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Practical quantum error correction with the XZZX code and Kerr-cat qubits

Andrew S. Darmawan, Benjamin J. Brown, Arne L. Grimsmo, David K. Tuckett, Shruti Puri

arXiv:2104.09539v2quant-ph

TL;DR

Scalable fault-tolerant computation requires error correction that is designed alongside its physical qubits. This paper combines Kerr-cat qubits with the XZZX surface code and reports a threshold near 6.5% CX infidelity under a parameter regime described as routinely achieved in superconducting circuits.

  • Problem

    Scalable fault-tolerant quantum error correction remains experimentally challenging, motivating codesign of the code and physical-qubit architecture.

    Source: §I · p. 1

  • Method

    The paper simulates a concatenated XZZX surface code and Kerr-cat qubits using circuit noise models reflecting identified Kerr-cat noise sources.

    Source: §I · p. 1

  • Results

    At |α|^2 = 6.25 and 8% thermal population, the threshold is κ/K = 2.5 × 10^-4, corresponding to ∼6.5% total infidelity for CX.

    Source: §I · p. 2

  • Takeaways & Limitations

    The authors present the architecture as a promising route to large-scale fault-tolerant quantum computing using parameters routinely achieved in superconducting circuits.

    Source: §I · p. 2, §VI · p. 16

    The surface-code simulations neglect residual leakage, leaving its effects in the Kerr-cat architecture for further investigation.

    Source: B. Kerr-cat noise thresholds · p. 10, §V · p. 13

Abstract

from arXiv · show

The development of robust architectures capable of large-scale fault-tolerant quantum computation should consider both their quantum error-correcting codes, and the underlying physical qubits upon which they are built, in tandem. Following this design principle we demonstrate remarkable error correction performance by concatenating the XZZX surface code with Kerr-cat qubits. We contrast several variants of fault-tolerant systems undergoing different circuit noise models that reflect the physics of Kerr-cat qubits. Our simulations show that our system is scalable below a threshold gate infidelity of $p_\mathrm{CX} \sim 6.5\%$ within a physically reasonable parameter regime, where $p_\mathrm{CX}$ is the infidelity of the noisiest gate of our system; the controlled-not gate. This threshold can be reached in a superconducting circuit architecture with a Kerr-nonlinearity of $10$MHz, a $\sim 6.25$ photon cat qubit, single-photon lifetime of $\gtrsim 64\mu$s, and thermal photon population $\lesssim 8\%$. Such parameters are routinely achieved in superconducting circuits.

I. INTRODUCTION

The paper pairs Kerr-cat qubits, whose noise is strongly biased, with the XZZX surface code to address the challenge of scalable fault-tolerant computation. Simulations report a threshold near 6.5% CX infidelity and a threshold advantage for XZZX under generic biased circuit noise.

I. INTRODUCTION

  • Kerr-cat qubits exponentially suppress bit flips as photon number increases, while XZZX has a high threshold when a particular Pauli error, such as Z, dominates.Their complementary noise and error-correction properties motivate the proposed concatenated architecture.Source: §I · p. 1
  • The study models circuit noise reflecting experimentally identified Kerr-cat processes so thresholds can be expressed in laboratory parameters.The introduction also reports exact small-code simulations that examine non-Pauli noise with optimal decoders.Source: §I · p. 2
  • At |α|^2 = 6.25 and 8% thermal population, the threshold is κ/K = 2.5 × 10^-4, corresponding to ∼6.5% total infidelity for the noisiest operation, CX.The stated superconducting-circuit example uses K/(2π) = 10 MHz and a single-photon lifetime κ^-1 = 63.6 µs.Source: §I · p. 2
  • Roughly a factor of two improvement in threshold error rate over the standard CSS surface code appears under generic biased circuit noise; with standard CX gates, XZZX has a 50% higher threshold.Source: §I · p. 2

II. THE XZZX SURFACE CODE

  • The XZZX code uses four-qubit XZZX stabilizers on a square lattice and encodes one logical qubit with open boundaries.Its smallest X and Z logical operators have lengths dx and dz, respectively, using n = O(dx dz) qubits.Source: §II · p. 3

A. Stabilizer measurements

The XZZX checks are measured by coupling prepared ancillas to neighboring data qubits and reading out the ancillas. The decoding and Kerr-cat descriptions explain how biased errors are handled and how the physical qubit is realized.

A. Stabilizer measurements

  • Each check is measured by preparing an ancilla in |+⟩, coupling it to the four neighboring data qubits, and measuring it in the X basis.The checks can be measured in parallel using a common coupling order, with boundary qubits left idle when a neighbor is absent.Source: A. Stabilizer measurements · p. 3

B. Decoding

  • MWPM matches syndrome defects using edge weights −log p, where p is the most likely error that could produce the defect pair.The syndrome graph is constructed from the error model and observed syndrome.Source: B. Decoding · p. 3
  • For XZZX, Z and X errors produce defect pairs displaced in perpendicular directions, making errors less ambiguous under strongly Z-biased noise.The decoder also assigns weights for circuit-level diagonal separations and hook errors.Source: B. Decoding · pp. 3–4

III. CAT QUBITS AND BIAS PRESERVING GATES

  • The Kerr-cat qubit encodes information in even- and odd-parity cat states separated from higher states by an energy gap approximately |ωgap| ∼ 4Kα^2 at large α.Its nonlinearity enables two-photon dissipation, which can be used for autonomous leakage correction.Source: §III · pp. 4–5
  • A SNAIL’s third-order nonlinearity lets a microwave drive at twice its resonance frequency generate the two-photon term in the Kerr-cat Hamiltonian.The cited experiment found that non-phase-flip errors decreased with increasing cat size.Source: §III · pp. 4–5

A. Kerr cats and dissipative cats

Kerr cats use their self-Kerr nonlinearity for fast gates and qubit operations without external coupling elements. The paper contrasts this with dissipative cats, whose additional nonlinear elements and slower operations complicate implementation.

A. Kerr cats and dissipative cats

  • Kerr-cat circuits can use their inherent nonlinearity for single- and multi-qubit operations and measurements, without external coupling elements.The cited typical energy gaps support fast gates, while circuit lifetimes are reported to be 100–1000 times longer than gate times.Source: A. Kerr cats and dissipative cats · p. 5
  • Dissipative cats have so far had two-photon dissipation strengths limited to 10–100 KHz, leading to gate operations slower than about 1 µs.Their X-basis readout is also described as slow and effectively independent of cat size.Source: A. Kerr cats and dissipative cats · p. 5
  • Additional nonlinear couplers in dissipative-cat systems may introduce dissipation that reduces noise bias, and gate operations may incur non-adiabatic phase-flip errors.The paper consequently focuses on Kerr-cat implementation given these challenges and current technology.Source: A. Kerr cats and dissipative cats · p. 5

B. Logical Cat Operations and Noise Channels

The logical cat operations include X-basis measurement and the gates S, S†, CZ, and CX, with qubits assumed to be pumped into the cat subspace. The S and S† gates use resonant microwave drives whose strength and duration are constrained by the energy gap.

B. Logical Cat Operations and Noise Channels

1. S, S† gates

  • Projecting the resonant-drive Hamiltonian into the logical cat subspace produces an effective Pauli-Z term.Source: §1 · p. 6
  • The S and S† gates use drives with opposite phase relative to the two-photon drive for time TS = π/8αε; increasing cat size or drive strength can shorten TS.The drive must satisfy |ε| ≪ |ωgap| so virtual transitions and leakage remain negligible.Source: §1 · p. 6

2. CZ gate

The section describes Kerr-cat gates and readout operations used to implement surface-code syndrome circuits. CZ uses a parametrically driven interaction, CX uses a bias-preserving conditional phase operation, and X-basis measurement combines ancilla rotation, controlled displacement, and homodyne readout.

2. CZ gate

  • Evolution under the CZ interaction for T_CZ = π/8Jα^2 realizes the gate, with larger α and J enabling faster operation subject to Jα ≪ |ωgap|.The interaction combines a beamsplitter term and a resonant drive on the control SNAIL.Source: §2 · p. 6

3. CX gate

  • CX conditions the target cat-state phase on the control state, using parametric interactions and an adiabatic phase sweep constrained by φ̇(t) ≪ |ωgap|.The gate temporarily leaves the cat-code subspace; a bias-preserving X gate provides the intuition for the conditional phase operation.Source: §3 · p. 6

4. MX, P|+⟩

  • X-basis readout rotates the ancilla states toward the computational basis, then uses controlled displacement and homodyne measurement to infer the qubit state.The ancilla rotation uses S followed by X(π/4), implemented by turning off the parametric drive and allowing self-Kerr evolution for π/2K; the readout displacement grows with cat size.Source: §4 · pp. 6–7, Figure 5 · p. 7, Figure 6 · p. 7
  • The readout interaction is engineered with two microwave drives whose beam-splitter and squeezing strengths are matched to produce Hreadout.In the large-cat limit, a single drive at the difference frequency is sufficient.Source: §4 · p. 7
  • After measurement, the ancilla is projected into a computational-basis state and reinitialized in |+⟩ or |−⟩ by applying X(−π/4) followed by S†.Source: §4 · p. 7

IV. THRESHOLDS

The paper estimates thresholds by simulating logical failure rates across code sizes under generic and Kerr-cat noise models. Under generic biased noise, XZZX outperforms CSS with either bias-preserving or standard CX gates, while simulation choices include a Pauli approximation and bias-tuned code aspect ratios.

IV. THRESHOLDS

  • Thresholds are estimated as the largest physical noise strengths for which logical error decreases exponentially with system size, using noisy syndrome rounds followed by noiseless readout.The decoder identifies failure when the physical error and decoder correction are homologically distinct; calculations use a Pauli noise approximation that smaller exact simulations found did not significantly affect logical error rates.Source: §IV · pp. 7–8
  • The XZZX code aspect ratio is tuned near threshold to balance logical X and Z error rates, then fixed as the code scales; changing it did not appreciably alter the threshold when fixed during scaling.Alternative boundary conditions likewise gave similar threshold values, despite large differences in logical error rates.Source: §IV · p. 8

A. Generic biased circuit noise thresholds

  • The generic biased-noise model assigns phase-flip errors probability pz and other two-qubit Pauli errors pz/ζ, with gate-specific assignments for standard and bias-preserving CX.At ζ = 100, bias-preserving CX has only negligibly higher total infidelity than the standard CX model.Source: A. Generic biased circuit noise… · p. 8
  • The standard CSS comparison uses a rotated layout with twice the code distance of unrotated XZZX at equal physical-qubit count, while the XZZX standard-CX aspect ratio may not be optimal.Source: A. Generic biased circuit noise… · p. 9
  • 0.98 ± 0.05% is the XZZX threshold pz with bias-preserving CX, versus 0.50 ± 0.05% for CSS and 0.80 ± 0.05% for XZZX with standard CX.The bias-preserving XZZX value corresponds to a two-qubit error rate close to 2%.Source: A. Generic biased circuit noise… · pp. 8–9, Figure 7 · p. 9

B. Kerr-cat noise thresholds

The XZZX Kerr-cat code reaches a CX error threshold of about 6.5% for a 6.25-photon cat, compared with 3.1% for a 1-photon cat. The simulations translate these thresholds into device parameters, while noting that residual leakage is omitted from the surface-code simulations.

B. Kerr-cat noise thresholds

  • Threshold noise channels are obtained from master-equation simulations of syndrome-extraction gates, measurement, and preparation for cat sizes α² = 1 and 6.25.The model includes thermal noise with nth = 8% and two-photon dissipation κ2 = K/10; the larger cat has the more biased noise channel.Source: B. Kerr-cat noise thresholds · p. 9
  • The simulated CX uses two steps, which roughly doubles its error probabilities relative to a one-step gate with time-dependent two-photon dissipation.The two-step procedure avoids the numerical and experimental complications of time-dependent two-photon dissipation during CX.Source: B. Kerr-cat noise thresholds · pp. 9–10
  • Residual leakage is suppressed at the physical level but omitted in the subsequent surface-code simulations, leaving its logical effects for further study.Leakage reduction units may be needed; leakage from one Kerr-cat qubit is described as unlikely to propagate to others during two-qubit gates.Source: B. Kerr-cat noise thresholds · p. 10
  • At α² = 6.25, the threshold is κ/K = 2.5 × 10⁻⁴, corresponding to a CX error threshold of ∼6.5%; at α² = 1, the thresholds are 2.5 × 10⁻⁴ and ∼3.1%, respectively.The α² = 1 threshold is κ/K = 1.1 × 10⁻⁴; at threshold, CX bias is ∼351 for α² = 6.25 and ∼16 for α² = 1.Source: Figure 8 · p. 11, B. Kerr-cat noise thresholds · pp. 11–12
  • With α² = 6.25, the non-bias-preserving CXR gate has a lower threshold, ∼1.84 × 10⁻⁴, than the bias-preserving CX despite its lower gate infidelity.The paper attributes the lower CXR threshold to its smaller noise bias.Source: Figure 8 · p. 11
  • At K/2π = 10 MHz, the α² = 6.25 threshold requires a SNAIL lifetime longer than 63.6 μs, while the α² = 1 threshold requires longer than 144.7 μs.The first experimental Kerr-cat demonstration had a SNAIL lifetime of ∼15.5 μs; the paper notes that 50–100 μs lifetimes are routinely achieved in superconducting Josephson-junction circuits.Source: B. Kerr-cat noise thresholds · pp. 11–12

1. Comparison with non-bias-preserving CX gate

The faster non-bias-preserving CXR gate reduces the XZZX threshold by about 40% relative to the bias-preserving CX gate, despite the latter having higher total infidelity.

1. Comparison with non-bias-preserving CX gate

  • The XZZX threshold with CXR is ∼40% lower than with bias-preserving CX, and its below-threshold logical error rate is significantly higher despite CXR's lower total infidelity.CXR is implemented as R CZ R†; a two-photon dissipation channel is applied after it to match residual leakage for the compared gates.Source: §1 · p. 12

2. Comparison with the CSS surface code

In the parameter region studied, the CSS surface code does not reach a threshold with Kerr-cat noise, unlike the XZZX code. The XZZX syndrome round can also be shorter because it uses fewer CX gates, the slowest gates at large α.

2. Comparison with the CSS surface code

  • At α = 2.5, the CSS code's logical error rate stays near 50% across the tested κ/K range and does not decrease with system size; at α = 1, no overall threshold is observed.For a fixed-width CSS code at α = 1, a threshold appears only for Z logical errors when X logical errors are neglected; the overall failure rate remains above threshold.Source: §2 · p. 12
  • At large α, XZZX syndrome-round entangling operations take ∼2TCX, roughly the repetition-code time and half the CSS surface-code time.XZZX checks use two CX and two parallelizable CZ gates; CSS X-check readout requires four CX gates.Source: §2 · p. 12

V. SMALL CODES

Exact small-code simulations compare XZZX, tailored surface, and repetition codes under Kerr-cat noise. XZZX performs well across the tested bias regimes and is relatively robust to decoder miscalibration and standard CX gates, while the other codes have advantages in narrower settings.

V. SMALL CODES

  • The n = 9 simulations use a 3 × 3 rotated layout for XZZX and the tailored surface code, plus a one-dimensional repetition code, with open boundaries.The tailored surface code measures weight-four Pauli-X and Pauli-Y checks; the smallest surface-code realization corrects at least one X and one Z error.Source: §V · p. 13
  • With open boundary conditions, the tailored surface code (TSC) tolerates up to (n - 1)/2 dephasing errors, whereas the 3 × 3 XZZX code can fail logically with as few as two Pauli-Z errors [41].Source: §V · p. 13
  • The repetition code can outperform surface codes when bias is very high relative to system size, but it provides no protection against bit-flip errors.The repetition code corrects up to four dephasing errors, but prior simulations found no increased suppression of logical failures beyond about n ∼10 under realistic biased noise.Source: §V · p. 13
  • XZZX performs well at both high and low bias and is relatively insensitive to decoder miscalibration and using standard rather than bias-preserving CX gates.The comparisons use exact simulations near the estimated threshold, a regime relevant to near-term experiments.Source: §V · p. 13
  • For the 3 × 3 layout, the smallest Z logical has weight 3 in XZZX versus 9 in the tailored surface and repetition codes, which better protect their Z logicals.Repetition-code failures are almost exclusively due to bit-flip errors in the simulations.Source: §V · p. 13
  • The exact surface-code simulations approximate cat qubits as two-level systems and neglect residual leakage, which the authors leave for future study.Leakage probability for the CX gate is stated to be an order of magnitude smaller than that of a non-dominant error.Source: §V · p. 13

A. Break-even error rates

Small-code simulations test break-even performance, gate bias preservation, and the Pauli-twirl approximation. XZZX reaches break-even at the tested high and low biases, while the logical channel is generally close to stochastic Pauli for the surface-code variants but not for repetition code.

A. Break-even error rates

  • At α = 2.5, XZZX and the tailored surface code have logical error rates of 5.41 ± 0.09% and 5.53 ± 0.10%, respectively, matching the CX gate's 5.43% within statistical error.At α = 1, XZZX is the only one of the three codes below the pseudo-threshold because it can detect and correct non-negligible bit-flip errors.Source: A. Break-even error rates · p. 14

B. The necessity of bias-preserving CX gates

  • At α = 2.5, non-bias-preserving CX increases logical error by around 70% for XZZX and nearly 400% for the tailored surface code.The authors suggest the larger tailored-code increase may reflect its use of twice as many CX gates; XZZX can still perform well without bias-preserving gates.Source: B. The necessity of bias-preserving CX… · p. 14
  • The repetition code depends on bias-preserving CX: added bit-flip errors from standard CX make its logical error rate roughly equal to the probability of a data-qubit X error during the circuit.Unlike the surface-code variants, the repetition code is designed only to correct phase-flip errors.Source: B. The necessity of bias-preserving CX… · p. 14

C. Interrogating the Pauli-twirl approximation

  • For surface-code variants, Pauli-twirled and full-noise simulations yield logical χ diagonal terms within statistical error, while off-diagonal terms are at least two orders of magnitude smaller.This supports using the Pauli-twirl approximation for the threshold calculations.Source: C. Interrogating the Pauli-twirl… · pp. 14–15
  • For the repetition code at α = 1 and κ/K = 8.33 × 10−5, the full noise model gives about 10% lower logical-channel fidelity than its Pauli twirl, with off-diagonal χ terms larger than the dominant diagonal terms.The authors relate this coherence to physical X noise remaining an undetectable logical X rotation.Source: C. Interrogating the Pauli-twirl… · p. 15

D. Miscalibrated decoding algorithms

At high bias, XZZX decoding is less sensitive to decoder miscalibration than TSC decoding, while lower bias reduces the effect for both. The discussion presents the architecture’s threshold promise alongside uncompleted resource estimates and noise sources to address as the system scales.

D. Miscalibrated decoding algorithms

  • At α = 2.5, decoder miscalibration raises XZZX logical error by less than 20%, versus over 100% for TSC.The authors suggest heuristic decoding may achieve near-optimal XZZX performance without precise noise characterization.Source: D. Miscalibrated decoding algorithms · p. 15
  • At α = 1, miscalibration increases logical error less for both XZZX and TSC, while repetition-code logical error appears almost unaffected at both biases.The authors suggest tailored decoding matters more as bias increases.Source: D. Miscalibrated decoding algorithms · pp. 15–16

VI. DISCUSSION

  • The simulations use a standard MWPM decoder without pulse optimization, though both decoder improvements and pulse optimization may improve performance.For the simulated parameters, the Kerr-cat anharmonicity is comparable to a transmon’s, leading to fast gates.Source: §VI · p. 16
  • Detailed resource costs for computation below a target logical failure rate remain unanalysed and are left to future work.The authors note that such estimates depend strongly on laboratory noise and may improve with code-layout, decoder, and hardware optimization.Source: §VI · p. 16
  • The proposal is motivated by on-chip superconducting SNAILs, and the authors suggest its scalability with existing technologies could offer a near-term route to fault-tolerant computing.They note that large-scale systems must also account for crosstalk, drive-induced heating, and leakage, and require better noise characterization and code refinement.Source: §VI · p. 16

Appendix A: Surface-code simulation method

The appendix uses Pauli simulations for threshold scaling and exact state-vector simulations for small codes, which can represent general noise and support optimal decoding. The exact method has exponential memory cost and simplifies check-measurement scheduling, with differences in error propagation from the parallelized circuit.

Appendix A: Surface-code simulation method

  • Pauli simulations sample circuit-level Pauli errors, propagate them through the Clifford circuit, and determine the resulting syndrome and residual data-qubit errors.The method supports system-size scaling analysis.Source: Appendix A · p. 16, §1 · pp. 16–17

1. Pauli simulation

  • Threshold simulations increase code distance and noisy measurement rounds together, counting logical failures to identify where increasing distance yields exponential failure-rate reduction.They assume noiseless initial code states and a final noiseless measurement round.Source: §1 · p. 17

2. Exact simulation

  • Exact simulation stores and updates the full density operator, allowing arbitrary noise models and optimal decoding, but its memory cost grows exponentially with qubit number.The paper notes tensor-network techniques can ease the memory burden, though their overall computational costs remain daunting.Source: §2 · p. 17
  • Exact simulations measure checks sequentially rather than in parallel, assigning idle noise to match per-qubit total strength while allowing error propagation to differ.Noise on qubits idle during CZ rounds is neglected in the exact simulations but included in Pauli simulations.Source: §2 · p. 17

Appendix B: Kerr cat and purely dissipative cat gate fidelities

The appendix compares Kerr-cat and purely dissipative cat S gates, focusing on leakage and phase-flip errors. It reports that the additional phase-flip errors associated with dissipative cats are noticeably absent in Kerr cats, and describes a hybrid gate simulation with added two-photon dissipation.

Appendix B: Kerr cat and purely dissipative cat gate fidelities

  • The figure caption attributes Kerr-cat leakage oscillations to off-resonant population transfer between cat and leakage spaces, and says smoother time-dependent pulses remove them.The average leakage is reported to agree with the caption’s analytic estimate.Source: Figure 10 · p. 18
  • Additional phase-flip errors from cooling excited states are described for purely dissipative cats, but are noticeably absent in Kerr cats, whose gates are reported to have higher fidelity.The dissipative-cat phase-flip amount scales as ∝ ε/κ2α3 and increases with ε.Source: Appendix B · p. 18
  • For the S-gate simulations, α = 2 and equal gate times were used; κ2 = 2K was chosen to make the energy and Lindbladian gaps, or residual leakage, approximately equal.Other decoherence sources were neglected in this comparison.Source: Appendix B · p. 18
  • The hybrid S-gate simulation adds small two-photon dissipation, κ2 = K/10, to the Kerr-cat Hamiltonian and reports no fidelity degradation in the studied parameter regime.The figure compares S-gate infidelity and leakage for Kerr cats, purely dissipative cats, and Kerr cats with two-photon dissipation.Source: Appendix B · p. 18, Figure 10 · p. 18

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