluctuations. However, careful inspection shows the presence of two cross terms, which actually imply the occurrence of a fluctuation during the Hadamard operation. Cross terms of this form are actually a manifestation of this superoperator formalism and should be removed without also removing any unitary evolution. It should be emphasised that this problem only occurs in the slow fluctuator limit, where there is a significant probability of there being no fluctuations during a two-qubit operation. As the fluctuation rate increases, the probability of a fluctuation occurring during the SINGLE qubit gate increases, which means that each application of the two-qubit gate becomes statistically independent. In this limit, the formalism as presented so far is perfectly correct and does not require any attention to cross terms.

6.5.1 Removing the cross terms

We wish to remove cross terms describing processes which should not occur, (nu) such as those in Eq. 6.53. When two or more applications of the Q0 (t) superoperator occur in succession, the introduction of these unphysical terms also occurs. It is possible to construct an approximate solution by carefully removing these cross terms. Later in this thesis, we will apply our superoperator formalism to determine the success with which a robust CNOT gate pulse scheme can be implemented to combat charge noise. The expectation is that such a control scheme will be effective in both the slow and fast fluctuator regimes. We will therefore require an approximation that is valid in these regimes. Cross terms in the large λ limit (nu) do not pose a problem as the Q>0 (t) superoperator provides the dominant contribution to Qtotal in this limit. It will therefore be most important to (nu) remove the cross terms due solely to the Q0 (t) superoperator - the zeroth order cross terms. In general, when many applications of the superoperators are required, there will be higher order cross terms. For example, the first order (nu) cross terms would contain the Q0 (t) superoperator and a single instance (nu) of the Q>0 (t) superoperator. Successive improvements to the approximate 6.6 Gate operations in the presence of charge noise 95 solution are achieved by removing these higher order cross terms. As more orders of these terms are removed the approximation improves for larger λ, with the actual region of improvement dependent on the Poissonian weighting of the cross terms being removed.

The zeroth order cross terms, denoted X0, are removed by ﬁrstly re-weighting

Qtotal using the Poisson distribution over the total process duration Ttotal (i.e., including non-commuting operations),

Qtotal = p0 (λTtotal) Qtotal + p>0 (λTtotal) Qtotal. (6.54)

The cross terms can now be removed

Q p (λT )(Q X )+ p (λT ) Q , (6.55) total ≈ 0 total total − 0 >0 total total taking care to not also remove any unitary evolution. The cross terms may also be removed from the second term, however the improvement from doing this is expected to be minimal due to the small contribution from the zeroth order terms at large λ. In the following section we analyse the eﬀects of a single charge ﬂuctuator on the feasible operation of a CNOT gate. A comparison of the relative performances of the robust CNOT gate, detailed in chapter 5 in the context of a noiseless environment, and an uncorrected CNOT gate is presented.

6.6 Gate operations in the presence of charge noise

Imperfections in the fabrication process may cause interface defects such as charge traps. These defects can be due to broken, dangling, elongated or strained bonds, interstitials or lattice vacancies. Much work has been performed on understanding interface characteristics, of which, to us the Si-SiO2 interface [135, 136] is of direct interest. As the name suggests charge traps can localise an electron at the interface hence altering the local electrostatic environment. The exchange interaction is sensitive to changes in the local elec- 96 Modelling eﬀects of charge noise on the exchange interaction trostatic environment due to the dependence of the interaction on electrostatic properties such as electron wave function overlap. Determining the response of quantum gate operations in a realistic noisy environment is a crucial step towards the development of a working quantum computer.

Current measurements of the defect density at the Si-SiO2 interface esti- 11 2 mate the number of charge traps to be of order 10 cm− [137, 138]. This amounts to approximately one defect per 1000 nm2. For donors separated by approximately 20 nm it is therefore resonable to assume that at most, a few charge traps will couple to the exchange interaction. Recently the effect of charge noise on exchange coupled qubits in a double quantum dot has been quantified [124]. For weakly coupled qubits (J 1 µeV), variations in the ≈ coupling of order δJ 0.01J are predicted due to charge fluctuations. This ≈ is in good agreement with the experimentally measured dependence of the exchange coupling on variations in bias voltage found in Ref. [139]. Using this result for the strength of the fluctuator, the effect of a single charge fluctuator on the fidelity of both an uncorrected and robust CNOT gate is now modelled as an application of our analytic solutions. Recall that in chapter 5 an analysis of the CNOT gate fidelity in a noiseless environment was presented. For computational simplicity, here we consider a CNOT gate constructed from the native Ising interaction, rather than isolating the Ising term from the exchange interaction as in chapter 5. The superoperator formalism applies to all Hamiltonians where the commutation relation in Eq. 6.12 exists, hence its direct applicability to the Ising interaction. To isolate the effect of the charge noise in this analysis, we also assume perfect and instantaneous single qubit unitaries. A more detailed analysis would include the gate operation time and error associated with performing single qubit unitaries, however for this first pass calculation analysing the accuracy of our analytic model and effects of the charge noise are of greater interest. The charge fluctuator couples to the Ising Hamiltonian analogously to Eq. 6.1 H (t)= J (t) σ σ , (6.56) ZZ Z ⊗ Z where JZZ (t)= JZZ + αη (t) describes the coupling variation due to the RTN 6.6 Gate operations in the presence of charge noise 97

process η (t), which couples with strength α to the bare Ising term, JZZ. The composite rotations, which were earlier used in a noiseless environment are not designed to correct time dependent error processes such as the RTN. Instead, they are specifically designed to correct for static errors, such as systematic under- or over-rotations. Based on this, the expectation is that the robust CNOT gate will outperform an uncorrected CNOT gate in the slow fluctuator limit, when fluctuations occur on a time scale much longer than the CNOT operation time. We would also expect the robust gate to outperform the uncorrected gate in certain instances when there is a degree of uncertainty in the bare coupling JZZ, as we will see later. In order to highlight each of the three important fluctuator regimes we have parameterised the rate in terms of the dimensionless quantity λTgate, where Tgate is the total gate operation time, different for both the robust and uncorrected implementations. We begin by considering a perfectly characterised system.

6.6.1 The perfectly characterised system

In the analysis that follows we compare two independent results. Firstly, we evaluate the performance of our analytic solutions - exact and approximate, with exact numerical results achieved by averaging over many noise histories. The uncorrected CNOT gate is described exactly by the analytic solution, whereas the robust CNOT gate can only be approximated by the analytic solution. This is due to the presence of non-commuting operations between the multiple superoperator applications required to simulate a robust CNOT operation, and requires us to remove the non-unitary cross terms as detailed in section 6.5.1. Secondly, we compare the relative performances of the two CNOT gate operations - uncorrected and robust. The CNOT fidelity of both gate implementations as a function of the fluctuator rate in a perfectly characterised system is shown in Fig. 6.3, including a magnified inset of the large λ regime (λT 1), showing the CNOT error. gate ≫ With respect to the robust gate operation, the region in which the approximate analytic solution breaks down due to the removal of only the zeroth order non-unitary cross terms X0, from the calculation is evident. Large deviations 98 Modelling effects of charge noise on the exchange interaction

1 0.9999 uncorrected 0.9998 −4 0.9997 10 −5 10 robust

F 0.9996 − 10 6

0.9995 −F −7

1 10 − 0.9994 10 8 −9 0.9993 10 102 103 104 0.9992 λTgate 0.9991 10 8 6 4 2 0 2 4 6 10− 10− 10− 10− 10− 10 10 10 10

λTgate

Figure 6.3: CNOT gate fidelity as a function of λ for the uncorrected and robust gates. The analytic simulations are represented by solid lines, whilst exact numerical simulations are represented by points. The robust gate clearly outperforms the uncorrected gate when λ is small (λT 1). When fluctua- gate ≪ tions occur on the time scale of the CNOT gate (λTgate 1), the robust gate fidelity drops, as the composite rotations do≈ not effectively correct when the coupling strength varies. In the large λ limit (λTgate 1), the averaging effect on JZZ causes a revival in the fidelity of≫ the robust gate, however it is still outperformed by the uncorrected gate (see inset for a close-up of the CNOT gate error for large λ). Very good agreement is achieved between the exact numerical and exact analytic results. The approximate analytic result breaks down when the fluctuations occur on the time scale of the CNOT gate. from the exact numerical solution occur when fluctuations are on roughly the same time scale as the CNOT operation, (λT 1). By removing higher gate ≈ order cross terms it is possible to improve the approximation in this region, as seen in Fig. 6.4, where we have also removed the first and second order cross terms X1 and X2. This is the only difference between these two figures. The approximation performs well for large λ, as each application of the (nu) superoperator is statistically independent, meaning Q0 (t) cross terms have 6.6 Gate operations in the presence of charge noise 99

0.9999 − 10 4 uncorrected 0.9998 − 10 5

F − 10 6 0.9997 −F −7

1 10 −8 robust 0.9996 10 − 10 9 0.9995 102 103 104 λTgate 0.9994 10 8 6 4 2 0 2 4 6 10− 10− 10− 10− 10− 10 10 10 10

λTgate

Figure 6.4: CNOT gate ﬁdelity as a function of λ for the uncorrected and robust gates. The analytic simulations are represented by solid lines, whilst exact numerical simulations are represented by points. The approximate analytic solution has been improved dramatically by removing the higher order cross terms.

negligible eﬀect on Qtotal. Although the robust gate achieves a high ﬁdelity in this limit, it is still outperformed by the uncorrected gate as seen in the inset of Figs. 6.3 and 6.4, where we show the CNOT gate error. The fast

fluctuations cause an averaging effect of JZZ, which results in a revival of the fidelity, however the robust gates longer exposure to the fluctuations slightly reduces the performance as compared to the uncorrected gate. The good agreement seen between the approximate analytic and numerical solutions in the slow fluctuator regime (λT 1) in Figs. 6.3 and 6.4 does gate ≪ not persist as α becomes large. Further analysis of the small λ limit, reveals that the analytic approximation used to determine the fidelity of the robust gate is only valid when α is also kept small. We highlight the breakdown of the approximate solution in Fig. 6.5, by plotting the fidelity as a function of the fluctuator strength α, for λT 1. The discrepancy between results gate ≪ can be attributed to incorrect scaling of the approximate solution with the fluctuator strength α. This is supported by the fact that the solution displays the expected characteristics, i.e., a broadening of the fidelity plot indicating 100 Modelling effects of charge noise on the exchange interaction

1 0.95 0.9 0.85 0.8

F 0.75 0.7 0.65 analytics numerics 0.6 uncorrected uncorrected 0.55 robust robust 0.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

α (JZZ )

Figure 6.5: CNOT gate ﬁdelity as a function of α in the small λ limit (λT 1), for the uncorrected and robust gates. The gate ≪ approximate analytic solution used to determine the robust gate ﬁdelity clearly breaks down for large α, deviating from the exact numerical result. In the case of the uncorrected gate, the analytic superoperator solution achieves perfect agreement with expected results, as for a single application of the two-qubit interaction the superoperator solution is exact.

the larger correction range of the robust gate as compared to the uncorrected gate, albeit on the wrong scale. Conversely, deviation between the analytic approximation and exact numerical solution is not seen as a function of α in the large λ limit (see Fig. 6.6). In this limit each application of the two-qubit gate becomes statistically independent, meaning the formalism is perfectly correct. Furthermore, it should be emphasised that the completely analytic superoperator solution, which is used to calculate the fidelity of the uncorrected gate, provides perfect agreement with the exact numerical solution. This is clearly visible in each of Figs. 6.3-6.6. We now consider the effect of the fluctuator on the CNOT implementations for a system which is not perfectly characterised. 6.6 Gate operations in the presence of charge noise 101

0.998

0.996

0.994 F 0.992

0.99 analytics numerics uncorrected uncorrected 0.988 robust robust

0.986 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

α (JZZ )

Figure 6.6: CNOT gate ﬁdelity as a function of α in the large λ limit (λT 1), for the uncorrected and robust gates. No de- gate ≫ viation from the expected results is observed with either analytic approach.

6.6.2 The imperfectly characterised system

In a perfectly characterised system, the robust CNOT gate only outperforms the uncorrected CNOT gate in the slow fluctuator regime. As outlined in chapter 5, imperfections in the fabrication process or drift in the Hamiltonian control parameters over time can lead to uncertainty in the coupling between qubits. Unless the system is perfectly characterised using a systematic two- qubit interaction identification procedure such as Hamiltonian characterisation [99, 100], an uncorrected CNOT gate would not operate with the very high fidelity required. We demonstrate this by including a small amount of uncertainty into the bare Ising coupling JZZ, the value assumed to be the coupling strength. The actual coupling strength is set to JZZ′ = 0.99JZZ. The fluctuations now occur about the offset coupling strength, hence the uncorrected gate is subject to systematic error due to under-rotation. In Fig. 6.7, we quantify this effect on the fidelity of both CNOT gates. Even for this very small uncertainty in the coupling, the robust CNOT gate now clearly outperforms in both the slow and fast fluctuator limits. This suggests that although 102 Modelling effects of charge noise on the exchange interaction

0.9999

0.9998 analytics F uncorrected 0.9997 robust 0.9996 numerics uncorrected 0.9995 robust

0.9994 10 8 6 4 2 0 2 4 6 10− 10− 10− 10− 10− 10 10 10 10

λTgate

Figure 6.7: CNOT gate ﬁdelity as a function of λ for the uncorrected and robust gates when a small amount of uncertainty exists

in the coupling strength (JZZ′ =0.99JZZ). The results show that the robust gate now clearly outperforms the uncorrected gate in both the small and large λ limits. This suggests that although the composite rotations are designed to correct for static errors, in a noisy system which is not perfectly characterised, using the composite rotations to construct robust gates may be advantageous. these composite rotations are designed to correct for static errors, in a noisy system, which is not perfectly characterised, using the composite rotations to construct robust operations may be advantageous. The superoperator formalism reproduces exactly the results achieved when using a numerical approach which sums over many noise histories. The only complication arises in systems such as the example shown in Fig. 6.2, when two or more applications of the superoperator separated by non-commuting operations is required. In this instance, as outlined in section 6.5.1, an approximate analytic solution can be constructed by removing the non-physical cross terms which effect the small λ results. This technique achieves reasonable agreement with expected results in the small λ regime provided α also remains small. Re- moving higher order cross terms markedly improves the approximation in the 6.7 Summary 103 region where fluctuations occur on the same time scale as the CNOT operation (λT 1), providing good agreement with expected results. In the large λ limit ≈ the approximate solution performs very well due to the small influence of the (nu) Q0 (t) superoperator on Qtotal and the statistical independence of repeated applications of the superoperator.

6.7 Summary

The exchange interaction is of fundamental importance for controllable quantum mechanics in solid-state systems. Its application to mediate spin flips or entanglement has particular importance in spintronics and QIP, hence the stability of the exchange interaction is crucial for precise time varying control. In solid-state spin systems this stability can be affected by the local charge environment, in particular charge fluctuators, due to the exchange couplings dependence on the electron wave function overlap. We have developed a model to describe the effect of charge fluctuators on the exchange interaction as a function of time, using superoperators dependent on the noise amplitude and rate. These superoperators can be included in time- dependent calculations of the state of the spin system to model the effect of the charge noise. Furthermore this analysis holds for other spin couplings, like the Ising interaction, where a commutation relation analogous to Eq. 6.12 exists. In the fast fluctuator limit we demonstrated how interaction with the environment via an exchange like decoherence channel leads to purely Markovian decoherence, although the decoherence operator leads to correlated noise across the two spins. The generalisation to multiple fluctuators means that the effect of charge fluctuators distributed according to a spectral distribution function can also be modelled. In the simpler case where only a small number of well defined fluctuators exist, the total superoperator is just the product of the individual fluctuator superoperators. As our model is completely analytic, the effects of exchange fluctuations can in most instances be included trivially in more sophisticated analyses, without the need to explicitly sum over noise histories. This is important for analysing 104 Modelling effects of charge noise on the exchange interaction the operation of spintronic devices as well as QEC and fault-tolerance for QIP. We used the performance of a robust CNOT gate pulse scheme in the presence of a single charge fluctuator to illustrate the application of our analytic solution. Throughout this analysis we compared both our analytic results and numerical simulations to evaluate the accuracy of our methods. The major findings of our analysis, for both the robust CNOT gate performance and the accuracy of our analytic method are now outlined. The analytic superoperator results for the uncorrected gate were in perfect agreement with the exact numerical results. When non-commuting operations separated two or more instances of the superoperators, such as for the robust CNOT gate, it was necessary to construct an approximate analytic solution by (nu) removing the non-physical cross terms caused by the Q0 (t) superoperator. These cross terms had the largest effect on results for small λ, the region where this superoperator provides the dominant contribution. Here, the approximate analytic solution remained in good agreement with numerical results provided α was also kept small. The robust CNOT gate easily outperformed the uncorrected CNOT gate in this regime as expected. When fluctuations occurred on roughly the same time scale as the CNOT gate the robust gate provided no advantage. In this regime very good agreement between the expected and approximate analytic results could be achieved by removing higher order cross terms. In the large λ limit the robust CNOT gate clearly outperformed the uncorrected gate provided a small amount of uncertainty existed in the strength of the bare coupling between qubits. The approximate results were in very good agreement with the expected results because as the likelihood of a fluctuation between successive applications of the superoperator increased, each application of the superoperator became statistically independent. In this regime cross term effects where negligible and the formalism perfectly correct. Chapter 7

Conclusions

The research covered in this thesis has dealt with the topics of read-out and two-qubit control for Si:P donor based quantum computing architectures, fo- cusing on the development of a new read-out protocol for nuclear spin qubits and the implementation of robust CNOT gates in donor electron spin devices. In the latter work we have considered the eﬀects of systematic errors and by developing an analytical model for charge noise, environmental errors also. On the topic of read-out, the work was comprised of two parts. In chapter 3 we began by presenting a review of the Kane adiabatic read-out protocol and in doing so, introduced the two electron read-out state known as the

D−. In the adiabatic read-out protocol, formation of the D− state via the 0 0 + transition D D D D− is reliant on the system being in the spin singlet, → which is in turn conditional on the state of the nuclear spin qubit being 0 . | i Appropriate biasing of the A gates provides a static electric field which causes this transition. The fragile nature of this state resulting from it low binding energy means it is susceptible to ionisation prior to read-out by the SET. This motivated the development of a theoretical model of the state. We began by modelling the host silicon lattice using Bloch functions. These Bloch functions could be used to calculate the band structure of silicon. Using the one-valley hydrogenic effective mass theory, a theory of shallow donor states, we were able to model both the donor qubit and read-out states. The binding energy of the D− state was calculated to be 0.9 meV, a result which is in reasonable 106 Conclusions agreement with the experimental value of 1.7 meV, however there exists room for substantial improvement of the model. Avenues for future extension of this work include the use of an anisotropic 0 envelope function for both the D and D− states and the inclusion of effects beyond the one-valley effective mass theory, such as the valley-orbit coupling. It is expected that the considering the mass anisotropy will allow a proper treatment of the non-spherical energy surfaces located at each of the six degenerate conduction band minima of silicon. Inclusion of valley-orbit coupling terms would result in a lifting of the degeneracy which remains in the one- valley effective mass theory, by accounting for interactions between electrons in equivalent and non-equivalent valleys. This would have a profound effect on the calculated binding energy of the D− state.

Preliminary calculations suggest that the lifetime of the D− state is not sufficiently long-lived to enable charge detection by the SET prompted the development of an alternative, optically driven read-out scheme for nuclear spin qubits. This work, which appeared in chapter 4, completed the work on read-out. The optically driven scheme was found to require fields much weaker than the equivalent dc fields required in the adiabatic protocol, due to the resonant nature of the charge transfer. High fidelity transfer on time scales faster than those required for single-shot measurement by an rf-SET was shown to be possible. In an electron spin paradigm this optical technique facilitates singlet-triplet read-out, which may be used in cluster state quantum computation. Spin resolved read-out and initialisation of electron spin qubits directly would require the degenerate energy levels to be spectrally resolved. This may be possible in a 2-D solid-state quantum computer with a separate read-out zone via a mechanism such as an inhomogeneous magnetic field. In chapter 5 the focus shifted to designing a robust CNOT gate despite the large exchange coupling variations that exist between qubits as a consequence of small donor misplacements. Using composite control sequences convention- ally used in the context of NMR we outlined how to optimally achieve a high fidelity gate considering restrictions imposed by the decoherence time of the system and fault tolerance requirements. With multiple concatenation of the composite control sequence we showed how in principle the CNOT gate could 107 be made arbitrarily accurate. This approach resulted in long gate operation times with respect to the decoherence time scale. In order to maintain a high fidelity operation and remain within the error threshold bounds required for fault-tolerant quantum computation, we combined a single level of the composite control sequence with a modest level of Hamiltonian characterisation. A number of opportunities exist for future work in this area. One obvious extension would be to include errors on the single qubit gates also and see the effect this has on the CNOT operation fidelity. Including these errors would allow us to determine a completely realistic approach towards implementing robust CNOT gates in solid-state devices, as single qubit unitaries may also require characterising or compensating. Developing robust pulse sequences for other entangling two-qubit gates, such as the √SWAP, the native gate for the isotropic Heisenberg Hamiltonian, would also be worthwhile. Finally, a quantitative comparison of this method to other quantum control techniques, such as numerically optimised algorithms like GRAPE[140], would help to map out the control landscape. In doing so we may begin to understand the best approach to any given application. In chapter 6 we studied the effects of RTN caused by a charge fluctuator on spins coupled via the exchange interaction. We developed an analytical model of this charge noise, as a function of fluctuator rate and strength, using time-dependent superoperators. Our approach was contingent on the interaction commuting with itself at all times and as such the model holds for other spin couplings where this commutation relation exists, such as the Ising interaction. We deduced that the fast fluctuator limit is equivalent to purely Markovian decoherence due to environmental interactions via an exchange like two-qubit decoherence channel, implying correlated errors across the two spins. The generalisation to multiple fluctuators, including spectral distributions of fluctuators completed the model. The model was then used to examine the effect of the charge noise from a single fluctuator on CNOT operation fidelity, comparing an uncorrected gate to the robust gate implemented in chapter 5. In the case of the uncorrected gate the analytic solution was in perfect agreement with expected results, achieved by numerically averaging over many noise histories. Simulation of the robust 108 Conclusions gate required applying multiple instances of the superoperators, each separated by non-commuting operations. This introduced non-physical cross terms into the analytic solution, with greatest effect in the small λ limit. By removing these cross terms we constructed an approximate analytic solution, which was in good agreement with expected results, particularly as λ became large. As expected, the robust gate clearly outperformed the uncorrected gate in the slow fluctuator regime. Provided a small amount of uncertainty existed in the coupling between qubits, the robust gate achieved a higher fidelity in the fast fluctuator regime also. No advantage was gained from the robust gate when fluctuations occurred on the time scale of the gate operation. Extensions of this work include applying the noise model to other two-qubit operations in a range of spin based solid-state architectures to determine the effects of charge noise in these systems. Testing the response of QEC codes to charge noise is of particular interest. A more immediate goal may be to relax some of the restrictions imposed in the comparison we have already performed. Bibliography

[1] B. E. Kane. A silicon-based nuclear spin quantum computer. Nature, 393(6681):133, 1998.

[2] C. D. Hill, L. C. L. Hollenberg, A. G. Fowler, C. J. Wellard, A. D. Greentree, and H. S. Goan. Global control and fast solid-state donor electron spin quantum computing. Phys. Rev. B, 72(4):045350, 2005.

[3] L. C. L. Hollenberg, C. J. Wellard, C. I. Pakes, and A. G. Fowler. Single- spin readout for buried dopant semiconductor qubits. Phys. Rev. B, 69(23):233301, 2004.

[4] M. J. Testolin, A. D. Greentree, C. J. Wellard, and L. C. L. Hollen- berg. Optically induced spin-to-charge transduction in donor-spin read- out. Phys. Rev. B, 72(19):195325, 2005.

[5] M. J. Testolin, C. D. Hill, C. J. Wellard, and L. C. L. Hollenberg. Robust controlled-NOT gate in the presence of large fabrication-induced variations of the exchange interaction strength. Phys. Rev. A, 76(1):012302, 2007.

[6] M. A. Nielsen and I. L. Chuang. Quantum computation and quantum information. Cambridge University Press, 2000.

[7] G. E. Moore. Cramming more components onto integrated circuits. Elec- tronics, 38(8):114, 1965.

[8] R. R. Schaller. Moore’s law: past, present, and future. IEEE Spectr., 34(6):52, 1997. 110 BIBLIOGRAPHY

[9] R. P. Feynman. Simulating physics with computers. Int. J. Theor. Phys., 21(6-7):467, 1982.

[10] D. Deutsch. Quantum-theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A, 400(1818):97, 1985.

[11] D. Deutsch and R. Jozsa. Rapid solution of problems by quantum computation. Proc. R. Soc. Lond. A, 439(1907):553, 1992.

[12] P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput., 26(5):1484, 1997.

[13] L. K. Grover. Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett., 79(2):325, 1997.

[14] A. R. Calderbank and P. W. Shor. Good quantum error-correcting codes exist. Phys. Rev. A, 54(2):1098, 1996.

[15] P. W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A, 52(4):R2493, 1995.

[16] A. M. Steane. Error correcting codes in quantum theory. Phys. Rev. Lett., 77(5):793, 1996.

[17] P. O. Boykin, T. Mor, M. Pulver, V. Roychowdhury, and F. Vatan. On universal and fault-tolerant quantum computing: a novel basis and a new constructive proof of universality for Shor’s basis. Proc. 40th Ann. Symp. on Foundations of Computer Science, page 486, 1999.

[18] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter. Elementary gates for quantum computation. Phys. Rev. A, 52(5):3457, 1995.

[19] P. W. Shor. Fault-tolerant quantum computation. Proc. 37th Ann. Symp. on Foundations of Computer Science, page 56, 1996. BIBLIOGRAPHY 111

[20] A. Y. Kitaev. Quantum computations: algorithms and error correction. Russ. Math. Surv., 52(6):1191, 1997.

[21] D. Deutsch, A. Barenco, and A. Ekert. Universality in quantum computation. Proc. R. Soc. Lond. A, 449(1937):669, 1995.

[22] S. Lloyd. Almost any quantum logic gate is universal. Phys. Rev. Lett., 75(2):346, 1995.

[23] D. P. DiVincenzo. The physical implementation of quantum computation. Fortschr. Phys., 48(9-11):771, 2000.

[24] I. L. Chuang and Y. Yamamoto. Simple quantum computer. Phys. Rev. A, 52(5):3489, 1995.

[25] E. Knill, R. Laﬂamme, and G. J. Milburn. A scheme for eﬃcient quantum computation with linear optics. Nature, 409(6816):46, 2001.

[26] J. I. Cirac and P. Zoller. Quantum computations with cold trapped ions. Phys. Rev. Lett., 74(20):4091, 1995.

[27] N. A. Gershenfeld and I. L. Chuang. Bulk spin-resonance quantum computation. Science, 275(5298):350, 1997.

[28] D. G. Cory, A. F. Fahmy, and T. F. Havel. Ensemble quantum computing by NMR spectroscopy. Proc. Natl. Acad. Sci. USA, 94(5):1634, 1997.

[29] J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. van der Wal, and S. Lloyd. Josephson persistent-current qubit. Science, 285(5430):1036, 1999.

[30] Y. Nakamura, Y. A. Pashkin, and J. S. Tsai. Coherent control of macroscopic quantum states in a single-Cooper-pair box. Nature, 398(6730):786, 1999.

[31] R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roy- chowdhury, T. Mor, and D. DiVincenzo. Electron-spin-resonance transistors for quantum computing in silicon-germanium heterostructures. Phys. Rev. A, 62(1):012306, 2000. 112 BIBLIOGRAPHY

[32] D. Loss and D. P. DiVincenzo. Quantum computation with quantum dots. Phys. Rev. A, 57(1):120, 1998.

[33] M. Friesen, P. Rugheimer, D. E. Savage, M. G. Lagally, D. W. van der Weide, R. Joynt, and M. A. Eriksson. Practical design and simulation of silicon-based quantum-dot qubits. Phys. Rev. B, 67(12):121301, 2003.

[34] P. Domokos, J. M. Raimond, M. Brune, and S. Haroche. Simple cavity- QED 2-bit universal quantum logic gate - the principle and expected performances. Phys. Rev. A, 52(5):3554, 1995.

[35] T. Pellizzari, S. A. Gardiner, J. I. Cirac, and P. Zoller. Decoherence, continuous observation, and quantum computing - a cavity QED model. Phys. Rev. Lett., 75(21):3788, 1995.

[36] L. C. L. Hollenberg, A. D. Greentree, A. G. Fowler, and C. J. Wellard. Two-dimensional architectures for donor-based quantum computing. Phys. Rev. B, 74(4):045311, 2006.

[37] H. S. Goan and G. J. Milburn. Silicon-based electron-mediated nuclear spin quantum computer. Unpublished, 2000.

[38] R. de Sousa, J. D. Delgado, and S. Das Sarma. Silicon quantum computation based on magnetic dipolar coupling. Phys. Rev. A, 70(5):052304, 2004.

[39] A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin, and A. M. Raitsimring. Electron spin relaxation times of phosphorus donors in silicon. Phys. Rev. B, 68(19):193207, 2003.

[40] D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui. Single spin detection by magnetic resonance force microscopy. Nature, 430(6997):329, 2004.

[41] A. D. Greentree, A. R. Hamilton, L. C. L. Hollenberg, and R. G. Clark. Electrical readout of a spin qubit without double occupancy. Phys. Rev. B, 71(11):113310, 2005. BIBLIOGRAPHY 113

[42] A. J. Skinner, M. E. Davenport, and B. E. Kane. Hydrogenic spin quantum computing in silicon: A digital approach. Phys. Rev. Lett., 90(8):087901, 2003.

[43] E. Pazy, E. Biolatti, T. Calarco, I. D’Amico, P. Zanardi, F. Rossi, and P. Zoller. Spin-based optical quantum computation via Pauli blocking in semiconductor quantum dots. Europhys. Lett., 62(2):175, 2003.

[44] J. M. Elzerman, R. Hanson, L. H. W. van Beveren, B. Witkamp, L. M. K. Vandersypen, and L. P. Kouwenhoven. Single-shot read-out of an individual electron spin in a quantum dot. Nature, 430(6998):431, 2004.

[45] F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup. Observa- tion of coherent oscillations in a single electron spin. Phys. Rev. Lett., 92(7):076401, 2004.

[46] M. Friesen, C. Tahan, R. Joynt, and M. A. Eriksson. Spin readout and initialization in a semiconductor quantum dot. Phys. Rev. Lett., 92(3):037901, 2004.

[47] S. D. Barrett and T. M. Stace. Two-spin measurements in exchange interaction quantum computers. Phys. Rev. B, 73(7):075324, 2006.

[48] R. Ionicioiu and A. E. Popescu. Single-spin measurement using spin- orbital entanglement. New J. Phys., 7:120, 2005.

[49] M. Taniguchi and S. Narita. D− state in silicon. Solid State Commun., 20(2):131, 1976.

[50] P. Norton. Formation of upper Hubbard band from negative-donor-ion states in silicon. Phys. Rev. Lett., 37(3):164, 1976.

[51] T. M. Buehler, D. J. Reilly, R. P. Starrett, A. D. Greentree, A. R. Hamil- ton, A. S. Dzurak, and R. G. Clark. Single-shot readout with the radio- frequency single-electron transistor in the presence of charge noise. Appl. Phys. Lett., 86(14):143117, 2005. 114 BIBLIOGRAPHY

[52] G. A. Thomas, M. Capizzi, F. DeRosa, R. N. Bhatt, and T. M. Rice. Optical study of interacting donors in semiconductors. Phys. Rev. B, 23(10):5472, 1981.

[53] R. L. Aggarwal and A. K. Ramdas. Optical determination of symmetry of ground states of group-V donors in silicon. Phys. Rev., 140(4A):A1246, 1965.

[54] S. T. Pantelides. Electronic-structure of impurities and other point- defects in semiconductors. Rev. Mod. Phys., 50(4):797, 1978.

[55] F. Bloch. Uber¨ die quantenmechanik der elektronen in kristallgittern. Z. Phys. A, 52:555, 1928.

[56] Peter Y. Yu and Manuel Cardona. Fundamentals of semiconductors: physics and materials properties. Springer, 2001.

[57] J. M. Luttinger and W. Kohn. Motion of electrons and holes in perturbed periodic ﬁelds. Phys. Rev., 97(4):869, 1955.

[58] W. Kohn and J. M. Luttinger. Theory of donor levels in silicon. Phys. Rev., 97(6):1721, 1955.

[59] W. Kohn and J. M. Luttinger. Theory of donor states in silicon. Phys. Rev., 98(4):915, 1955.

[60] W. Kohn. Shallow impurity states in silicon and germanium. Solid State Physics-Advances in Research and Applications, 5:257, 1957.

[61] C. Kittel and A. H. Mitchell. Theory of donor and acceptor states in silicon and germanium. Phys. Rev., 96(6):1488, 1954.

[62] M. A. Lampert. Ground state of impurity atoms in semiconductors having anisotropic energy surfaces. Phys. Rev., 97(2):352, 1955.

[63] R. L. Aggarwal. Optical determination of the valley-orbit splitting of the ground state of donors in silicon. Solid State Commun., 2(6):163, 1964. BIBLIOGRAPHY 115

[64] D. M. Larsen. Stress dependence of the binding-energy of D− centers in Si. Phys. Rev. B, 23(10):5521, 1981.

[65] C. J. Wellard, L. C. L. Hollenberg, F. Parisoli, L. M. Kettle, H. S. Goan, J. A. L. McIntosh, and D. N. Jamieson. Electron exchange coupling for single-donor solid-state spin qubits. Phys. Rev. B, 68(19):195209, 2003.

[66] M. J. Testolin. Donor electron wave functions and read-out in the Kane quantum computer. Honours report, The University of Melbourne, 2002.

[67] A. Baldereschi. Valley-orbit interaction in semiconductors. Phys. Rev. B, 1(12):4673, 1970.

[68] G. Feher. Electron spin resonance experiments on donors in silicon. I. Electronic structure of donors by the electron nuclear double resonance technique. Phys. Rev., 114(5):1219, 1959.

[69] D. K. Wilson and G. Feher. Electron spin resonance experiments on donors in silicon. III. Investigation of excited states by application of uniaxial stress and their importance in relaxation processes. Phys. Rev., 124(4):1068, 1961.

[70] K.-M. C. Fu, T. D. Ladd, C. Santori, and Y. Yamamoto. Optical detection of the spin state of a single nucleus in silicon. Phys. Rev. B, 69(12):125306, 2004.

[71] A. Shabaev, A. L. Efros, D. Gammon, and I. A. Merkulov. Optical readout and initialization of an electron spin in a single quantum dot. Phys. Rev. B, 68(20):201305, 2003.

[72] A. A. Larionov, L. E. Fedichkin, and K. A. Valiev. A silicon-based nuclear magnetic resonance (NMR) quantum computer using resonant transfer of a single electron for the inter-qubit interaction. Nanotechnol- ogy, 12(4):536, 2001.

[73] T. Rudolph and S. S. Virmani. A relational quantum computer using only two-qubit total spin measurement and an initial supply of highly mixed single-qubit states. New J. Phys., 7:228, 2005. 116 BIBLIOGRAPHY

[74] A. B. Fang, Y. C. Chang, and J. R. Tucker. Eﬀects of J-gate potential and uniform electric ﬁeld on a coupled donor pair in Si for quantum computing. Phys. Rev. B, 66(15):155331, 2002.

[75] P. L. Knight and P. W. Milonni. The Rabi frequency in optical-spectra. Phys. Rep., 66(2):21, 1980.

[76] H. Sigg, H. J. A. Bluyssen, and P. Wyder. New laser lines with wavelengths from λ = 61.7 µm down to λ = 27.7 µm in optically pumped

CH3OH and CD3OH. IEEE J. Quantum Electron., 20(6):616, 1984.

[77] S. G. Pavlov, R. K. Zhukavin, E. E. Orlova, V. N. Shastin, A. V. Kir- sanov, H. W. Hubers, K. Auen, and H. Riemann. Stimulated emission from donor transitions in silicon. Phys. Rev. Lett., 84(22):5220, 2000.

[78] G. D. J. Smit, S. Rogge, J. Caro, and T. M. Klapwijk. Stark eﬀect in shallow impurities in Si. Phys. Rev. B, 70(3):035206, 2004.

[79] M. F. Doty, B. E. Cole, B. T. King, and M. S. Sherwin. Wavelength- speciﬁc laser-activated switches for improved contrast ratio in generation of short THz pulses. Rev. Sci. Instrum., 75(9):2921, 2004.

[80] F. A. Hegmann and M. S. Sherwin. Generation of picosecond far-infrared pulses using laser activated semiconductor reﬂection switches. Proc. SPIE, 2842:90, 1996.

[81] A. G. Fowler. (unpublished).

[82] E. Abe, K. M. Itoh, T. D. Ladd, J. R. Goldman, F. Yamaguchi, and Y. Yamamoto. Solid-state silicon NMR quantum computer. J. Super- cond., 16(1):175, 2003.

[83] J. R. Goldman, T. D. Ladd, F. Yamaguchi, and Y. Yamamoto. Magnet designs for a crystal-lattice quantum computer. Appl. Phys. A-Mater. Sci. Process., 71(1):11, 2000. BIBLIOGRAPHY 117

[84] A. D. Greentree, J. H. Cole, A. R. Hamilton, and L. C. L. Hollenberg. Coherent electronic transfer in quantum dot systems using adiabatic passage. Phys. Rev. B, 70(23):235317, 2004.

[85] D. A. Lidar and J. H. Thywissen. Exponentially localized magnetic ﬁelds for single-spin quantum logic gates. J. Appl. Phys., 96(1):754, 2004.

[86] B. Koiller, X. D. Hu, and S. Das Sarma. Exchange in silicon-based quantum computer architecture. Phys. Rev. Lett., 88(2):027903, 2002.

[87] C. J. Wellard, L. C. L. Hollenberg, L. M. Kettle, and H. S. Goan. Voltage control of exchange coupling in phosphorus doped silicon. J. Phys.: Condens. Matter, 16(32):5697, 2004.

[88] B. Koiller, X. D. Hu, and S. Das Sarma. Strain eﬀects on silicon donor exchange: Quantum computer architecture considerations. Phys. Rev. B, 66(11):115201, 2002.

[89] S. R. Schoﬁeld, N. J. Curson, M. Y. Simmons, F. J. Ruess, T. Hallam, L. Oberbeck, and R. G. Clark. Atomically precise placement of single dopants in Si. Phys. Rev. Lett., 91(13):136104, 2003.

[90] J. A. Jones. Robust Ising gates for practical quantum computation. Phys. Rev. A, 67(1):012317, 2003.

[91] C. D. Hill. Robust controlled-NOT gates from almost any interaction. Phys. Rev. Lett., 98(18):180501, 2007.

[92] H. K. Cummins and J. A. Jones. Use of composite rotations to correct systematic errors in NMR quantum computation. New J. Phys., 2:1, 2000.

[93] E. Collin, G. Ithier, A. Aassime, P. Joyez, D. Vion, and D. Esteve. NMR- like control of a quantum bit superconducting circuit. Phys. Rev. Lett., 93(15):157005, 2004. 118 BIBLIOGRAPHY

[94] J. J. L. Morton, A. M. Tyryshkin, A. Ardavan, K. Porfyrakis, S. A. Lyon, and G. A. D. Briggs. High ﬁdelity single qubit operations using pulsed electron paramagnetic resonance. Phys. Rev. Lett., 95(20):200501, 2005.

[95] M. Riebe, H. Haﬀner, C. F. Roos, W. Hansel, J. Benhelm, G. P. T. Lancaster, T. W. Korber, C. Becher, F. Schmidt-Kaler, D. F. V. James, and R. Blatt. Deterministic quantum teleportation with atoms. Nature, 429(6993):734, 2004.

[96] F. Schmidt-Kaler, H. Haﬀner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt. Re- alization of the Cirac-Zoller controlled-NOT quantum gate. Nature, 422(6930):408, 2003.

[97] S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Haﬀner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt. Implementa- tion of the Deutsch-Jozsa algorithm on an ion-trap quantum computer. Nature, 421(6918):48, 2003.

[98] S. G. Schirmer, A. Kolli, and D. K. L. Oi. Experimental Hamil- tonian identiﬁcation for controlled two-level systems. Phys. Rev. A, 69(5):050306, 2004.

[99] J. H. Cole, S. J. Devitt, and L. C. L. Hollenberg. Precision characteriza- tion of two-qubit Hamiltonians via entanglement mapping. J. Phys. A: Math. Gen., 39(47):14649, 2006.

[100] S. J. Devitt, J. H. Cole, and L. C. L. Hollenberg. Scheme for direct measurement of a general two-qubit Hamiltonian. Phys. Rev. A, 73(5):052317, 2006.

[101] 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 Hamil- tonian to arbitrary accuracy. Phys. Rev. A, 71(6):062312, 2005. BIBLIOGRAPHY 119

[102] 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. Phys. Rev. A, 73(6):062333, 2006.

[103] M. H. Levitt. Composite pulses. Prog. Nucl. Magn. Reson. Spectrosc., 18:61, 1986.

[104] S. Wimperis. Broad-band, narrow-band, and passband composite pulses for use in advanced NMR experiments. J. Magn. Reson. Ser. A, 109(2):221, 1994.

[105] M. J. Bremner, J. L. Dodd, M. A. Nielsen, and D. Bacon. Fungible dynamics: There are only two types of entangling multiple-qubit interactions. Phys. Rev. A, 69(1):012313, 2004.

[106] H. K. Cummins, G. Llewellyn, and J. A. Jones. Tackling systematic errors in quantum logic gates with composite rotations. Phys. Rev. A, 67(4):042308, 2003.

[107] L. Xiao and J. A. Jones. Robust logic gates and realistic quantum computation. Phys. Rev. A, 73(3):032334, 2006.

[108] A. S. Dzurak, M. Y. Simmons, A. R. Hamilton, R. G. Clark, R. Brenner, T. M. Buehler, N. J. Curson, E. Gauja, R. P. McKinnon, L. D. Macks, M. Mitic, J. L. O’Brien, L. Oberbeck, D. J. Reilly, S. R. Schoﬁeld, F. E. Stanley, D. N. Jamieson, S. Prawer, C. Yang, and G. J. Milburn. Con- struction of a silicon-based solid state quantum computer. Quantum Inf. Comput., 1:82, 2001.

[109] D. N. Jamieson, C. Yang, T. Hopf, S. M. Hearne, C. I. Pakes, S. Prawer, M. Mitic, E. Gauja, S. E. Andresen, F. E. Hudson, A. S. Dzurak, and R. G. Clark. Controlled shallow single-ion implantation in silicon using an active substrate for sub-20-keV ions. Appl. Phys. Lett., 86(20):202101, 2005. 120 BIBLIOGRAPHY

[110] C. J. Wellard and L. C. L. Hollenberg. Donor electron wave functions for phosphorus in silicon: Beyond eﬀective-mass theory. Phys. Rev. B, 72(8):085202, 2005.

[111] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac. Improvement of frequency standards with quantum entanglement. Phys. Rev. Lett., 79(20):3865, 1997.

[112] Daniel Gottesman. Stabilizer Codes and Quantum Error Correction. PhD thesis, California Institute of Technology, 1997.

[113] P. Aliferis, D. Gottesman, and J. Preskill. Quantum accuracy threshold for concatenated distance-3 codes. Quantum Inf. Comput., 6(2):97, 2006.

[114] A. M. Stephens, A. G. Fowler, and L. C. L. Hollenberg. Universal fault tolerant quantum computation on bilinear nearest neighbor arrays. Quantum Inf. Comput., 8:330, 2008.

[115] R. Kubo. Statistical-mechanical theory of irreversible processes. 1. Gen- eral theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn., 12(6):570, 1957.

[116] U. Fano. Description of states in quantum mechanics by density matrix and operator techniques. Rev. Mod. Phys., 29(1):74, 1957.

[117] R. Zwanzig. Ensemble method in the theory of irreversibility. J. Chem. Phys., 33(5):1338, 1960.

[118] U. Fano. Pressure broadening as a prototype of relaxation. Phys. Rev., 131(1):259, 1963.

[119] R. Zwanzig. On the identity of three generalized master equations. Phys- ica, 30(6):1109, 1964.

[120] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger. Spintronics: A spin-based electronics vision for the future. Science, 294(5546):1488, 2001. BIBLIOGRAPHY 121

[121] I. Zutic, J. Fabian, and S. Das Sarma. Spintronics: Fundamentals and applications. Rev. Mod. Phys., 76(2):323, 2004.

[122] R. de Sousa, X. D. Hu, and S. Das Sarma. Eﬀect of an inhomogeneous external magnetic ﬁeld on a quantum-dot quantum computer. Phys. Rev. A, 64(4):042307, 2001.

[123] W. A. Coish and D. Loss. Exchange-controlled single-electron-spin rotations in quantum dots. Phys. Rev. B, 75(16):161302, 2007.

[124] X. D. Hu and S. Das Sarma. Charge-ﬂuctuation-induced dephasing of exchange-coupled spin qubits. Phys. Rev. Lett., 96(10):100501, 2006.

[125] M. Mottonen, R. de Sousa, J. Zhang, and K. B. Whaley. High-ﬁdelity one-qubit operations under random telegraph noise. Phys. Rev. A, 73(2):022332, 2006.

[126] Richard E. Barlow, Frank Proschan, and Larry C. Hunter. Mathematical theory of reliability. SIAM, 1996.

[127] T. Aven and U. Jensen. Stochastic models in reliability. Springer, 1999.

[128] Randolph Nelson. Probability, stochastic processes, and queueing theory: the mathematics of computer performance modelling. Springer-Verlag, 1995.

[129] L. Tak´acs. On certain sojourn time problems in the theory of stochastic processes. Acta Math. Hung., 8(1):169, 1957.

[130] D. Perry, W. Stadje, and S. Zacks. Contributions to the theory of ﬁrst- exit times of some compound processes in queueing theory. Queueing Syst., 33(4):369, 1999.

[131] S. Zacks. Generalized integrated telegraph processes and the distribution of related stopping times. J. Appl. Probab., 41(2):497, 2004.

[132] W. Happer and A. C. Tam. Eﬀect of rapid spin exchange on magnetic- resonance spectrum of alkali vapors. Phys. Rev. A, 16(5):1877, 1977. 122 BIBLIOGRAPHY

[133] G. Lindblad. Generators of quantum dynamical semigroups. Commun. Math. Phys., 48(2):119, 1976.

[134] C. W. Gardiner. Quantum noise. Springer-Verlag, 1991.

[135] J. F. Verwey, E. A. Amerasekera, and J. Bisschop. The physics of SiO2 layers. Rep. Prog. Phys., 53(10):1297, 1990.

[136] C. R. Helms and E. H. Poindexter. The silicon silicon-dioxide system: its microstructure and imperfections. Rep. Prog. Phys., 57(8):791, 1994.

[137] D. R. McCamey, M. Francis, J. C. McCallum, R. Hamilton, A. D. Green-

tree, and R. G. Clark. Donor activation and damage in Si-SiO2 from low-dose, low-energy ion implantation studied via electrical transport in MOSFETs. Semicond. Sci. Technol., 20(5):363, 2005.

[138] G. Scappucci, F. Ratto, D. L. Thompson, T. C. G. Reusch, W. Pok, F. J. Ruess, F. Rosei, and M. Y. Simmons. Structural and electrical charac-

terization of room temperature ultra-high-vacuum compatible SiO2 for gating scanning tunneling microscope-patterned devices. Appl. Phys. Lett., 91(22):222109, 2007.

[139] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard. Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science, 309(5744):2180, 2005.

[140] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbruggen, and S. J. Glaser. Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. J. Magn. Reson., 172(2):296, 2005.

Minerva Access is the Institutional Repository of The University of Melbourne

Author/s: Testolin, Matthew J.

Title: Realistic read-out and control for Si:P based quantum computers

Date: 2008

Citation: Testolin, M. J. (2008). Realistic read-out and control for Si:P based quantum computers. PhD thesis, Faculty of Science, Physics, University of Melbourne.

Publication Status: Unpublished

Persistent Link: http://hdl.handle.net/11343/39492

File Description: Realistic read-out and control for Si:P based quantum computers

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