Andreas Dewes

204 Pages·2013·21.94 MB·English

Demonstrating Quantum Speed-Up with a
Two-Transmon Quantum Processor

PhD Thesis, 2012
Andreas Dewes

Quantronics Group - CEA Saclay
Université Pierre et Marie Curie
Ecole Doctorale de Physique de la Région Parisienne - ED107
CoverIllustration: KingAegeusaskingthepriestessPythia,theOracleofDelphi,foradvice
concerning family matters. Analogously, in this work we make use of a quantum Oracle to find
the solution to a mathematical problem faster than possible when using classical computing1.
SimilartothepropheciesutteredbytheancientOracle,thesolutionsthatweobtainalsotendto
beslightlyambiguous.
1Incidentally,weusealanguagecalledPythontoconveythemessagesofourOracle.
Thèse de Doctorat de l’Université Pierre et Marie Curie
Specialité: Physique de la matière condensée
Ecole Doctorale de Physique de la Région Parisienne - ED 107
Demonstrating Quantum Speed-Up with a
Two-Transmon Quantum Processor
Présentée par
Andreas Dewes
Pourobtenirlegradede“Docteurdel’UniversitéPierreetMarieCurie”
Soutenue le 15 novembre 2012 devant le jury composé de:
Prof. Alexey Ustinov (rapporteur)
Dr. Olivier Buisson (rapporteur)
Prof. Jean-Michel Raimond
Prof. David DiVincenzo
Dr. Denis Vion
Thèse préparée au sein du Service de Physique de l’Etat Condensé,
CEA Saclay

Andreas Dewes
Acknowledgments
This thesis work would not have been possible without the invaluable help of my col-
leagues at the CEA Saclay. Special thanks go to my advisers Denis Vion, Patrice
Bertet and Daniel Esteve, without them this thesis work would simply not exist. Count-
less times they provided me with invaluable scientific and personal advise and were
always patient, optimistic and supportive, even at times when I was not.
In addition, I thank Florian Ong, who worked as a Post-Doc in the group during
the first year of my thesis and fabricated the original qubit chip with which most of the
measurements discussed in this thesis have been made. Also, I want to thank Agustin
Palacios-Laloy, who introduced me to his setup and helped me to get started with my
measurements. Furthermore,IwanttothankNicolasBoulant,who,withhisexperience
inquantumstate&processtomography,helpedmetounderstandthedataoftheiSWAP
experiment.
Last but not least, I want to thank all of my colleagues for their help and support,
as well as for the countless interesting discussions I had with them. They made the
Quantronics lab a very special and enjoyable place to work at.

Andreas Dewes
Short Summary
Demonstrating Quantum Speed-Up with a
Two-Transmon Quantum Processor
The thesis work discusses the design, realization, characterization and operation of
a two-qubitprocessor implemented usingcapacitively coupled tunablesuperconducting
qubits of the Transmon type. Each qubit can be manipulated and read out individually
using a non-destructive single-shot readout. In addition, a universal-two qubit gate can
be implemented using the interaction between the qubits. The processor implements
therefore all basic building blocks of a universal two-qubit quantum processor. Using it,
√
we implement the universal iSWAP two-qubit gate, characterizing the gate operation
by quantum process tomography and obtaining a gate fidelity of 90 %. We use this gate
tocreateentangledtwo-qubitBellstatesandperformatestoftheCHSHBellinequality,
observingaviolationoftheclassicalboundaryby22standarddeviationsaftercorrecting
for readout errors.
Usingtheimplementedtwo-qubitgate,weruntheso-calledGroversearchalgorithm:
For two-qubits, this algorithm finds among four elements x ∈ {00,01,10,11} the one
element y that solves a search problem encoded by a function f for which f(y) = 1
and f(x 6= y) = 0. Our implementation retrieves the correct answer to the search
problem after a single evaluation of the search function f(x), with a success probability
between 52 % and 67 %, therefore outperforming classical algorithms that are bound
to a success probability of 25 %. This constitutes therefore a proof-of-concept of the
quantum speed-up for superconducting quantum processors.
Finally,weproposeascalablearchitectureforasuperconductingquantumprocessor
that can potentially overcome the scalability issues faced by today’s superconducting
qubit architectures.

Andreas Dewes
Contents
1 Introduction & Summary 13
1.1 Quantum Computing with Superconducting Circuits . . . . . . . . . . . . 13
1.2 Realizing a Two-Qubit Quantum Processor . . . . . . . . . . . . . . . . 16
1.3 Demonstrating Simultaneous Single-Shot Readout . . . . . . . . . . . . 17
1.4 Generating and Characterizing Entanglement . . . . . . . . . . . . . . . 19
1.5 Realizing a Universal Two-Qubit Quantum Gate . . . . . . . . . . . . . . 21
1.6 Running a Quantum Search Algorithm . . . . . . . . . . . . . . . . . . . 22
1.7 Demonstrating Quantum Speed-Up . . . . . . . . . . . . . . . . . . . . 24
1.8 Towards a Scalable Multi-Qubit Architecture . . . . . . . . . . . . . . . . 25
2 Theoretical Foundations 27
2.1 Classical & Quantum Information Processing . . . . . . . . . . . . . . . 27
2.2 Principles of "Conventional" Quantum Computing . . . . . . . . . . . . . 28
2.2.1 Quantum Bits and Registers . . . . . . . . . . . . . . . . . . . . 29
2.2.2 Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.3 Quantum Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.4 Quantum Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.5 Realization of a Quantum Computer . . . . . . . . . . . . . . . . 33
2.3 Superconducting Quantum Circuits . . . . . . . . . . . . . . . . . . . . 33
2.3.1 The Josephson Junction . . . . . . . . . . . . . . . . . . . . . . 33
2.3.2 Quantization of Electrical Circuits . . . . . . . . . . . . . . . . . . 34
2.3.3 The LCR Resonator . . . . . . . . . . . . . . . . . . . . . . . . . 37
Coplanar Waveguide Resonators . . . . . . . . . . . . . . . . . . 38
Quantization of the Resonator . . . . . . . . . . . . . . . . . . . 40
2.3.4 The Cooper Pair Box . . . . . . . . . . . . . . . . . . . . . . . . 41
The Transmon Qubit . . . . . . . . . . . . . . . . . . . . . . . . . 45
Decoherence of the Transmon . . . . . . . . . . . . . . . . . . . 46
2.4 Circuit Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . 49
2.4.1 Qubit Driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4.2 Dispersive Limit & Qubit Readout . . . . . . . . . . . . . . . . . . 51
2.4.3 The Josephson Bifurcation Amplifier . . . . . . . . . . . . . . . . 52
Operation Principle . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4.4 Qubit-Qubit Interaction . . . . . . . . . . . . . . . . . . . . . . . 55
Direct Capacitive Coupling . . . . . . . . . . . . . . . . . . . . . 55
Coupling Bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.4.5 Qubit Decoherence in CQED . . . . . . . . . . . . . . . . . . . . 57
2.5 Master Equation Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5.1 Simulation of the Two-Qubit Processor . . . . . . . . . . . . . . . 60
3 Realizing a Two-Qubit Processor 63
3.1 Introduction & Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.1 Processor Operation . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 Qubit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.1 Qubit Frequency & Junction Asymmetry . . . . . . . . . . . . . . 66
3.2.2 Single-Qubit Gates . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.3 Qubit-Qubit Coupling . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.4 Relaxation and Dephasing . . . . . . . . . . . . . . . . . . . . . 72
Qubit Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Qubit Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3 Readout Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4 Summary: Qubit and Readout Parameters. . . . . . . . . . . . . . . . . 76
3.5 Processor Layout & Fabrication . . . . . . . . . . . . . . . . . . . . . . . 76
3.6 Electromagnetic Simulation of the Qubit Chip . . . . . . . . . . . . . . . 79
3.7 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4 Measurement Setup & Techniques 83
4.1 Chip Mounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Signal Generation & Acquisition . . . . . . . . . . . . . . . . . . . . . . 84
4.2.1 Driving and Measurement of the Qubits . . . . . . . . . . . . . . 85
4.2.2 X-Y Pulse Generation by Microwave Single Sideband Mixing . . . 86
4.2.3 Fast Magnetic Flux Pulse Generation and Calibration . . . . . . . 89
4.2.4 Microwave and DC Pulse Synchronization . . . . . . . . . . . . . 91
4.3 Measurement Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3.1 Qubit Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3.2 Qubit Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3.3 Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3.4 Relaxation Time Measurement . . . . . . . . . . . . . . . . . . . 96
4.3.5 Dephasing Time Measurement . . . . . . . . . . . . . . . . . . . 96