Abstract Controlling Error-Correctable Bosonic Qubits Philip Reinhold 2019 The harmonic oscillator is a ubiquitous system in physics, describing a wide range of phenom- ena, both classically and quantum mechanically. While oscillators are relatively straightforward to control classically, they present much more of a challenge in the quantum realm where such systems, modeled as Bosonic modes, have many more degrees of freedom. Controlling Bosonic modes is a crucial task in light of proposals to use these systems to encode quantum information in a way that is protected from noise and dissipation. In this thesis a variety of approaches to controlling such systems are discussed, particularly in the superconducting microwave domain with cavity resonators. In the first part, an experiment demonstrates how a simple dispersively coupled auxiliary system results in universal control, and therefore allows the synthesis of arbi- trary manipulations of the system. This approach is employed to create and manipulate states that constitute an error-correctable qubit. The main drawback of this approach is the way in which errors and decoherence present in the auxiliary system are inherited by the oscillator. In the second part, I show how these effects can be suppressed using Hamiltonian engineering to produce a simple form of first-order ”fault-tolerance.” This approach allows us to demonstrate versions of cavity measurements and manipulations that are protected from dominant error mechanisms. Controlling Error-Correctable Bosonic Qubits A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy by Philip Reinhold Dissertation Director: Robert J. Schoelkopf December 2019 © 2020 by Philip Reinhold All rights reserved. ii Contents Contents iii List of Figures vi List of Tables ix List of Abbreviations and Symbols x Acknowledgments xiv 1 Introduction 1 1.1 Outline of thesis . .5 1.2 Background and suggested reading . .7 2 Controlling cavities 9 2.1 The limits of control in linear cavities . 10 2.2 Control in the Jaynes-Cummings model . 13 2.3 Control in the dispersive regime . 14 2.3.1 Applications: Parity map . 17 2.3.2 Applications: Wigner tomography . 19 2.3.3 Applications: qcMAP . 21 2.4 SNAP and universality . 22 2.5 Optimal control . 27 3 Doing more with less: error correction with harmonic oscillators 29 3.1 Error correction criteria . 30 3.2 Error models for qubits . 31 3.3 Making error correction work in practice . 32 3.4 Error models for cavities . 34 3.5 Error correction in damped harmonic oscillators . 35 3.6 Cat codes . 37 3.6.1 Choosing α ................................ 43 3.6.2 “No-jump” errors and autonomous stabilization . 44 3.7 Alternate cavity encodings . 46 3.7.1 Binomial codes . 46 3.7.2 Cat code generalizations . 48 3.7.3 Numerically optimized codes . 49 iii 3.7.4 GKP codes . 51 4 Numerical quantum optimal control 53 4.1 Defining the problem . 54 4.2 Calculating the gradient . 55 4.3 Cost function variations . 59 4.3.1 Open system GRAPE . 60 4.3.2 Robust control . 61 4.3.3 Gauge degrees of freedom . 62 4.4 Constraints and penalties . 63 4.4.1 Limiting the pulse amplitude . 63 4.4.2 Limiting the bandwidth . 64 4.5 Limiting the intermediate photon number . 66 4.6 Troubleshooting optimization convergence . 67 4.7 What if it doesn’t work?: Debugging optimal control pulses . 68 4.8 Closed-loop optimization methods . 71 5 Meet the samples 73 5.1 The seamless storage cavity . 73 5.2 The antenna transmon . 77 5.3 The stripline readout . 81 5.4 Coupling pins . 82 5.5 Putting it all together . 86 5.5.1 Fabrication . 86 5.5.2 Attenuators and filtering . 86 5.5.3 Readout amplification . 93 5.5.4 Electronics . 94 6 Universal control of a cat-code qubit 97 6.1 First things first: Characterizing the system . 98 6.2 Showing off a bit: Creating distant Fock states from scratch . 103 6.3 Alternating Hilbert spaces: Encode and decode . 105 6.4 Testing encoded operations . 109 6.5 Empirical tuning . 113 7 Venturing forth in frequency space: sideband drives 116 7.1 Four-wave mixing: A cornucopia of terms . 117 7.2 Q-Switching for faster system reset . 120 7.3 Creating photons one at a time . 121 7.4 Engineered dissipation . 123 8 A Fault-Tolerant Parity Measurement 125 8.1 Error Transparency: A paradigm for hardware-efficient fault-tolerance . 127 8.2 Cancelling χ: The simplest useful symmetry . 129 8.2.1 Choosing drive parameters . 132 8.3 Extending the cavity lifetime by protecting it from the transmon . 134 8.4 Parity measurement using |fi ........................... 137 iv 8.5 Postselecting on errors: Identifying transmon-induced cavity dephasing . 138 8.6 Performance analysis . 143 9 Fault-tolerant SNAP 147 9.1 An interaction picture for SNAP . 148 9.2 Analyzing fault propagation in SNAP . 150 9.3 Raman SNAP . 153 9.4 Some assembly required: The FT SNAP protocol . 153 9.5 Tuneup procedure . 156 9.6 Characterizing the FT SNAP . 159 10 Conclusion and Prospects 167 Bibliography 171 A Identities and derivations 183 A.1 Changing frames . 183 A.2 Commutator relations . 184 A.3 Rotating frame . 185 A.4 Displaced frame . 187 A.5 Dispersive Hamiltonian for a multi-level ancilla . 188 A.6 Damped harmonic oscillator master equation . 191 A.7 Fr´echet Derivative of the matrix exponential . 193 A.8 Approximate time-independent Hamiltonians . 194 A.8.1 Floquet formalism . 195 A.8.2 Block Diagonalization . 196 B Construction of unitary operations from a universal control set 198 C Tomographies, large and small 200 C.1 State tomography . 200 C.1.1 Optimizing tomography . 202 C.1.2 Wigner tomography . 203 C.2 Process tomography . 203 C.2.1 Pauli transfer representation . 206 C.3 Gate set tomography . 207 C.4 Randomized benchmarking . 208 D Taming the Black Mamba: Structure and software for the Yngwie FPGA quan- tum controller 211 D.1 Understanding the hardware . 211 D.2 The Yngwie VHDL logic . 213 D.2.1 Analog output chain . 213 D.2.2 Analog input chain . 215 D.2.3 Digital inputs and outputs . 216 D.2.4 Tables . 218 D.2.5 CPU . 219 D.3 The Yngwie python libraries . 221 v D.3.1 Basic sequencing . 221 D.3.2 Master table . 222 D.3.3 Analog tables . 223 D.3.4 Digital tables . 223 D.3.5 Result records . 224 D.4 The Yngwie FPGA instrument class . 224 D.4.1 Running experiments . 226 D.5 The fpga lib.dsl sequencing language . 227 D.5.1 Output . 228 D.5.2 Acquisition . 228 D.5.3 Control flow . ..