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Encoding a Qubit Into an Oscillator with Near-Term Experimental Devices Delft University of Technology Encoding a Qubit into an Oscillator with Near-Term Experimental Devices Weigand, D.J. DOI 10.4233/uuid:72abf99f-dd2d-42a1-8c59-7a83870c9d3c Publication date 2020 Document Version Final published version Citation (APA) Weigand, D. J. (2020). Encoding a Qubit into an Oscillator with Near-Term Experimental Devices. https://doi.org/10.4233/uuid:72abf99f-dd2d-42a1-8c59-7a83870c9d3c Important note To cite this publication, please use the final published version (if applicable). Please check the document version above. Copyright Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim. This work is downloaded from Delft University of Technology. For technical reasons the number of authors shown on this cover page is limited to a maximum of 10. ENCODING A QUBIT INTO AN OSCILLATOR WITH NEAR-TERM EXPERIMENTAL DEVICES Dissertation for the purpose of obtaining the degree of doctor at Delft University of Technology by the authority of the Rector Magnificus, Prof. dr. ir. T.H.J.J. van der Hagen, chair of the Board for Doctorates to be defended publicly on Wednesday 2 December 2020 at 10:00 o’clock by Daniel Jonathan WEIGAND Master of Science RWTH Aachen University, RWTH Aachen University, Germany, born in Neustadt an der Weinstraße, Germany. This dissertation has been approved by the promotor. Composition of the doctoral committee: Rector magnificus, chairman Prof. dr. B. M. Terhal, Delft University of Technology, promotor Independent Members: Dr. P.Campagne-Ibarcq INRIA Paris Prof. dr. A. L. Grimsmo University of Sydney Prof. dr. R. Hanson Delft University of Technology Prof. dr. M. Müller RWTH Aachen University Prof. dr. G. A. Steele Delft University of Technology Keywords: Quantum Error Correction, Bosonic Codes, Circuit QED Front & Back: The Wigner function of an approximate GKP state arising from squeezed vacuum states. Copyright © 2020 by D. J. Weigand ISBN 978-94-6421-139-9 An electronic version of this dissertation is available at http://repository.tudelft.nl/. CONTENTS Summary ix Samenvatting xi Preface xiii 1 Introduction1 1.1 Error Correction..............................4 1.1.1 Classical Error Correction......................4 1.1.2 Quantum Error Correction......................5 1.1.3 Quantum Error Correction with Superconducting Qubits......7 1.2 Bosonic Encodings of Qubits........................9 1.3 Applications Related to the Gottesman - Kitaev - Preskill (GKP) Code.... 13 1.3.1 Maximal Violation of Bell Inequalities................ 14 1.3.2 The von Neumann Lattice...................... 14 1.3.3 Displacement Sensing Below the Heisenberg Limit......... 15 1.3.4 Protecting Gaussian Information.................. 15 1.4 Outline of the Thesis............................ 16 2 Theoretical Background 23 2.1 Fundamental Definitions and Derivations................. 24 2.1.1 Quantum Optics........................... 24 2.1.2 The Gottesman-Kitaev-Preskill (GKP) Code............. 25 2.1.3 Stabilizers, Logical Operations, and Code States of the GKP Code.. 27 2.1.4 Logical and Physical Qubits..................... 27 2.2 Representations of GKP Code States.................... 28 2.2.1 Position or Momentum Wave Function............... 28 2.2.2 Shifted Code States.......................... 30 2.2.3 GKP Code States in the Fock Basis.................. 32 2.3 Quality Measures for the GKP Code..................... 33 2.3.1 Ideal Decoder............................ 33 2.3.2 Why the Overlap is Not a Good quality Measure........... 34 2.3.3 Holevo Phase Variance........................ 35 2.3.4 Characterizing Wave Functions in a Quadrature Basis........ 37 3 Encoding a GKP Qubit using Phase Estimation 43 3.1 Phase Estimation.............................. 44 3.1.1 Standard Phase Estimation...................... 44 3.1.2 Phase Estimation by Repetition................... 46 3.1.3 Adaptive Phase Estimation...................... 46 3.1.4 Maximal Information Gain Strategy................. 49 v vi CONTENTS 3.2 Implementations of a Controlled Displacement.............. 50 3.2.1 Direct Controlled Displacement................... 50 3.2.2 Controlled Displacement by Rotation................ 51 3.2.3 Controlled Displacements in Trapped Ions.............. 55 3.3 Numerical Analysis of Noise During Preparation and Measurement.... 55 3.3.1 Choice of Phase Estimation..................... 56 3.3.2 Hamiltonian and Noise Model.................... 56 3.3.3 Simulation Results.......................... 58 4 Generating Grid States from Schrödinger Cat States without Post-Selection 63 4.1 Background................................. 65 4.1.1 Grid States.............................. 65 4.1.2 Effective Squeezing Parameters................... 67 4.1.3 Adaptive Phase Estimation...................... 68 4.2 Breeding.................................. 69 4.2.1 Slow Breeding............................ 70 4.2.2 Efficient Breeding.......................... 71 4.3 Analytic Discussion of Breeding....................... 74 4.3.1 Scale-Dependent Shifted Grid States................. 75 4.3.2 Breeding Shifted Grid States..................... 76 4.3.3 Choice of Wave Function £(u,v)................... 77 4.3.4 Choice for Wave Function V (u)................... 79 4.4 Asymptotic Behavior............................ 80 4.5 Simulation................................. 83 4.6 Discussion................................. 85 5 Realizing Modular Quadrature Measurements via a Tunable Photon-Pressure Coupling in Circuit-QED 89 5.1 Preliminaries................................ 91 5.2 Overview of Measurement Protocol..................... 92 5.2.1 GKP Qubit Readout......................... 94 5.2.2 Why Probing the Ancilla Oscillator’s Frequency Reveals the Wrong Information............................. 94 5.3 Circuit-QED Setup............................. 96 5.3.1 Previous Circuit-QED Work on Photon-Pressure and Longitudinal Coupling............................... 96 5.3.2 Mechanical Analogue........................ 97 5.3.3 Circuit Analysis and Approximations................. 98 5.3.4 Release of Ancilla Oscillator State.................. 104 5.4 Parametric Flux Drive............................ 104 5.4.1 Parametric Flux Drive........................ 105 5.4.2 Use of a Microwave Drive...................... 107 5.5 Modeling The Modular Quadrature Measurement............. 107 5.5.1 Effective Squeezing......................... 108 5.5.2 Measurement Squeezing Strength.................. 109 CONTENTS vii 5.6 Heterodyne Measurement via Release of Coherent Oscillator State into a Transmission Line............................. 112 5.7 Effective Squeezing with Vacuum Input State................ 116 5.8 Noise and Imperfections.......................... 117 5.8.1 Photon Loss in Ancilla Oscillator during Photon-Pressure Interac- tion................................. 118 5.8.2 Comparison with Sequential-Qubit Phase Estimation Measurement and Photon Loss on Target Oscillator................ 119 5.8.3 Readout Loss............................. 121 5.8.4 Expansion of the Circuit Hamiltonian................ 122 5.8.5 Third-Order Nonlinearity...................... 123 5.8.6 Flux Noise.............................. 125 5.8.7 Flux Noise with Reduced Drive Amplitude.............. 126 5.9 Discussion................................. 126 6 Conclusion and Outlook 133 6.1 Conclusion................................. 134 6.2 Outlook................................... 136 A Appendix 139 A.1 Details of Numerical Simulations...................... 140 A.1.1 Stabilizer Measurements using Phase Estimation.......... 140 A.1.2 Simulation of Breeding........................ 140 A.1.3 Simulations of Modular Quadrature Measurements......... 142 A.2 Relating a Photon-Pressure Coupling to Phase Estimation......... 143 A.2.1 Initial State.............................. 143 A.2.2 Photon-Pressure Hamiltonian.................... 144 A.2.3 Heterodyne Measurement...................... 145 Curriculum Vitæ 149 List of Publications 151 SUMMARY A universal, large-scale quantum computer would be a powerful tool with applications of high value to mankind. For example, such a computer could significantly speed up the search for new medications or materials. However, the error rates of current qubit designs are simply too large to enable interesting computations. Therefore, both error correction and improved designs of qubits are needed. In 2001, Gottesman, Kitaev and Preskill proposed an encoding (GKP code) where a qubit is stored in a harmonic oscillator — a system that can be controlled and manu- factured with high precision, and therefore have comparatively high coherence times. Moreover, the code offers good protection against losses, a simple gate set, and error correction circuits that are comparatively easy to implement. The drawback is that en- coding a qubit into a GKP code state is a challenging task. In this thesis, we develop efficient schemes to encode a GKP qubit. Bosonic codes, where a qubit is stored in an oscillator, and in particular the GKP code are still relatively unknown. Therefore, we will start the thesis with an overview of the field, and provide the reader with the tools to analyze a GKP code, as these are quite different from standard
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