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Continuous Feedback on Quantum Superconducting Circuits By Continuous Feedback on Quantum Superconducting Circuits by William Livingston A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Irfan Siddiqi, Chair Professor Birgitta Whaley Assistant Professor Norman Yao Spring 2021 Continuous Feedback on Quantum Superconducting Circuits Copyright 2021 by William Livingston 1 Abstract Continuous Feedback on Quantum Superconducting Circuits by William Livingston Doctor of Philosophy in Physics University of California, Berkeley Professor Irfan Siddiqi, Chair Quantum measurement theory describes the dynamics of a quantum system when it transfers information to an observer. Within the framework of continuous measurement, this outflow of information and resultant back-action occur at a finite rate. Continuously measured sys- tems do not instantaneously collapse to eigenstates, but rather undergo stochastic evolution highly correlated with the observer’s received information. When the measurement rate is on the same timescale as the processing speed of classical control electronics, we can per- form feedback on a quantum system during its collapse process. We experimentally realize such a protocol by using an adaptively controlled quantum amplifier to implement a canon- ical phase measurement on a flying photon. Continuous measurements can also be used as a tool in quantum error correction. Error correction typically requires discrete rounds of measurement, often using entangling gates and ancilla qubits. In contrast, continuous error correction is implemented via always-on measurements of parity syndromes. Using direct parity measurements of pairs of qubits in a three-qubit system along with a custom classical controller, we implement the first continuous quantum error correction code. i To my family ii Contents Contents ii List of Figures iv List of Tables xii 1 Introduction 1 1.1 Quantum measurement and feedback . 1 1.2 Quantum computation and error correction . 2 1.3 Superconducting qubits . 3 1.4 Thesis overview . 4 2 Circuit Quantum Electrodynamics 5 2.1 Classical resonators . 5 2.2 Quantum circuits . 16 2.3 Dispersive coupling . 19 2.4 Black box quantization . 22 3 Quantum Measurement 31 3.1 Discrete maps . 31 3.2 Continuous measurement . 34 3.3 Qubit readout with a cavity . 47 4 Canonical Phase Measurement with Adaptive Feedback 62 4.1 Background on canonical phase measurements . 62 4.2 Implementation of the canonical phase measurement . 65 4.3 Outlook . 77 5 Continuous Quantum Error Correction 78 5.1 Background on quantum error correction . 78 5.2 Experimental demonstration . 84 5.3 Outlook and conclusion . 97 iii A Making a Continuous Feedback Controller 99 A.1 FPGA basics . 99 A.2 A board for control . 104 A.3 Writing an AWG for feedback . 107 A.4 Computer interface . 111 Bibliography 113 iv List of Figures 2.1 Lumped element oscillators. (a) Parallel LC oscillator with no loss. (b) LCR oscillator. (c) Resonator connected to a transmission line in a reflection geometry. A small input capacitance weakens the coupling to the transmission line. 7 2.2 Physical layout for two varieties of resonators with a common measurement feed- line. The left resonator is primarily a lumped element design. A meandering wire at the top acts as an inductor and the interdigitated fingers at the bottom form the capacitor. This element supports one low frequency mode and has a large frequency gap to the next resonance. On the right is a distributed element resonator, which supports many modes roughly evenly spaced in frequency. The design is that of a quarter wave (open on one end and terminated on the other). 10 2.3 (a) 4-port directional coupler with weak scattering . (b) Hanger resonator geom- etry. Port 3 is terminated in an open, represented by an x. Port 4 is connected to a resonant circuit and, for illustration purposes grounded on the far end. (In principle, there could be additional coupling to ground.) . 12 2.4 Resonator in transmission. The central red section is the resonator. The capac- itors act as beamsplitters which partially transmit and partially reflect signals with coupling constant 1 and 2. The incoming and outgoing waves at each port are shown in blue. Although the transmission lines and the resonator often have the same impedance, they do not have to. Effects of mismatched port impedances can be absorbed into the coupling constants. 13 2.5 Summary of typical styles of resonators. The scattering parameters and resonator geometry are shown, with the resonator in red. In the middle are plots of complex sample scattering parameters for different internal kappas of the resonators. The bottom plots show the phase and Log Mag responses of the resonators as a func- tion of detuning from resonance. Notice in that for the resonator in reflection, as the system transitions from κi > κe to κe < κi, the phase response starts to wrap a full 2π......................................... 14 2.6 Fitting resonance data. The resonator is measured in a hangar geometry, with measured S21 parameters in light blue. The green dashed line is the fit guessed from the linear regression, and the red solid line is the result of the full regression using the parameters found in the linear fit as an initial guess. 16 v 2.7 Transmon qubit. (a) The transmon consists of a capacitor and a Josephson junction indicated by a box with an x. The junction allows a jump in BCS phase from 'a in purple to 'b in pink. (b) Physical images of a transmon. i) The transmon’s footprint is primarily capacitor pads. A coupling resonator enters at the top right and a control line enters from the bottom. ii) Leads connecting the capacitor pads to a junction. iii) Josephson junction formed from a thin layer of aluminum oxide between two layers of aluminum. (c) Energy potential in the phase basis for EJ =EC = 50, with the first six eigenfunctions plotted. Dotted lines represent the periodicity of the potential and eigenfunctions as they are defined over a phase variable. 20 2.8 Transmon in brown coupling to a resonator in black through a capacitor with coupling strength g. ................................. 21 2.9 Parameters extracted for a coupled resonator-transmon system with a single junc- tion as a function of Josephson inductance (LJ ). The “Symplectic” method takes the quartic approximation in the transformed frame and the “Sym w/ND” method diagonalizes the two excitation subspace using the quartic (and quadratic) terms in the transformed frame. The bare frequencies and participation ratios for the resonator and qubit respectively are (6.379 GHz, .02) and (5.751 GHz, .965). These values correspond to a subsystem of a fabricated parity chip with a tun- able qubit and a resonator of frequency 6.314 GHz. After tuning the qubit so that the system match the numerical-BBQ expected resonator-qubit detuning, we measure qubit anharmonicity α and dispersive shift χ. This single data point is plotted as a red star. A more comprehensive comparison of theory to experiment over a wide range of participation ratios may be found in [32]. 30 3.1 Interaction of a quantum system and a transmission line. (a) An example of an interacting system: a cavity couples to a transmission line with decay constant γ. (b) Coupling between the system and the waveguide can be modeled as a series of harmonic oscillators which serially interact with the cavity each for time ∆T . The outgoing modes can then be measured by a detector. 34 3.2 Simulating decay under photodetection measurement. (a) Single shot decay of a cavity (S = a) starting in a Fock state, and starting in a coherent state. (b) Decay of a qubit (S = σ) starting in a pure state with initial population hσyσi ≈ :96 under different trajectories of a photon loss. The dashed line shows the ensemble average decay. 37 3.3 Decay under homodyne measurement. (a) Single shot decay of a cavity (S = a) starting in a Fock state, and starting in a coherent state. (b) Decay of a qubit (S = σ) starting in a pure state with initial population hσyσi ≈ :96 under different trajectories. For an individual trajectory, the state stays pure for the duration of the measurement. 42 3.4 Sample Wiener processes. Such processes occur in many contexts, including when integrating vacuum noise coming down a transmission line. 43 vi 3.5 Cavity states separating during during readout. The centers of the circles repre- sent the coherent states αj. Dynamics of the cavity states associated with each qubit eigenvector are classical in that they remain coherent states throughout the measurement. The labeled vectors on each cavity state are the different com- ponents of α_ j. The cavity drive E moves both states in the same direction, the dispersive term χ rotates the states in opposite directions, and the dissipative term κ pulls the states back towards the ground state at the center of the IQ plane. In steady state (not shown), the vectors acting on each coherent state add to zero. Maximal qubit state information is obtained by measuring along the real axis. 52 3.6 Simulation of a qubit dephasing from a measurement. Parameters used are χ = κ = 1 MHz. (a) In-phase (real) and quadrature (imaginary) response of the cavity to a square drive pulse in the rotating frame of the drive. Only α0 dynamics are shown, but since the drive term E is real, under the cavity equations of motion ∗ in Eq. 3.78a, α1 = α0 for all time. The dotted line shows the drive E in units −1 ~ of µs . (b) Measurement-induced (Γφ) and irreversible dephasing (Γφ) of the qubit during the readout pulse. The measurement induced dephasing dips below zero in the final transient, indicating coherence revival of the ρ.
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