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Protecting a Bosonic Qubit with Autonomous Quantum Error Correction

Jeffrey M. Gertler, Brian Baker, Juliang Li, Shruti Shirol, Jens Koch, Chen Wang

arXiv:2004.09322v2quant-ph

TL;DR

Autonomous, continuous quantum error correction through tailored dissipation remains challenging, while existing demonstrations rely on discrete syndrome measurements and adaptive recovery. This paper encodes a logical qubit in a superconducting cavity and uses continuously applied dissipation to recover photon-number parity. The corrected logical qubit lifetime was 288 ± 5 µs, more than twice that of its uncorrected counterpart, but remained below QEC break-even.

  • Problem

    In four-component cat codes, repetitive parity checks for single-photon-loss correction are not compatible with the continuous driven dissipation used for phase-space stabilization.

    Source: Outlook · p. 5

  • Method

    The paper encodes a logical qubit in an odd-parity cavity subspace and applies PReSPA as a continuously applied dissipation process to correct single-photon loss.

    Source: Error correction code and strategy · p. 1, AQEC performance · p. 5

  • Results

    288 ± 5 µs was the corrected logical qubit lifetime, more than twice the uncorrected T4C qubit's lifetime, but it had not reached QEC break-even.

    Source: AQEC performance · p. 4

  • Takeaways & Limitations

    Autonomous parity recovery can extend storage lifetime in a cavity-encoded cat qubit, although this experiment's corrected qubit did not reach QEC break-even.

    Source: AQEC performance · p. 4

    Correction remains limited by PReSPA's finite success probabilities in correcting single-photon loss.

    Source: AQEC performance · p. 5

Abstract

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To build a universal quantum computer from fragile physical qubits, effective implementation of quantum error correction (QEC) is an essential requirement and a central challenge. Existing demonstrations of QEC are based on a schedule of discrete error syndrome measurements and adaptive recovery operations. These active routines are hardware intensive, prone to introducing and propagating errors, and expected to consume a vast majority of the processing power in a large-scale quantum computer. In principle, QEC can be realized autonomously and continuously by tailoring dissipation within the quantum system, but this strategy has remained challenging so far. Here we encode a logical qubit in Schrödinger cat-like multiphoton states of a superconducting cavity, and demonstrate a corrective dissipation process that directly stabilizes an error syndrome operator: the photon number parity. Implemented with continuous-wave control fields only, this passive protocol realizes autonomous correction against single-photon loss and boosts the coherence time of the multiphoton qubit by over a factor of two. Notably, QEC is realized in a modest hardware setup with neither high-fidelity readout nor fast digital feedback, in contrast to the technological sophistication required for prior QEC demonstrations. Compatible with other error suppression and phase stabilization techniques, our experiment suggests reservoir engineering as a resource-efficient alternative or supplement to active QEC in future quantum computing architectures.

Error correction code and strategy · AQEC technique and device implementation · Characterization of dissipation operator

The paper encodes a cavity qubit in odd-parity cat-like states and uses engineered dissipation to recover parity after photon loss. Its PReSPA protocol converts even-parity states into odd-parity states, with measured photon-addition coherence and conversion dynamics characterizing the process.

Error correction code and strategy

  • The logical qubit is encoded in an odd-parity, truncated four-component cat code, whose photon-number parity acts as a stabilizer against single-photon loss.The code uses approximately balanced average photon numbers of 3.5, and the cat-state amplitude is approximately 1.87.Source: Error correction code and strategy · p. 1
  • PReSPA uses four targeted dissipative processes to add a photon after a parity jump, stabilizing parity while preserving coherence without implementing a true parity operator.The construction works around the challenge of continuous quantum non-demolition parity projection.Source: Error correction code and strategy · p. 1

Characterization of dissipation operator

AQEC technique and device implementation

  • Two continuous-wave frequency combs, each with four tones spaced by 2χq, selectively drive even-number states; reservoir emission returns them to the odd-parity code space without which-path information.Selectivity requires λ ≪ χq, with the rates following λ < Ω < κ < χq.Source: Figure 1 · p. 2, AQEC technique and device implementation · p. 2

Error correction code and strategy

  • Instantaneous, exact Πeo operations are theoretically estimated to reduce the T4C code’s logical error rate from single-photon loss by approximately 50 times.The estimate could be larger for an encoding with larger |α|.Source: Error correction code and strategy · p. 2

Characterization of dissipation operator

  • PReSPA converts even-parity superpositions with nearly zero additional phase and approximately 67% average fidelity.Wigner tomography shows interference fringes in the output states, indicating coherence persists after photon addition.Source: Characterization of dissipation operator · p. 4
  • The four conversion paths have well-matched temporal profiles and a convergence half-time of approximately 8 µs; conversion fidelity decreases at higher photon numbers.Source: Characterization of dissipation operator · p. 3

AQEC performance

PReSPA preserves nonclassicality and more than doubles the encoded qubit’s lifetime, but the demonstrated AQEC remains below break-even. Its performance is limited by transmon excitations and imperfect single-photon-loss correction.

AQEC performance

  • Over 143 µs, PReSPA preserves Wigner negativity in six cavity multiphoton states to a greater extent than uncorrected free evolution.The six states correspond to the cardinal points of the logical Bloch sphere.Source: AQEC performance · p. 4
  • The corrected logical qubit’s lifetime is 288 ± 5 µs, more than twice its uncorrected counterpart, but it has not reached the QEC break-even point.The measured longitudinal relaxation time is 365 ± 8 µs for pole states and transverse relaxation time is 258 ± 6 µs for equator states. Decoding maps both code words and their orthogonal vectors to the same transmon states, making retrieval insensitive to encoding size.Source: AQEC performance · p. 4
  • Transmon thermal excitation rises from 1.4 ms−1 to 2.0 ms−1 during AQEC and accounts for ∼80% of Γl and ∼50% of Γt.Spontaneous transmon excitation can activate sequential two-photon gain under PReSPA, causing incoherent logical bit flips; filtering and shielding could reduce this effect.Source: AQEC performance · p. 4
  • PReSPA’s estimated single-photon-loss correction success probabilities are Sl = 89% and St = 76%; an equator state is expected to recover after 25 µs with fidelity F ≈Ste−tΓt = 69%.Prominent correction failures include a second photon loss or ancilla decay during PReSPA, virtually activated pumping, and higher-order cavity nonlinearity.Source: AQEC performance · pp. 4–5
  • The required success-probability condition for break-even without undesirable transmon excitation is 1⁄3Sl + 2⁄3St > 1 − 2⁄3 n̄−1 = 81%.The authors estimate that modest improvements in ancilla and cavity lifetimes could raise Sl and St above this threshold under realistic experimental conditions.Source: AQEC performance · p. 5

Outlook · METHODS · Approximate AQEC and decoding unitary

PReSPA enables autonomous photon-loss correction for cat-code qubits, while the T4C analysis explains its approximate decoding limits and points to ways of extending continuous protection.

Outlook

  • Unlike repetitive parity checks that conflict with driven stabilization [5,6,16,25], PReSPA permits photon-loss correction alongside logic or odd-parity stabilization; modest four-photon dissipation can address cat-size changes and phase drifts [16,32].Together, these techniques could complete fully passive first-order protection of a single-cavity logical qubit.Source: Outlook · p. 5
  • Path-independent operations on code and error spaces can enable autonomous gates [17,18,41], while error-transparent transmon processes [17] or biased-noise ancillas [40,42] can suppress propagated ancilla-relaxation errors.The cascaded pumping technique for path erasure can also generalize to other dissipation operators [11,13,20].Source: Outlook · p. 5
  • Continuous AQEC bridges intrinsically protected physical qubits and redundancy-based QEC, suggesting reservoir engineering can apply code-based correction to improve physical-qubit robustness with minimal resource overhead.The demonstrated cavity qubit is protected by a driven-dissipative environment in quasi-equilibrium, described by a time-independent rotating-frame Hamiltonian and dissipation operators.Source: Outlook · p. 5

Approximate AQEC and decoding unitary

  • T4C approximates a Schrödinger cat code at small photon numbers, and its practical odd-Fock codewords stop at n ≤ 7 while PReSPA extends to an infinite parity-selective raising-operator series.In the idealized large-photon-number limit with instantaneous loss operations, the cat code would permit perfect retrieval of quantum information.Source: Approximate AQEC and decoding unitary · p. 5
  • Decoding is imperfect because unequal loss-event coefficients and rotations in the H15 and H37 subspaces distort amplitudes, but the θj and ϕj rotations partially cancel as loss-event count changes.Jump and no-jump evolution rotate states within the subspaces, and the angles depend on both time and the number of loss events.Source: Approximate AQEC and decoding unitary · p. 6
  • Averaged over trajectories, photon number remains exactly constant under combined loss and PReSPA, making the T4C encoding self-sustained in energy with orthogonality loss only through a second-order diffusive process.Unlike parity-measurement-based experiments, the cat size does not deterministically shrink on a timescale of 1/2κ.Source: Approximate AQEC and decoding unitary · p. 6
  • Theoretically, corrected logical qubits incur no longitudinal relaxation, with transverse relaxation rate Γt 40 times lower than single-photon loss rate n̄κ and intrinsic QEC failure rate 1 − St = 2.5%.The theoretical comparison considers PReSPA-corrected T4C against uncorrected T4C and Fock-state encoding.Source: Approximate AQEC and decoding unitary · p. 6
  • Balancing the codewords at n̄ = 3.5 would improve St by only 0.08%, versus experimental 1 − St ≈24%; with this hardware, choosing n̄ close to 3 can improve AQEC.The intrinsic QEC failure trade-off is minimized at n̄ = 4, while photon-loss rate is minimized at n̄ = 3.Source: Approximate AQEC and decoding unitary · pp. 6–7

Device and fabrication · Driven Josephson circuit Hamiltonian under PReSPA

The device combines a 3D-planar-hybrid circuit QED architecture with a driven Josephson-junction scheme for PReSPA. The method uses two frequency combs to activate four-wave mixing and derives transition rates that account for direct and parametric driving.

Device and fabrication

  • The 3D-planar-hybrid device houses two re-entrant quarterwave resonators in an aluminum cavity and uses a sapphire chip with a thin-film aluminum transmon and low-Q readout resonator.The transmon has a single Al-AlOx-Al junction fabricated by the Dolan bridge technique.Source: Device and fabrication · p. 7
  • Cavity mode A is the storage mode used; mode B remains in vacuum because its 6.5 MHz transmon coupling is too high for PReSPA, while its 1%–2% thermal population slightly dephases A.The cavity has two high-Q storage modes; mode B is at 6.089 GHz.Source: Device and fabrication · p. 7

Driven Josephson circuit Hamiltonian under PReSPA

  • The circuit Hamiltonian models cavity A, transmon q, and reservoir R as weakly anharmonic modes, with Josephson-junction phase fluctuations set by each mode’s zero-point phase across the junction.The junction energy and phase operator define the nonlinear Hamiltonian.Source: Driven Josephson circuit Hamiltonian… · p. 7
  • PReSPA uses a transmon-excitation comb and a mixing comb to convert transmon excitations into cavity-A and reservoir excitations through Josephson-junction four-wave mixing.The mixing-tone treatment displaces the qubit annihilation operator to account for four mixing frequencies.Source: Driven Josephson circuit Hamiltonian… · p. 7
  • After expanding the cosine, transforming to a rotating frame, and canceling second-order terms dispersively, the relevant fourth- and sixth-order Hamiltonian terms include driven and non-driven contributions.The transmon is treated as a two-level system; reservoir R is restricted to its ground or first excited state outside readout.Source: Driven Josephson circuit Hamiltonian… · p. 7
  • The mixing tones use constant spacing η, and the transmon drives use the same spacing magnitude with opposite sign to preserve path independence; parametric mixing renormalizes rates Ωn and λn.Individual mixing tones Stark-shift the transmon, while cross terms between tones also produce slowly rotating Stark shifts; the transmon drives are weak and do not significantly Stark-shift it.Source: Driven Josephson circuit Hamiltonian… · pp. 8–9
  • Four mixing tones drive the |2n, e, 0⟩↔|2n + 1, g, 1⟩ transitions for n = 0, 1, 2, 3 through resonant single-tone and multi-tone parametric pathways.The parametric contribution comes from transmon-frequency modulation by Stark shifts from tone pairs; relative phases can make it constructive or destructive, and bosonic enhancement means higher-photon transitions typically need smaller ξn.Source: Driven Josephson circuit Hamiltonian… · p. 8

PReSPA spectroscopy and rates

PReSPA spectroscopy calibrates drive-comb frequencies to match transitions and measures the four conversion paths. The measurements show that equal conversion rates require different microwave amplitudes across the combs, in agreement with theory.

PReSPA spectroscopy and rates

  • η = 2.679 MHz and Δ ≈ 2.9 MHz are calibrated to keep every tone within 10 kHz of its corresponding transition.The initial comb-spacing assumption is η = 2χ_q; experimental corrections account for transmon Stark shifts and photon-occupation-dependent quadratic shifts.Source: PReSPA spectroscopy and rates · p. 9
  • Photon-addition probabilities and transmon excitation fits across four even states yield Ω_n and λ_n for each conversion path.The QuTiP model simulates a two-stage pumping process across four relevant levels, including drive detuning, cavity and reservoir decay, and transmon relaxation, dephasing, and spurious two-frequency switching.Source: PReSPA spectroscopy and rates · p. 9
  • Equal Ω_n and λ_n across the four conversion paths require significantly different microwave amplitudes within each comb, in quantitative agreement with theory.A discrepancy remains in the global prefactor for Ω_n magnitude, possibly due to coarse estimates of zero-point fluctuation parameters.Source: PReSPA spectroscopy and rates · p. 9

Cavity and transmon Ramsey under PReSPA

PReSPA Ramsey measurements track multiphoton-state frequency, coherence, and phase, and help tune conversion phases and logical decoding. A transmon Ramsey measurement shows that the strong off-resonant PReSPA drives do not affect transmon coherence.

Cavity and transmon Ramsey under PReSPA

  • PReSPA Ramsey tracks multiphoton-state frequency and coherence by reading out displacement-enabled interference, with an exponential fit envelope indicating coherence decay.The measurements calibrate state frequencies and phases under PReSPA; the protocol is useful at high photon numbers because PReSPA approximately preserves cavity photon-number distributions.Source: Cavity and transmon Ramsey under PReSPA · pp. 9–10, Figure 4 · p. 10
  • After more than 25 µs of PReSPA, even-to-odd conversion is nearly complete, allowing Ramsey oscillations to reveal the converted-state phase against the corresponding odd-parity initial state.The measured photon-addition phases guide microwave-comb phase and amplitude adjustments to minimize imparted phases.Source: Cavity and transmon Ramsey under PReSPA · pp. 9–10
  • PReSPA Ramsey measures logical-codeword phase accumulation, enabling calculation of the correct decoding angle when |0⟩L is stationary.Source: Cavity and transmon Ramsey under PReSPA · p. 10
  • T∗2q = 17 µs, and the transmon coherence time is not affected by PReSPA’s strong off-resonant drives.A regular qubit Ramsey measurement in the presence of PReSPA mixing combs measures the transmon Stark shift, because the mixing comb approximately accounts for it.Source: Cavity and transmon Ramsey under PReSPA · p. 10

GRAPE methods · Cavity Kerr effect on decoding

GRAPE optimizes cavity-state preparation and decoding controls by minimizing state or unitary infidelity. The cavity’s self-Kerr evolution complicates decoding, while pulse reuse with a logical Z correction produces slightly different decoding performance and alternating T4C fidelities.

GRAPE methods

  • GRAPE minimizes state or process infidelity by gradient-descent optimization of transmon and cavity drive envelopes, following QOC methods adapted to arbitrary cavity-state control [33, 53–55, 56].The cost functional quantifies deviations of the realized state or unitary from its target.Source: GRAPE methods · p. 11
  • For state preparation, envelope-modulated drives at ωA and ωq are optimized to maximize Fprep = |⟨ψT|ψ⟩|^2 under closed-system rotating-frame evolution.The drives act on the storage cavity and transmon, respectively.Source: GRAPE methods · p. 11
  • The decoding unitary is constrained to map the four relevant T4C basis states as specified in Eq. (7), with three additional optimization constraints in the seven-dimensional subspace S.The additional constraints are discretionary apart from preserving unitarity.Source: GRAPE methods · p. 11
  • The decoding unitary is unconstrained outside S, so its uncharacterized action there may slightly help or hurt decoded-qubit fidelity; further constraints may yield small improvements.The study does not characterize the transformation on states orthogonal to S.Source: GRAPE methods · p. 11

Cavity Kerr effect on decoding

  • Self-Kerr drives deterministic collapse-revival without directly causing logical decay, apart from a small QEC-success reduction, but requires time-dependent decoding; a logical Z rotation enables pulse reuse every 72 µs.The Kerr cycle is π/K or 288 µs, while the reused pulse interval is π/4K.Source: Cavity Kerr effect on decoding · p. 11
  • The two decoding pulses differ slightly in performance, producing alternating T4C state and process fidelities over time.The pulses are used at t = 0 and 36 µs modulo 72 µs and differ by a minus sign on the |7⟩ coefficient in the second transformation.Source: Cavity Kerr effect on decoding · p. 12

Simulation of experimental AQEC

Master-equation simulations reproduce the experiment’s corrected logical-qubit lifetime and project substantial AQEC gains under improved coherence, while identifying error channels that constrain those gains.

Simulation of experimental AQEC

  • Simulations predict QEC break-even for a T4C encoding with n̄ = 3.4 and T1A = 1 ms, T1q = Tφ = 100 µs33,50, if spontaneous transmon excitation γ↑ is kept reasonably low.Source: Figure 6 · p. 13
  • The master-equation model includes single-photon loss, spurious transmon excitation, ancilla relaxation and dephasing, and cavity dephasing, then evaluates decoded-state fidelity over time.It captures relative photon-number-state dynamics on the ∼χq timescale and accounts for cavity Kerr effects in the decoding phases.Source: Simulation of experimental AQEC · pp. 12–13
  • The simulation predicts a 290 µs corrected logical-qubit lifetime, in excellent agreement with the experimental Γ_t^-1 = 255 µs.Sensitivity analysis of individual Hamiltonian and loss parameters is consistent with the estimated contributions of individual error channels.Source: Simulation of experimental AQEC · p. 13
  • With improved cavity and transmon coherence, transmon-decay and second-photon-decay logical errors are expected to fall quadratically, but selectivity and higher-order nonlinear errors do not scale down equally.The simulations therefore motivate reducing χq, κ, Ω, and λ to trade some coherence-related gain for suppression of those other error channels.Source: Simulation of experimental AQEC · p. 13
  • At T1A > 2 ms, the modeled gain factor decreases because off-resonant transmon excitation from imperfect comb frequency selectivity becomes more significant.Using |2n + 1, f, 0⟩ as the first intermediate level could reduce frequency-selectivity errors by a factor of 4, at the cost of 2× faster transmon decay.Source: Simulation of experimental AQEC · p. 14

1. BOSONIC CODE FOR APPROXIMATE AQEC

The section analyzes recovery after photon loss for large Schrödinger-cat codes and a truncated four-component cat (T4C) code. It derives conditions for approximate information retrieval with T4C and describes a short-time decoding unitary.

1.1. Applying PReSPA to Schr¨odinger-cat code (|α| ≫1)

  • For large-photon-number cat codes, single-photon loss followed by ideal PReSPA preserves the ability to recover the initially stored quantum information with an appropriate unitary.Source: §1.1 · p. 17

1.2. Evolution of truncated 4-component cat (T4C) states under photon loss and PReSPA

  • The T4C encoding is a single-cavity binomial code, also interpretable as a cat code with |α|^2 ≈ 3.5 truncated at photon number 7; its loss-and-recovery trajectory is independent of jump times.The jump operator commutes with the deterministic evolution operator.Source: §1.2 · pp. 17–18

1.3. T4C decoding unitary for approximate quantum information retrieval

  • For short times, matching the mean and second photon-number moments across T4C subspaces makes the no-jump distortion unity through second order, fixing optimal amplitudes C1 = C7 = 1/2 and C3 = C5 =The experimental amplitudes C1 = C7 = 0.1 yield mean photon numbers 3.6 and 3.4 and second moments 16.6 and 13, a reasonable but non-optimal choice; higher-order and nonzero-jump conditions cannot all be satisfied.Source: §1.3 · p. 18
  • For t/T1A ≪ 1, the decoding unitary uses the original codewords as basis states and yields a good short-time approximation to the intended transmon state, though larger jump numbers cause imperfections.The fidelity can in principle be improved by optimizing the decoding basis, which this analysis omits.Source: §1.3 · p. 19

2. OPTIMAL CONTROL FOR UNITARY OPERATIONS AND STATE TRANSFER

The section describes quantum optimal control for preparing and decoding states by optimizing control pulses to maximize state-transfer or unitary fidelity. Its implementation discretizes the dynamics and combines fidelity objectives with pulse and state constraints.

2. OPTIMAL CONTROL FOR UNITARY OPERATIONS AND STATE TRANSFER

  • Quantum optimal control uses gradient-based pulse optimization to steer the system toward target states or unitaries, using an automatic-differentiation extension of GRAPE.The cost functional incorporates process or state infidelity and additional terms for realistic pulse shapes.Source: §2 · p. 19
  • Control fields are discretized at δt = 1 ns, and optimization targets either process fidelity for unitary operations or state-transfer fidelity for state preparation.The decoding operation uses a subspace of dimension D = 7.Source: §2 · p. 19
  • The full evolution is computed from short-time propagators under a Hamiltonian comprising a static system term and transmon and cavity controls.The controls are {σx, σy, xA, pA}; the static Hamiltonian includes self-Kerr, fourth-order cross-Kerr, and sixth-order cross-Kerr terms.Source: §2 · p. 19
  • The weighted cost combines fidelity infidelity with penalties for rapid pulse-envelope changes, overall pulse power, and occupation of forbidden higher states.The forbidden-state penalty can limit transmon leakage beyond the {|g⟩, |e⟩} logical subspace and avoid population near the Hilbert-space truncation level.Source: §2 · p. 20
  • Optimization uses ADAM with a decaying learning rate and TensorFlow automatic differentiation; typical convergence takes 1,000–2,000 iterations.Gaussian white noise initializes pulses; durations are 1 µs for preparation and 2 µs for decoding, with computations requiring a few hours on a CPU.Source: §2 · p. 20

3. ERROR SOURCES OF THE CORRECTED LOGICAL QUBIT

The section accounts for errors that remain uncorrected or arise during PReSPA correction, finding that their combined effect agrees with the measured logical-qubit relaxation times. It also quantifies photon-loss correction failures and identifies hardware and process limitations that constrain performance.

3. ERROR SOURCES OF THE CORRECTED LOGICAL QUBIT

  • The estimated error contributions agree within 5% with measured logical-qubit longitudinal and transverse relaxation times of 366 ± 10 µs and 256 ± 7 µs.The contributions comprise errors unrelated to single-photon loss and failure modes of the correction scheme.Source: §3 · p. 20

3.1. Photon-loss unrelated errors

  • Ancilla excitation γ↑ can dephase the cavity and trigger two-photon addition with about 50% probability, causing pole-state bit flips or fully dephasing equator states.The scheme is modestly more susceptible to ancilla γ↑ than a bare cavity storing information in Fock states |0⟩ and |1⟩.Source: §3.1 · p. 20
  • Long-time mixing-comb pumping raises the transmon excitation rate γ↑ by about 30%, from 1.4 ms−1 at idle to 1.8 ms−1, with extrinsic heating considered a likely contributor.The authors cannot rule out non-perturbative driven-circuit effects; better thermalization may reduce the heating, and a better-optimized protocol is expected to use weaker pumping.Source: §3.1 · p. 21
  • Off-resonant transmon excitation in the odd-parity code space is simulated below 0.2 ms−1 and scales with the cube of QEC operation speed when λ, Ω, and κ scale proportionally.This intrinsic effect creates a trade-off in QEC operation speed.Source: §3.1 · p. 21

3.2. Photon-loss correction efficiency

  • Photon-loss correction succeeds with 89% probability for pole states and 76% for equator states, versus ideal success rates of 100% and 97.5%, respectively.The corresponding experimental failure rates are 11% and 24%; the ideal implementation fully eliminates longitudinal errors and corrects transverse errors with 97.5% success.Source: §3.2 · p. 21
  • Ancilla relaxation and second-photon loss contribute to correction failures: transmon T1 decay has 7% probability, while second-photon decay occurs with 5.6% probability and reduces pole-state success by about 11%.These rates reflect competition between correction speed and hardware coherence, and are expected to improve with longer transmon and cavity lifetimes.Source: §3.2 · p. 21
  • Virtual activation by a neighboring mixing tone causes cavity dephasing with an estimated 3% probability, scaling as (κ/2χq)^2; reducing κ by at least about 20% could improve performance.Stochastic correction time also creates phase uncertainty through the cavity self-Kerr, producing logical dephasing.Source: §3.2 · pp. 21–22
  • Ancilla dephasing does not directly dephase the cavity during photon addition; it resets the process and slightly reduces correction rate, while low-frequency ancilla noise has no first-order effect.Continuous AQEC automatically repeats error detection until correction succeeds, with minimal overhead.Source: §3.2 · p. 22

4. ERROR RECOVERY TEST

The test evaluates whether PReSPA restores an ideal single-error state to the logical state, measuring the recovered-state fidelity against an encoding-and-decoding control. The result is consistent with the reported average preservation of phase coherence and one-time PReSPA success probability.

4. ERROR RECOVERY TEST

  • An OCT pulse prepares the ideal error state, which ideal PReSPA corrects to |X⟩L after 25 µs, with phases compensating for cavity self-Kerr accumulation.Source: §4 · p. 22
  • F = 67 ± 2% normalized fidelity was measured for the recovered state, consistent with 67% average phase-coherence preservation and St = 76% one-time success probability after accounting for 10% fidelity loss.The 10% loss comes from compounded transverse error channels.Source: §4 · p. 22
  • After 25 µs of PReSPA, decoding maps the logical state to the transmon for σ̂x projection, compared with an encode-and-immediately-decode control measurement.The comparison compensates for fidelity loss due to encoding and decoding.Source: §4 · p. 22

5. PRESPA OPTIMIZER FOR LOGICAL EQUATOR STATES

An empirical optimization routine tunes PReSPA control amplitudes and phases using logical-equator-state fidelities. The optimized PReSPA-2 parameters slightly improve logical coherence time over PReSPA-1, though the rates are less well matched.

5. PRESPA OPTIMIZER FOR LOGICAL EQUATOR STATES

  • The routine iteratively varies PReSPA microwave-tone amplitudes and phases, evaluating fidelity after 144 µs of PReSPA through state decoding and transmon tomography.Its cost function averages fidelities of |X⟩L, |−X⟩L, |Y⟩L, and |−Y⟩L because equator states are more sensitive to small parameter changes than the pole states |Z⟩L and |−Z⟩L.Source: §5 · p. 22
  • PReSPA-2 achieved a slight gain in logical coherence time over PReSPA-1, although its Ωn and λn rates were less well matched.PReSPA-1 parameters were chosen using calibration experiments; further work is needed to understand how a dissipative operator deviating from Eq. (2) can improve AQEC performance.Source: §5 · pp. 22–23

6. ALL ABOUT READOUT

Readout relies on heterodyne measurements and averaged state calibration rather than confident single-shot discrimination. The section also details corrections for contrast and preparation errors that qualify reported measurements and comparisons.

6. ALL ABOUT READOUT

  • Heterodyne readout lacks sufficient signal-to-noise for confident single-shot discrimination, so averaged ground- and excited-state signals calibrate raw transmission into excited-state populations.The signal is amplified by a TWPA, HEMT, and room-temperature amplifier.Source: §6 · p. 23
  • The 5% equilibrium excited-state population scales reported fidelities up by 10% versus a pure target, but does not affect relative comparisons or Fig. 4b relaxation rates.The nominal ground state is prepared by waiting more than 10T1q; the excited state is prepared with a π pulse.Source: §6 · p. 23
  • Cross-Kerr coupling to cavity A causes photon-number-dependent readout-contrast reduction; Wigner parity measurements use paired maps that swap even/odd cavity states between transmon ground and excited states.Removing the selective transmon rotation allows the cavity-occupation-dependent contrast shift to be measured.Source: §6 · p. 23
  • For Fig. 2f, a background measurement accounts for excitation-dependent contrast, enabling accurate photon-addition probabilities and transmon-excitation estimates used to calculate λ and Ω.The latter calculations are shown in Extended Data Fig. 3.Source: §6 · p. 23
  • OCT state-preparation-error corrections renormalize Fig. 2f and Extended Data Figs. 3 and 8, but are not applied to AQEC figures such as Fig. 4d.The corrections exclude preparation inaccuracies from figures characterizing the PReSPA process.Source: §6 · p. 23
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