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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

Bosonic QEC seeks hardware-efficient protection for cavity-encoded qubits without repetitive parity checks that conflict with continuous phase-space stabilization. This work uses a truncated four-component cat encoding and PReSPA autonomous correction, with simulations indicating substantial logical-lifetime improvement while identifying photon-loss and transmon-related error channels.

  • Problem

    Four-component cat codes offer first-order error protection, but repetitive parity checks for correcting single-photon loss are incompatible with continuous driven dissipation for phase-space stabilization.

  • Method

    The approach encodes a logical qubit in an odd-parity subspace using a truncated four-component cat code and applies PReSPA pumping to engineer corrective dissipation.

  • Results

    A simulated T4C code with mean photon number 3.4 achieves a logical lifetime 7–9 times longer than the physical photon-loss time, or 40–80% above breakeven, under specified parameters.

  • Takeaways & Limitations

    PReSPA provides a route to autonomous bosonic error correction while retaining compatibility with continuous stabilization, although performance depends on balancing coherence improvements against other error channels.

  • Takeaways & Limitations

    Ancilla excitation can induce logical errors, including an estimated 50% probability of adding two photons after an ancilla excitation event.

Abstract

from arXiv · show

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

The paper encodes a logical qubit in an odd-parity truncated four-component cat code and uses continuously driven dissipation to convert photon-loss errors back into the code space while preserving coherence.

  • Code and error model: The logical qubit is encoded in an odd-parity T4C subspace of a superconducting cavity, with single-photon loss mapping odd Fock components to even ones.The encoding uses four Fock-state components and balances the logical code words at an average photon number near 3.5.
  • Autonomous correction: PReSPA continuously stabilizes photon-number parity by automatically adding a photon whenever loss causes a parity jump.The operator is implemented as four targeted dissipative processes rather than a direct continuous parity measurement.
  • Autonomous correction: Four-tone continuous-wave combs selectively address the four even-photon states, while reservoir decay returns the cavity to the odd-parity code space.Level selectivity requires λ ≪ χq, with the rate hierarchy λ < Ω < κ < χq.
  • Experimental implementation: The experiment uses a storage cavity, transmon ancilla, and rapidly decaying reservoir resonator, with spectroscopy and Wigner tomography to characterize population conversion and coherence.The protocol is implemented using continuous-wave control fields and ancilla-mediated measurements.
  • Operator characterization: The photon-addition paths have a convergence half-time of approximately 8 µs, with lower conversion fidelity at higher photon numbers.The four paths show well-matched temporal profiles, while faster photon loss reduces fidelity for larger photon numbers.
  • Operator characterization: PReSPA preserves coherence during conversion, producing nearly zero additional phase and an average fidelity of approximately 67% for even-parity superpositions.Interference fringes in Wigner functions demonstrate persistence of coherence after photon addition.

AQEC performance

PReSPA preserved nonclassical cavity states and extended the corrected logical qubit lifetime, but experimental imperfections kept performance below the QEC break-even point.

  • AQEC performance: PReSPA preserved Wigner negativity over 143 µs more effectively than uncorrected free evolution.The experiment used six cavity multiphoton states at the cardinal points of the logical Bloch sphere.
  • AQEC performance: 288 ± 5 µs corrected logical-qubit lifetime was more than twice the uncorrected counterpart.The measured transverse relaxation time for equator states was 258 ± 6 µs.
  • AQEC performance: The corrected qubit did not reach QEC break-even because redundant information storage caused the uncorrected T4C qubit to decay over three times faster than the bare-cavity qubit.The bare-cavity comparison used the lowest-energy Fock states.
  • AQEC performance: Ancilla thermal excitation increased during AQEC and accounted for approximately 80% of longitudinal and 50% of transverse logical-qubit error rates.Spontaneous transmon excitation can activate an incoherent sequential two-photon-gain process.
  • AQEC performance: PReSPA correction success probabilities were estimated as S_l = 89% and S_t = 76%, with failures arising from additional losses, ancilla decay, virtual pumping, and cavity nonlinearity.A logical equator state recovering after photon loss was expected to reach fidelity F ≈ 69% after 25 µs.

Outlook

The four-component cat-code approach positions PReSPA as a noninvasive route to photon-loss correction that can coexist with dissipative stabilization and broader autonomous error-correction schemes.

  • Outlook: The four-component cat code offers a hardware-efficient route to universal computation with first-order error protection in a single cavity.Its repetitive parity checks had previously conflicted with continuous driven dissipation for phase-space stabilization.
  • Outlook: PReSPA provides noninvasive photon-loss correction compatible with concurrent logic operations or dissipative stabilization within the odd-parity subspace.The approach addresses the incompatibility between repetitive parity checks and continuous driven dissipation described for earlier cat-code schemes.
  • Outlook: A second reservoir providing four-photon dissipation could be combined with PReSPA to correct cat-size changes and phase drifts.This combination is proposed as a route toward fully passive first-order protection of a single cavity.
  • Outlook: The demonstration connects intrinsically protected qubits and redundancy-based QEC through a driven-dissipative environment operating in quasi-equilibrium.The implementation is described by a time-independent rotating-frame Hamiltonian and dissipation operators.
  • Outlook: The T4C code is an approximate small-photon-number Schrödinger cat code and can also be viewed as a binomial-code instance.Its practical design uses odd-numbered Fock states with n ≤ 7.
  • Outlook: The encoding keeps its average photon-number expectation constant under the effective loss-and-recovery operator, making the cat size self-sustained on average.This differs from cat states in parity-measurement-based bosonic QEC experiments, where the cat states decrease over time.

Device and fabrication

The experiment uses a hybrid circuit-QED device whose Josephson nonlinearity and two frequency combs implement selective PReSPA transitions between cavity, ancilla, and reservoir states.

  • Device and fabrication: The device combines a high-Q cavity, a transmon ancilla, and a low-Q stripline resonator in a 3D-planar-hybrid cQED architecture.The cavity is machined from high-purity aluminum, while the transmon and readout resonator use thin-film aluminum on sapphire.
  • Device and fabrication: The circuit-QED Hamiltonian is derived from three weakly anharmonic oscillator modes with dispersive couplings, Kerr nonlinearity, and Josephson-junction phase fluctuations.The transmon is treated as a two-level system and higher reservoir levels are neglected outside readout.
  • Device and fabrication: PReSPA uses a transmon comb and a mixing comb to activate four-wave mixing that converts transmon excitation into cavity and reservoir excitations.The mixing comb exploits the Josephson-junction nonlinearity and is represented through a displaced qubit operator.
  • Device and fabrication: The spectroscopy prepares even-parity Fock states, applies PReSPA for 12 µs, and measures photon-addition likelihood across transmon- and mixing-comb detunings.A 1 µs reservoir-relaxation wait precedes conditional transmon readout, and the procedure is repeated for four transitions.
  • Device and fabrication: Calculated and measured complex PReSPA transition rates are compared using calibrated comb amplitudes, phases, Rabi oscillations, and Stark shifts.The relevant rates are λ_n and Ω_n, with mixing-tone amplitudes converted approximately through the measured Stark shift.

PReSPA spectroscopy and rates

PReSPA parameters were calibrated by spectroscopy and time-dependent photon-addition measurements, with fitted transition rates for four conversion paths. Equalizing the rates requires path-dependent microwave amplitudes, broadly matching theory despite a global prefactor discrepancy.

  • Spectroscopic calibration: PReSPA comb frequencies were corrected for Stark shifts and calibrated so corresponding transitions remained within 10 kHz of resonance.The calibrated values were η = 2.679 MHz and Δ ≈ 2.9 MHz.
  • Rate extraction: Photon-addition and transmon-excitation trajectories were fitted with a two-stage pumping model to extract Ω_n and λ_n for each conversion path.The model included detunings, cavity and reservoir decay, and transmon relaxation, dephasing, and switching.
  • Rate matching: Equal Ω_n and λ_n across the four paths required significantly different microwave amplitudes within each comb, quantitatively agreeing with theoretical predictions.A discrepancy remained in the global Ω_n magnitude prefactor, possibly from coarse zero-point-fluctuation estimates.

Cavity and transmon Ramsey under PReSPA

Ramsey, Wigner, and photon-number tracking measurements characterized phase evolution, coherence, and transition dynamics under PReSPA. These measurements supported phase calibration while monitoring the transmon and logical-qubit behavior during the protocol.

  • PReSPA Ramsey: PReSPA Ramsey measurements extracted the frequency and coherence of odd-parity superpositions and tuned phases in the PReSPA operator.The experiment compared converted odd-parity states with corresponding odd-parity initial states.
  • Wigner characterization: Wigner measurements reconstructed cavity-state coherence from photon-number parity measurements after variable displacements.The measured Wigner function rotated at a rate set by the frequency difference between |1⟩ and |5⟩ in the rotating frame.
  • Phase calibration: The amplitudes and phases of the microwave combs were adjusted to minimize phases imparted during photon addition.This calibration targeted phase preservation in the converted superpositions.
  • Logical phase tracking: The logical decoding angle was calculated from the phase accumulated by |1⟩_L while |0⟩_L was held stationary.Process-fidelity measurements used a frame calibrated by PReSPA Ramsey.
  • Transmon Ramsey: A transmon Ramsey experiment found that PReSPA’s strong off-resonant drives did not affect transmon coherence, with T∗_2q = 17 µs.The measurement isolated the mixing comb’s contribution to the transmon Stark shift.
  • AQEC tracking: Extended Data Fig. 5 tracks corrected logical-qubit decay under two PReSPA parameter sets alongside cavity T2A.Process-fidelity decay was extracted from measurements of all six cardinal points of the logical Bloch sphere.

GRAPE methods

GRAPE-based quantum optimal control prepared cavity states and constructed decoding unitaries for the T4C encoding. The decoding design was constrained on relevant basis states, while its action outside the relevant subspace remained incompletely characterized.

  • State and unitary optimization: Quantum optimal control minimizes deviations between realized and target cavity states or unitaries by optimizing transmon and cavity drive envelopes.The optimization quantifies deviations using state or process infidelity.
  • State preparation: GRAPE maximizes preparation fidelity F_prep = |⟨ψ_T|ψ⟩|^2 for target cavity states using two envelope-modulated drives.Closed-system evolution is simulated in the rotating-frame Hamiltonian.
  • Decoding: The decoding unitary U_d maps the T4C-encoded cavity state to the transmon while satisfying required transformations on four relevant basis states.Additional numerical constraints were used because the minimal requirements did not uniquely determine the unitary.
  • Decoding limitation: The action of U_d outside the seven-dimensional relevant subspace was not characterized and may slightly improve or reduce decoded-state fidelity.Additional constraints could produce small future improvements in logical fidelity.
  • Kerr compensation: Kerr-induced collapse-revival requires dedicated decoding pulses at arbitrary times to account for accumulated phase.Two decoding pulses were used for different time classes, and their performance difference produced alternating fidelity patterns.

Simulation of experimental AQEC

Master-equation simulations modeled the driven transmon–cavity–reservoir system and its experimental error channels. They reproduced the measured corrected lifetime and projected substantial AQEC gains, while identifying transmon excitation and frequency-selectivity effects as important boundaries.

  • Error channels: The simulation included single-photon loss, spurious ancilla excitation, relaxation, dephasing, and other cavity-dephasing mechanisms.The model used T1A, T1r, T1q, γ↑, and γφ to parameterize these processes.
  • Experimental agreement: The simulated corrected logical-qubit lifetime was 290 µs, compared with an experimental Γ_t^-1 = 255 µs.The simulation agreed closely with experiment and reproduced the displayed error-channel contributions.
  • Scaling with coherence: With improved cavity and transmon coherence, transmon-decay and second-photon-loss logical error rates are expected to decrease quadratically for this first-order QEC protocol.Higher-order nonlinearity and drive-tone selectivity do not scale down equally, requiring parameter trade-offs.
  • Projected performance: Simulations predict a logical lifetime 7–9 times longer than physical photon-loss time for a T4C code with n̄ = 3.4.This corresponds to 40–80% above breakeven using state-of-the-art cavity and transmon parameters of the experimental style.
  • Heating mechanism: Spurious transmon excitations can trigger sequential two-photon gain and incoherent logical bit flips in the simulated PReSPA system.The associated heating mechanism is illustrated in Extended Data Fig. 7.
  • High-coherence boundary: At the longest cavity lifetimes, T1A > 2 ms, the QEC gain decreases because imperfect transmon-comb selectivity increases off-resonant excitation errors.Reducing λ and using an alternative intermediate level are proposed ways to suppress these errors.

Supplementary Materials for Protecting a Bosonic Qubit with Autonomous Quantum Error Correction

The supplementary analysis models the T4C code under photon-loss and recovery trajectories, then constructs an approximate decoding unitary for short-time quantum-information retrieval. It identifies codeword-moment matching as the source of optimal short-time distortion suppression while quantifying residual decoding limitations.

  • Loss and recovery trajectories: Photon loss followed by PReSPA produces jump trajectories whose effects can be analyzed through parity-preserving cavity states and jump-number probabilities pj(t).The trajectory evolution is piecewise deterministic, with photon-loss events instantaneously followed by PReSPA.
  • T4C encoding: The T4C encoding is a single-cavity binomial code that can also be viewed as a cat code with |α|2 ≈3.5 truncated at photon number 7.The logical state is encoded as |ψ(0)⟩= x|0L⟩+ y|1L⟩.
  • Approximate decoding: For t/T1A ≪1, decoding uses a unitary selected on the four relevant basis states to approximately recover the transmon state x|g⟩+ y|e⟩.The construction focuses on small jump numbers, which dominate at short times.
  • Distortion suppression: Matching the mean and second photon-number moments between codeword subspaces makes n^kl_0 = 1 up to second order in t/T1A.Together with normalization, these conditions determine the optimal codeword amplitudes described in the supplementary analysis.
  • Limitations: Higher-order expansions and jump numbers j > 0 require matching moments above second order, which cannot be satisfied by the finite T4C codeword choices.The supplementary analysis also identifies optimizing the decoding basis as future work.
  • Decoding fidelity: The decoded reduced transmon state is a good short-time approximation to the intended state, with deviations quantified by process fidelity averaged over six Bloch-sphere cardinal points.The process fidelity is rescaled from the state-transfer fidelity range to 1/4–1.

2. OPTIMAL CONTROL FOR UNITARY OPERATIONS AND STATE TRANSFER

The control protocol uses quantum optimal control to shape discretized transmon and cavity pulses for high-fidelity state transfer or unitary operations. Its objective combines fidelity with hardware-oriented penalties, and ADAM optimization updates the pulses through automatic differentiation.

  • Control objective: Quantum optimal control iteratively adjusts control pulses to minimize deviation between the realized and target unitary or state transfer.The method is designed to maximize the fidelity of the desired operation.
  • Pulse parametrization: Control fields are discretized into N ≫1 time intervals, yielding a finite optimization vector of pulse amplitudes.The experiments use δt = 1 ns for this discretization.
  • Numerical evolution: The realized evolution is computed as a product of short-time propagators exp(−iH_nδt) evaluated at successive time steps.Each propagator uses the Hamiltonian at its corresponding time tn.
  • System model: The control Hamiltonian includes transmon and cavity operators {σx, σy, xA, pA} together with the static system Hamiltonian and Kerr terms.The model includes self-Kerr, fourth-order cross-Kerr, and sixth-order cross-Kerr contributions.
  • Experimental constraints: The cost function combines process or transfer infidelity with penalties for rapid pulse changes, pulse power, and occupation of forbidden higher states.These secondary terms constrain bandwidth, power, leakage, and proximity to the Hilbert-space truncation.
  • Optimization procedure: ADAM performs momentum-accelerated gradient descent with a decaying learning rate and TensorFlow automatic differentiation.Gaussian white-noise pulses initialize the optimization, which typically converges after 1,000–2,000 iterations.

3. ERROR SOURCES OF THE CORRECTED LOGICAL QUBIT

The corrected logical qubit’s errors arise from uncorrected non-loss mechanisms and failures during photon-loss recovery. These include ancilla excitation, finite correction speed, unintended transitions, dephasing, and higher-order dynamics.

  • The total error budget matches measured logical relaxation times of 366 ± 10 µs longitudinally and 256 ± 7 µs transversely within 5%.Errors comprise mechanisms unrelated to single-photon loss and failure modes of the AQEC process.
  • Ancilla excitation can dephase the cavity and trigger erroneous two-photon additions, making the logical qubit modestly more susceptible than a bare |0⟩/|1⟩ cavity encoding.The estimated probability of the erroneous process is about 50%.
  • The current PReSPA scheme leaves residual errors from off-resonant transmon excitation, finite correction-time phase uncertainty, and higher-order nonlinear effects.These effects limit the success rate needed for break-even beyond the ideal photon-loss benchmark.
  • PReSPA is designed for photon loss occurring on a timescale of approximately 150 µs, which can induce both longitudinal and transverse logical errors.
  • Ideal PReSPA reaches 97.5% transverse-error correction success, whereas the experiment estimates 11% pole-state and 24% equator-state failure rates.Ancilla relaxation and second-photon loss contribute substantially, with estimated probabilities of 7% and 5.6%, respectively.
  • Unintended virtual transitions cause about 3% cavity-dephasing probability, while ancilla dephasing is largely tolerated and mainly reduces the correction rate.The virtual-transition contribution scales as (κ/2χq)^2.

4. ERROR RECOVERY TEST

The experiment directly tests whether PReSPA restores an intentionally prepared photon-loss error state. After 25 µs of correction, the recovered logical state retains 67 ± 2% normalized fidelity, consistent with the broader coherence-preservation measurement.

  • An ideal error state is prepared so that PReSPA should return it to |X⟩L after 25 µs.The phases φ3, φ5, and φ7 compensate cavity self-Kerr phase accumulation.
  • 67 ± 2% normalized fidelity is measured for the recovered state after 25 µs of PReSPA.The state is decoded to the transmon and evaluated through a σx projection, with encoding and decoding loss calibrated separately.
  • The result agrees with 67% average phase-coherence preservation and is consistent with a 76% one-time equator-state success probability after other transverse errors are included.

5. PRESPA OPTIMIZER FOR LOGICAL EQUATOR STATES

The authors optimize PReSPA control parameters empirically for logical equator states by minimizing process-fidelity decay. The optimized version slightly improves logical coherence time, while leaving rate matching imperfect.

  • An empirical routine varies PReSPA microwave amplitudes and phases to reduce process-fidelity decay after 144 µs of operation.The state is decoded and assessed by transmon tomography during optimization.
  • PReSPA-2 achieves a slight gain in logical coherence time over PReSPA-1, whose parameters were selected only through calibration experiments.
  • PReSPA-2 has less well-matched Ωn and λn rates, and further work is needed to understand how a deviating dissipative operator can improve AQEC performance.

6. ALL ABOUT READOUT

Readout uses heterodyne transmission through a cavity and calibrated averaged signals rather than single-shot transmon assignment. Several contrast and preparation corrections are characterized, with state-preparation errors excluded from AQEC figures.

  • The transmon is read out through heterodyne transmission, but insufficient signal-to-noise prevents confident single-shot state assignment.Averaged signals are scaled against nominal ground and excited references instead.
  • A 5% equilibrium excited-state population means reported state fidelities are scaled up by 10% relative to a pure target state.
  • Cross-Kerr between the readout mode and cavity A produces photon-number-dependent reductions in transmon readout contrast.The effect also appears in Wigner parity measurements and is quantified using complementary parity mappings.
  • Transmon excitation probabilities are corrected using background measurements so photon-addition probabilities and control parameters can be estimated accurately.
  • State-preparation errors are renormalized in process-characterization figures but are not compensated in figures discussing AQEC.
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