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Stabilized Cat in Driven Nonlinear Cavity: A Fault-Tolerant Error Syndrome Detector

Shruti Puri, Alexander Grimm, Philippe Campagne-Ibarcq, Alec Eickbusch, Kyungjoo Noh, Gabrielle Roberts, Liang Jiang, Mazyar Mirrahimi, Michel H. Devoret, Steven M. Girvin

arXiv:1807.09334v1quant-ph

TL;DR

Ancilla errors during syndrome extraction can propagate to encoded data, while fault-tolerant approaches can require substantial overhead. This paper proposes a parametrically driven cat-cavity ancilla with strongly biased noise and demonstrates syndrome-extraction procedures for multiple code types. The authors describe this as a hardware-efficient route to fault-tolerant error syndrome extraction.

  • Problem

    Ancilla errors during syndrome extraction can propagate to encoded data and produce errors beyond those correctable by the code.

    Source: §I · p. 1

  • Method

    The paper proposes using a pumped-cat ancilla in a parametrically driven nonlinear cavity, whose noise is dominated by bit-flips while phase-flips are exponentially suppressed.

    Source: §I · p. 2

  • Results

    The proposed protocol applies to qubit-based toric and bosonic codes, and the authors describe the PCC as resolving the tension between noise bias and efficient control.

    Source: §VI · pp. 10–11

  • Takeaways & Limitations

    The authors present the PCC as a realistic, hardware-efficient approach to fault-tolerant syndrome extraction.

    Source: §VI · p. 11

    In the discussed dissipative-cat realization, cross-Kerr coupling exceeds the Liouvillian gap and heating causes phase-flips, making the noise channel unbiased; the authors note alternate realizations may overcome this.

    Source: §VI · p. 11

Abstract

from arXiv · show

In quantum error correction, information is encoded in a high-dimensional system to protect it from the environment. A crucial step is to use natural, low-weight operations with an ancilla to extract information about errors without causing backaction on the encoded system. Essentially, ancilla errors must not propagate to the encoded system and induce errors beyond those which can be corrected. The current schemes for achieving this fault-tolerance to ancilla errors come at the cost of increased overhead requirements. An efficient way to extract error syndromes in a fault-tolerant manner is by using a single ancilla with strongly biased noise channel. Typically, however, required elementary operations can become challenging when the noise is extremely biased. We propose to overcome this shortcoming by using a bosonic-cat ancilla in a parametrically driven nonlinear cavity. Such a cat-qubit experiences only bit-flip noise and is stabilized against phase-flips. To highlight the flexibility of this approach, we illustrate the syndrome extraction process in a variety of codes such as qubit-based toric codes, bosonic cat- and Gottesman-Kitaev-Preskill (GKP) codes. Our results open a path for realizing hardware-efficient, fault-tolerant error syndrome extraction.

I. INTRODUCTION

Fault-tolerant syndrome extraction must prevent ancilla errors from propagating to encoded data, but established approaches carry substantial overhead. The paper proposes a pumped-cat ancilla whose biased noise supports a low-overhead extraction protocol across several code types.

I. INTRODUCTION

  • Low-weight couplings can synthesize high-weight stabilizer measurements, but ancilla errors during entanglement can propagate into uncorrectable errors on encoded data.An ancilla bit-flip can instead cause only a syndrome readout error, whereas dephasing or amplitude damping may propagate as a high-weight data error.Source: §I · pp. 1–2
  • Shor, Steane, and Knill extraction use multiple ancillas or encoded ancilla blocks, increasing entangling-gate and hardware overhead.Other cited cat-code schemes offer only first-order ancilla-error protection or require challenging high-weight stabilizer engineering.Source: §I · p. 2
  • A pumped-cat ancilla with strongly asymmetric noise is proposed for fault-tolerant syndrome extraction across toric, bosonic-cat, and GKP codes.The proposed interactions use the ancilla's inherent nonlinearity, without additional coupling elements.Source: §I · pp. 2–3
  • The pumped cat is dominated by bit-flip errors that grow linearly with |β|², while phase-flips and amplitude damping are exponentially suppressed.The pump creates an energy barrier that prevents phase rotations between the coherent states |β⟩ and |−β⟩.Source: §I · p. 2
  • The syndrome is mapped onto the cat state, then a Q-Switch conditionally displaces a readout cavity whose homodyne signal reveals the syndrome.The cat is rotated using low-weight local interactions before readout.Source: Figure 1 · p. 3, §I · p. 3
  • The proposal illustrates syndrome extraction for qubit-based toric, bosonic-cat, and GKP codes, and notes possible extension beyond stabilizer codes.The desired data-ancilla interactions are described as realizable through the ancilla's inherent nonlinearity, without extra coupling elements.Source: §I · p. 3

II. PUMPED-CAT SYNDROME DETECTOR

The detector uses a two-photon-driven Kerr cavity to host a cat-state ancilla and couple it to the encoded system. The stabilizer-conditioned cat flip carries the syndrome, which is obtained by reading out the cat.

II. PUMPED-CAT SYNDROME DETECTOR

  • The two-photon-driven Kerr cavity has degenerate cat states |C±β⟩, with β = √(P/K), and the cat subspace is separated by an energy gap.The gap scales approximately as 4Kβ², while at β = 0 it becomes 2K.Source: §II · p. 3
  • Within the cat subspace, the cavity interaction approximates an ancilla X entangling interaction; small couplings suppress excitations out of that subspace.The energy gap between the cat subspace and its complement suppresses spurious excitations for small couplings.Source: §II · p. 3
  • At time T, the cat flips conditioned on whether the stabilizer is +1 or −1, so reading the cat yields the error syndrome.The coupling strengths and duration are chosen to produce this conditional flip.Source: §II · pp. 3–4
  • A physically convenient coupling of the form L_i a† + L†_i a can also synthesize stabilizer measurements when L_i + L_i† = M_i.The paper uses this coupling form for GKP syndrome extraction.Source: §II · p. 4

III. SINGLE-PHOTON LOSS

Within the cat subspace, single-photon loss primarily produces bit-flips and exponentially suppresses phase-flips. The bit-flips impair syndrome-readout accuracy but do not cause backaction on the encoded system.

III. SINGLE-PHOTON LOSS

  • Single-photon loss primarily causes bit-flips, accompanied by an exponentially small phase-flip error proportional to exp(−2β²).This description assumes dynamics remain confined to the cat subspace.Source: §III · p. 4
  • Because bath coupling commutes with the interaction Hamiltonian, photon-loss bit-flips do not cause backaction on encoded data, though they reduce readout accuracy.Repeating the measurement and taking a majority vote can recover accuracy.Source: §III · p. 4
  • The analysis assumes a bath without thermal excitations; rare leakage out of the cat subspace can impede fault tolerance, though photon loss can autonomously correct leakage errors.In the assumed bath, the PCC loses photons but does not gain them.Source: §III · p. 4

A. Four-qubit stabilizer ˆσz,1ˆσz,2ˆσz,3ˆσz,4 in toric codes

The four-qubit toric-code stabilizer can be measured using two-body interactions with a parametrically driven cat ancilla, rather than a direct five-body interaction. Simulations show that PCC photon loss reduces syndrome-readout accuracy while leaving the qubits unaffected.

A. Four-qubit stabilizer ˆσz,1ˆσz,2ˆσz,3ˆσz,4 in toric codes

  • The stabilizer is measured using two-body qubit–PCC interactions instead of the challenging direct five-body interaction.The four-qubit σz stabilizer has code and error eigenspaces; the PCC state after the interaction indicates the syndrome.Source: A. Four-qubit stabilizer… · pp. 4–5
  • Known unequal interaction strengths can be accommodated by adjusting each qubit’s interaction duration; equal strengths are not required.An alternative is to keep durations fixed and use appropriately timed bit-flip pulses.Source: A. Four-qubit stabilizer… · p. 5
  • At κ = K/200, the PCC indicates the four-qubit stabilizer syndrome with ∼93% probability while the qubits remain unaffected.The figure compares PCC and qubit state probabilities during measurement with κ = 0 and κ = K/200.Source: Figure 2 · p. 5, A. Four-qubit stabilizer… · p. 5
  • At κ = K/10, syndrome-mapping probability falls to 0.52, but the qubits remain in their initial state with probability 0.9999.At κ = K/200, repeating the protocol five times raises the probability of correctly mapping the syndrome to 99.7%.Source: A. Four-qubit stabilizer… · p. 6
  • The example uses χ small compared with the cat-subspace energy gap; large χ/ωgap can cause qubit phase diffusion.The authors also state that this approach can measure the four-qubit σx stabilizer using a Jaynes–Cummings-type interaction.Source: A. Four-qubit stabilizer… · p. 6

B. Cat code stabilizer eiπˆa†

For a bosonic cat code, the PCC measures storage-cavity photon-number parity through a longitudinal interaction. Simulations indicate that PCC photon loss reduces syndrome-readout accuracy without disturbing the storage cat, provided the interaction remains in the stated regime.

B. Cat code stabilizer eiπˆa†

  • The scheme replaces dispersive coupling to a qubit ancilla with coupling between the storage cavity’s photon number and the PCC.The interaction is described as a tunable longitudinal interaction.Source: B. Cat code stabilizer eiπˆa† · p. 6
  • The PCC state records the storage cat’s photon-number parity without revealing its photon statistics when χ is small and PCC dynamics stay within the stabilized cat manifold.Finite χ/Kβ^2 can excite states outside that manifold and cause phase diffusion in the storage cat; a counter-drive can partially correct this effect.Source: B. Cat code stabilizer eiπˆa† · p. 7
  • In the example, β decreases to 0.90, while the qubits are in state |ψe⟩ with probability 0.9999; the storage cavity remains in |ψo⟩ with probability 0.9999.Source: B. Cat code stabilizer eiπˆa† · p. 7
  • The authors state that the approach can extend to binomial and pair-cat stabilizers and can be adapted to a bias-preserving CNOT between PCCs.They note that this CNOT promises to improve threshold requirements in quantum error-correcting codes.Source: B. Cat code stabilizer eiπˆa† · p. 7

C. Gottesman-Kitaev-Preskill (GKP) code stabilizers

The section applies adaptive phase estimation (APE) with a stabilized cat ancilla to estimate GKP stabilizer eigenvalues through conditional displacements. Simulations find that the protocol approaches ideal-ancilla performance for large cat size, while PCC loss does not worsen phase variance under the reported conditions.

C. Gottesman-Kitaev-Preskill (GKP) code stabilizers

  • GKP stabilizers encode displacement errors u and v in their eigenvalues, and their Holevo phase variances quantify estimation uncertainty.The ideal GKP state has zero Holevo variance, while large uncertainty in u or v makes the variance diverge; the code corrects low-rate phase-space displacements.Source: C. Gottesman-Kitaev-Preskill (GKP)… · pp. 7–8
  • APE estimates the GKP stabilizer eigenvalues by sequentially applying ancilla-conditioned displacements, with X-axis feedback rotations chosen from the previous measurement record.The eigenvalue estimates improve as the number of rounds increases; the protocol estimates S_p and S_q using their respective gates and feedback phases.Source: C. Gottesman-Kitaev-Preskill (GKP)… · p. 8, Figure 4 · p. 8
  • A successful APE round reduces the variance of the measured quadrature while leaving the other quadrature unchanged.For a round estimating S_p, V′_p decreases and V′_q remains unchanged; the Holevo variance is defined from the stabilizer expectation magnitudes.Source: C. Gottesman-Kitaev-Preskill (GKP)… · pp. 8–9
  • For large β, APE with the PCC produces the same decrease in Holevo variance as an ideal two-level ancilla; at small β, non-idealities reduce the decrease.Figure 5 compares percent changes in quadrature variances for PCC cat sizes and an ideal ancilla.Source: C. Gottesman-Kitaev-Preskill (GKP)… · p. 9
  • A bit-flip during APE can give an incorrect estimate, but the resulting displacement error can be corrected by repeating APE without increasing phase variance.For β = 2, the reported phase variance remains negligible even at κT = 1; in contrast, relaxation of a two-level ancilla increases the GKP state’s phase variance.Source: C. Gottesman-Kitaev-Preskill (GKP)… · p. 9
  • For β = 2 and κT = 1, the PCC-induced phase variance V^m_q,p is negligible (< 10^-5), while a two-level ancilla with γT_ideal = 1 gives V^m,ideal_q = +9.82.The authors identify the absence of increased phase variance as a hallmark of fault-tolerance.Source: C. Gottesman-Kitaev-Preskill (GKP)… · p. 9

V. READING OUT THE ANCILLA CAT

The PCC’s X-basis state reveals the mapped error syndrome. The proposed readout uses a Q-switched low-Q cavity for fast homodyne detection without requiring additional system nonlinearities.

V. READING OUT THE ANCILLA CAT

  • The proposed X-axis readout avoids additional nonlinearities, while direct single-shot cat-state readout using a transmon has demonstrated fidelity above 99%.The paper presents the homodyne strategy as an alternative readout approach.Source: §V · p. 9
  • The syndrome is read from the PCC’s X-axis state by homodyne detection after Q-switching it into a low-Q readout cavity.The PCC-codeword interaction can be turned off during probing, so readout need not be quantum non-demolition to prevent ancilla errors from propagating to the encoded system.Source: §V · p. 9

A. Rotating the PCC cat states to coherent states

The PCC cat states are rotated into near-coherent states by turning off the two-photon pump and allowing Kerr evolution. Single-photon loss during the pump-off interval can cause phase errors, which remain readout errors addressable by majority vote.

A. Rotating the PCC cat states to coherent states

  • With the two-photon pump off, Kerr evolution rotates the parityless cats into near-coherent states, after which the pump is reapplied to restore phase-flip stabilization.The described Kerr evolution lasts T = π/2K.Source: A. Rotating the PCC cat states to… · p. 10
  • Single-photon loss while the two-photon drive is off can induce phase errors, but these cause only readout errors that majority voting can overcome.Source: A. Rotating the PCC cat states to… · p. 10

B. Q-Switching

Q-switching couples the PCC to a low-Q readout resonator so homodyne detection can measure the PCC’s X-basis state. The measurement rate is constrained by the cat-subspace energy gap, while unwanted Y rotations are exponentially suppressed relative to that rate.

B. Q-Switching

  • Q-switching displaces the readout resonator conditioned on the PCC state, allowing homodyne detection of the PCC’s X basis and thus the error syndrome.The single-photon exchange coupling is activated by external drive(s) that compensate for the PCC–resonator frequency difference.Source: B. Q-Switching · p. 10
  • The homodyne measurement rate is limited by the energy gap between the cat subspace C and C⊥, while unwanted Y rotations are exponentially suppressed relative to the measurement rate.At steady state, Rideal = 8g^2β^2/κr, and the confinement condition is (g^2/κr) ≪ 2Kβ.Source: B. Q-Switching · p. 10

VI. DISCUSSION

The discussion presents the parametrically driven nonlinear cavity (PCC) as a controllable cat ancilla for fault-tolerant syndrome extraction, then explains its energy gap and projected photon operators.

VI. DISCUSSION

  • PCC-based syndrome extraction targets diverse codes with a single strongly biased cat ancilla, while the driven cavity offers controllable stabilization and a Q-switch readout path.An alternative engineered-dissipation cat has not yet shown noise bias; the PCC avoids an external nonlinear element, with cross-Kerr to spurious modes suppressible below ωgap. The alternative uses two-photon dissipation.Source: §VI · pp. 10–11

A. Energy gap

  • The excitation gap is approximately 4Kβ2 at large β, but the exact gap tends to 2K as β decreases, so the approximation fails at small β.The approximate expression converges to the exact gap for large β; at β = 0, the exact gap is 2K.Source: A. Energy gap · pp. 11–12, Figure 7 · p. 12

B. Photon annihilation and creation operators in the cat-subspace

C. Master equation with single-photon loss

For single-photon loss, the paper compares effective and bosonic descriptions with a finite-linewidth bath model, and reports decreasing phase-flip error as β increases.

C. Master equation with single-photon loss

  • When bath coupling is below the cat-subspace gap and thermal excitations cannot bridge it, PCC dynamics remain confined to the cat subspace.Source: C. Master equation with single-photon… · p. 12
  • Numerical comparisons use an effective two-level master equation, a standard bosonic master equation, and a finite-linewidth cavity model; the latter reduces to the effective equation in the weak-coupling limit.The effective master equation assumes a flat spectral density around ωpcc (Markov approximation).Source: C. Master equation with single-photon… · p. 13
  • At t = 2/κC, phase-flip error falls from 0.018 to 0.0067 and 5×10^-7 across increasing-β cases, while bit-flip error rises with β.Source: C. Master equation with single-photon… · p. 13

D. Master equation with single-photon gain

For single-photon gain, the paper compares effective two-level dynamics with a finite-linewidth bath model and identifies spectral and linewidth conditions on that approximation.

D. Master equation with single-photon gain

E. Autonomous correction of out-of-subspace excitation with two-photon dissipation

This section describes phase selection for adaptive phase estimation, the effects of discarding PCC measurement outcomes, and readout and error handling during Kerr rotation and Q-switching. It examines two-photon dissipation for refocusing out-of-subspace cat-cavity excitations, reporting that intermediate photon-loss errors can be detected while finite-rate effects leave residual leakage.

E. Autonomous correction of out-of-subspace excitation with two-photon dissipation

  • Two-photon dissipation can return spurious states from C⊥ to C through environment-induced quantum-Zeno dynamics, provided its excitation rate remains below the C-to-C⊥ gap.For sufficiently weak two-photon coupling, the process introduces no errors within the PCC code subspace.Source: E. Autonomous correction of… · p. 14
  • ≈3 × 10^-4 residual out-of-subspace excitation remains as two-photon dissipation increases, despite the simulated decrease in leakage.The simulations compare κ2ph = 0 and 0.05K with β = 1; correction is imperfect when two-photon coupling itself causes excitations.Source: E. Autonomous correction of… · pp. 14–15, Figure 11 · p. 15

F. Master equation with pure dephasing

  • Increasing cat amplitude β exponentially suppresses the phase-flip rate from pure dephasing.The dephasing-induced probability of leaving the cat subspace is small when gφ,kβ ≪ ωgap.Source: F. Master equation with pure dephasing · p. 15

codes

  • For qubit measurements, phase diffusion scales as (χ/ωgap)^2 at small χ and can be reduced by increasing ωgap; it falls below 10^-4 when χ′/Kβ^2 < 0.045.The first-order theory predicts no phase diffusion for odd-parity inputs, but higher-order effects can produce it as χ increases.Source: H. Phase diffusion during the… · pp. 16–18
  • Storage-photon-number-dependent phases cause phase diffusion, but a counter-drive can correct the mean phase while the PCC measures the stabilizer without revealing photon statistics.For the Gaussian pulse, phase diffusion is below 10^-4 when χ′/Kβ^2 < 0.02.Source: H. Phase diffusion during the… · p. 18

J. Estimating the feedback phases for phase estimation

  • For one phase-estimation round with an initial GKP state approximately in the u = v = 0 stabilizer eigenstate, the feedback phase is chosen as φ = π/2 to maximize sensitivity.In adaptive phase estimation, the feedback phase depends on whether the PCC reached |C+⟩ in the previous round.Source: J. Estimating the feedback phases for… · p. 19

K. Holevo phase variance in the presence of single-photon loss in the PCC

L. One round of APE with a qubit

  • The readout-resonator field displacement is conditioned on the PCC state.The figure shows evolution of the readout cavity's I-quadrature for PCC states; the accompanying text states that the displacement depends on the PCC state.Source: L. One round of APE with a qubit · p. 19, Figure 16 · p. 19

M. Errors during rotation under Kerr-evolution

  • Intermediate single-photon loss during Kerr rotation can move the PCC outside the cat subspace, where Q-switching and homodyne measurement reveal the error; two-photon dissipation can refocus out-of-subspace excitations into the cat manifold.Loss at the start instead gives an incorrect measurement result that can be corrected by repeating the protocol and taking a majority vote; loss at the end leaves the state correct. Reinitialization can be supplemented by the quantum-Zeno effect of two-photon loss.Source: M. Errors during rotation under… · p. 19

N. Evolution during Q-Switch operation

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