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Controlling Quantum Information Devices Controlling Quantum Information Devices by Felix Motzoi A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 2012 c Felix Motzoi 2012 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Quantum information and quantum computation are linked by a common math- ematical and physical framework of quantum mechanics. The manipulation of the predicted dynamics and its optimization is known as quantum control. Many tech- niques, originating in the study of nuclear magnetic resonance, have found common usage in methods for processing quantum information and steering physical systems into desired states. This thesis expands on these techniques, with careful atten- tion to the regime where competing effects in the dynamics are present, and no semi-classical picture exists where one effect dominates over the others. That is, the transition between the diabatic and adiabatic error regimes is examined, with the use of such techniques as time-dependent diagonalization, interaction frames, average- Hamiltonian expansion, and numerical optimization with multiple time-dependences. The results are applied specifically to superconducting systems, but are general and improve on existing methods with regard to selectivity and crosstalk problems, filter- ing of modulation of resonance between qubits, leakage to non-compuational states, multi-photon virtual transitions, and the strong driving limit. iii Acknowledgements This research would not have been possible without the support and hard work of my collaborators and supervisors. I would like to thank my supervisor Frank Wilhelm for his openness to taking a chance on me on as a graduate student. His leadership, methodology, and insight have all had a significant impact on my personal and academic development. My secondary supervisor, Jay Gambetta, has also been instrumental to the main lines of thought in this research and his vision and guidance as well as his perseverence have opened many opportunities (and driven some home run balls!). I would also like to thank Seth Merkel for his guidance and feedback. I have also been fortunate to help assist other graduate students in common research. Peter Groskowski, Botan Khani, Amira Eltony, Yuval Sanders, Likun Hu, Victor Vetch, and Jerry Chow have all collaborated in quantum control projects that have generated many stimulating and valuable conversations. Finally, I would like to thank the fellow graduate students in 2117 for some freshness and brightness in a room that had neither. This research was made possible by NSERC through the discovery grants, QuantumWorks, and a PGS scholarship. iv Dedication This thesis is dedicated to my three muses: my mother Ina who would have written it herself if she needed to, my sister Clairneige who tirelessly listened to rants and ravings about the endless "wonders" of quantum physics growing up, and my girlfriend Kanako who carried me through these arduous Ph.D. years. v Contents List of Figures xi 1 Introduction1 2 Universal Quantum Computation 10 2.1 Quantum gates.............................. 10 2.1.1 Superposition and measurement................. 10 2.1.2 Unitary evolution......................... 11 2.1.3 Hamiltonians........................... 13 2.1.4 Non-unitary evolution...................... 14 2.2 Quantum software............................ 16 2.2.1 Assembly............................. 16 2.2.2 Outlook on algorithms...................... 17 2.3 Quantum hardware............................ 18 2.3.1 Candidate implementations................... 19 2.3.1.1 Superconducting qubits................ 19 2.3.2 Controllability........................... 20 2.3.3 Coupling topologies........................ 22 2.3.4 Coupling mechanisms....................... 22 2.4 Quantum errors.............................. 24 2.4.1 Worst-case error.......................... 24 2.4.2 Threshold theorem........................ 25 2.4.3 Average error........................... 25 3 Elements of Quantum Control 27 3.1 Time-dependent fields.......................... 29 3.2 Selection error............................... 30 3.3 Off-resonant operator error........................ 31 3.4 Resonant operator error......................... 33 vi CONTENTS 3.5 Static non-deterministic error...................... 33 3.6 Fast non-deterministic error....................... 34 4 Frame Transformations 35 4.1 General theory.............................. 35 4.1.1 Transformation formula..................... 36 4.1.2 Equivalent frames......................... 37 4.1.3 Interaction frames......................... 38 4.2 Phase transformations.......................... 38 4.2.1 Phase tracking.......................... 38 4.2.2 Rotating frames.......................... 39 4.2.3 Phase ramping.......................... 40 4.3 Diagonalization.............................. 41 4.3.1 Adiabatic expansion....................... 41 4.3.2 Interaction frame......................... 42 4.4 Magnus expansion............................ 42 4.4.1 Interaction frame......................... 43 5 Numerical Simulation and Optimization 44 5.1 Time-slicing................................ 44 5.2 Cost functions............................... 45 5.2.1 Overlap fidelity.......................... 45 5.2.2 Subspace fidelity......................... 46 5.2.3 Ensemble fidelity......................... 46 5.2.4 Penalties.............................. 47 5.3 Gradient Ascent.............................. 48 5.3.1 Non-unitary control........................ 50 5.3.2 Robust control.......................... 51 5.4 Convergence improvements........................ 51 6 Waveform Shaping 53 6.1 Introduction................................ 53 6.2 Waveform generator example....................... 54 6.3 Numerical optimization with fast fields................. 55 6.4 Response functions............................ 56 6.4.1 Cubic spline interpolation.................... 57 6.4.2 Filter function........................... 59 6.4.3 Fourier components........................ 60 6.5 Filtered amplitude-modulated coupling................. 61 vii CONTENTS 6.6 Filtered frequency-modulated coupling................. 64 6.7 Application to 2-qubit coupling..................... 66 6.7.1 Physical example: capacitively coupled phase qubits...... 67 6.7.2 Reduction to single qubit analytic treatment.......... 69 6.7.3 Numerical optimization..................... 69 6.8 Summary................................. 70 7 Crosstalk 72 7.1 Frequency crosstalk........................... 73 7.1.1 Selectivity criteria......................... 74 7.1.2 Semiclassical spectrum analysis................. 76 7.1.2.1 Doublet analysis.................... 77 7.1.2.2 Multiplet analysis................... 77 7.1.3 Quantum operator diagonalization............... 80 7.1.3.1 Rotating frame..................... 80 7.1.3.2 Block diagonal frame.................. 81 7.1.3.3 Solution basis of higher derivatives.......... 83 7.1.3.4 Doublet analysis.................... 84 7.1.3.5 Multiplet analysis................... 87 7.2 Spatial crosstalk.............................. 89 7.2.1 Physical Model.......................... 90 7.2.2 Coupling landscape........................ 92 7.2.3 Optimization of off-resonant coupling.............. 94 7.2.4 Optimization of on-resonance coupling............. 96 8 Leakage 98 8.1 Introduction................................ 98 8.2 Mathieu equation............................. 100 8.3 Qutrit approximation........................... 101 8.3.1 Coupling strength from dressing................. 102 8.4 Conventional control........................... 105 8.5 Adiabatic diagonalization......................... 106 8.5.1 Exact solution to the qutrit................... 110 8.6 Generalized transformation........................ 112 8.6.1 Zeroth order solutions...................... 113 8.6.2 First order solutions....................... 113 8.6.3 Second order solutions...................... 116 8.7 Numerical optimization.......................... 117 viii CONTENTS 8.7.1 Prefactor optimization...................... 117 8.7.2 Full optimization......................... 119 8.8 Incoherent effects............................. 120 8.8.1 Relaxation............................. 120 8.8.2 Dispersion............................. 122 8.8.2.1 Optimization...................... 124 8.9 Implementation.............................. 127 9 Multi-Channel Leakage 133 9.1 Duffing oscillator............................. 133 9.1.1 SNO approximation........................ 134 9.1.2 Numerical solutions........................ 136 9.1.3 Analytical solutions........................ 139 9.2 Multiple leakages from one level..................... 142 9.3 2-Qubit leakage.............................. 144 9.3.1 General Model.......................... 144 9.3.2 Numerical solution........................ 146 9.3.3 Analytical solution........................ 147 10 Virtual Transitions 150 10.1 Raman transitions............................ 151 10.2 2-Qubit coupling............................. 154 10.2.1 Ansatz solution.......................... 154 10.2.2 Optimization with two tones................... 156 10.3 Driving through a cavity......................... 157 10.4 Summary................................. 159 11 Strong Coupling 161 11.1 Analytic solution............................
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