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Leveraging biased noise for more efficient quantum error correction at the circuit-level with two-level qubits

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Native bias-preserving CZ gates let the XZZX code exploit biased noise, raising its threshold from 0.66% to 1.27% and cutting qubit footprints by up to 75%.

desk verdict A plausible and useful central idea, but the 1.27% threshold number depends on a residual CNOT error channel the paper never actually specifies. read the letter →

arxiv 2505.17718 v1 pith:GWPQGJKV submitted 2025-05-23 quant-ph

Josu Etxezarreta Martinez , Paul Schnabl , Javier Oliva del Moral , Reza Dastbasteh , Pedro M. Crespo , Ruben M. Otxoa This is my paper
classification quant-ph MSC 81P7381P68
keywords biasednoiseXZZXsurfacecodecircuit-levelbias-preservingCZgatetwo-levelqubitsdephasingerrorsquantumerrorcorrectionqubitfootprint
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Quantum error correction tailored to biased noise works well in idealized code-capacity models, but a no-go theorem says that two-level qubits cannot have bias-preserving CNOT gates, which seemed to rule out circuit-level bias tailoring for most hardware. This paper argues that the bias can still be exploited if the syndrome extraction circuit's CZ gates are implemented natively in a bias-preserving way and if the unavoidable CNOT gates retain even a modest residual bias. Under the proposed hybrid biased-depolarizing circuit-level noise model, the rotated XZZX surface code's threshold rises from about 0.66% to 1.27% (a roughly 90% improvement), and the qubit footprint needed to reach Mega-, Giga-, and Teraquop error rates falls by up to about 75%. If the CZ gates are not bias-preserving, the threshold barely moves, so the native CZ implementation is the decisive ingredient.

What carries the argument

The load-bearing object is the hybrid biased-depolarizing (HBD) circuit-level noise model, which classifies each syndrome-extraction element as either bias-preserving or depolarizing: native CZ gates are followed by two-qubit Pauli noise with bias $\eta$ (pure dephasing errors $\eta$ times more likely than the others), while CNOT, Hadamard, preparation, and measurement are followed by depolarizing or symmetric noise, with every element failing at the same rate $p$. The second pillar is a Lindblad master-equation computation of the CNOT noise channel for the interaction $H_{\mathrm{CNOT}} = V[(\mathbb{I}+Z)/2\otimes \mathbb{I} + (\mathbb{I}-Z)/2\otimes X]$, which shows that the phase-flip bias is reduced but not removed, saturating near $\eta_{\mathrm{CNOT}}\approx 5$ for system bias above $\eta\sim 1000$. The XZZX syndrome extraction pattern (XZZX rather than ZXXZ) is then chosen so that the bias-preserving CZ gates carry most of the syndrome extraction and the residually biased CNOT gates boost the overall biased noise, which is what moves the threshold.

What would settle it

Measure the full two-qubit Pauli error channel of a natively implemented CZ gate and of a CNOT gate in a dephasing-biased platform, such as weak-exchange silicon spin qubits or Rydberg atoms; if the CZ channel's ratio of Z errors to X/Y errors is near 1, or if the CNOT channel's residual bias falls well below $\eta\approx 5$, then the predicted 0.66% to 1.27% threshold jump and the 75% footprint reduction should fail to appear in a distance-scaling memory experiment.

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Extended reading notes

Core claim

The paper establishes that the no-go theorem for bias-preserving CNOT gates does not, by itself, prevent two-level qubits from leveraging biased noise in quantum error correction. The key discovery is that bias-preserving CZ gates are sufficient, provided they are performed natively in a way that commutes with the dominant dephasing channel, and that CNOT gates still carry a residual bias that saturates near $\eta_{\mathrm{CNOT}}\approx 5$. With these ingredients, the rotated XZZX surface code under the proposed hybrid biased-depolarizing circuit-level noise model reaches a threshold of about 1.27% for system bias $\eta\geq 1000$, compared with 0.66% for standard depolarizing circuit-level noise; when the CZ gates are depolarizing, the threshold saturates near 0.7%. Footprint estimates at physical error rate $p=0.003$ show reductions of 57% up to 76% for system biases of $\eta\geq 10$, and at $p=0.001$ the reductions are between 33% and 54%.

Load-bearing premise

The gains rest on the availability of a native CZ gate whose noise channel remains strongly biased toward Z errors; if the CZ gate in a real device is depolarizing, for example because it is compiled from CNOT plus Hadamard or implemented with strong-exchange interactions, the threshold improvement essentially disappears.

Editorial extensions

If this is right

  • With bias-preserving CZ gates and residually biased CNOT gates, the rotated XZZX code's threshold rises from 0.66% for depolarizing circuit-level noise to about 1.27% at system bias $\eta\geq 1000$; in the HBD model without CNOT residual bias the improvement is smaller, from 0.66% to about 0.93%.
  • If CZ gates are implemented in a non-bias-preserving way, the threshold saturates near 0.7%, so the native bias-preserving CZ implementation is the decisive ingredient rather than the XZZX code alone.
  • At physical error rate $p=0.003$, the qubit footprint for Mega-, Giga-, and Teraquop operation falls by up to roughly 75% (for the Teraquop target, from 8977 qubits to as few as 2177) once the system bias exceeds $\eta\approx 100$.
  • At $p=0.001$, footprint reductions are smaller, between 33% and 54%, and the leading-order space-time cost, which scales as $d^3$ in the code distance, drops by 45% to 88% depending on bias and target regime.
  • When the CNOT gates are compiled as CZ plus Hadamard layers (the only-entangling-gate-is-CZ compilation), the threshold still improves, from about 0.53% to 0.8%, showing that the effect survives with CZ-only syndrome extraction circuits.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If a platform's CNOT pulses preserve more than the $\eta_{\mathrm{CNOT}}\approx 5$ seen for the paper's specific interaction Hamiltonian, the threshold gains would be larger than the 1.27% reported here, making the residual CNOT bias a design target rather than a fixed limitation.
  • At realistic non-identically distributed error rates, where single-qubit Hadamard gates fail much less often than two-qubit gates, the CZ-only compilation's extra Hadamard layers would cost less than in the equal-rate HBD model, so the 0.53% to 0.8% result is likely a conservative estimate for real devices.
  • The same design rule, use native ZZ-type interactions for syndrome extraction and avoid compiling CZ through CNOT plus Hadamard, should transfer to bias-tailored qLDPC codes, whose circuit-level syndrome extraction is also entangling-gate-dominated, potentially giving even larger overhead reductions than the surface-code numbers.
  • A direct test on a single device would be to run the same XZZX memory experiment twice, once with a native bias-preserving CZ gate and once with a compiled depolarizing CZ gate; the paper predicts the depolarizing version's threshold should stay near 0.7% while the native version approaches 1.27%.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper addresses whether biased noise can be leveraged for quantum error correction with two-level qubits despite the no-go theorem against bias-preserving CNOT gates. The authors use Lindblad simulations to show that CNOT gates retain a residual bias of about 5 for high system bias, and they argue that CZ gates can be implemented natively in a bias-preserving manner in several platforms. They introduce a hybrid biased-depolarizing (HBD) circuit-level noise model and numerically study the rotated XZZX surface code. The central results are that bias-preserving CZ gates are critical (threshold improves from 0.66% to 0.93% at high bias) and that including the residual CNOT bias raises the threshold to about 1.27% (a ~90% improvement over the standard depolarizing model), with qubit footprint reductions up to 75% at p=0.003. The paper also discusses implications for specific platforms and provides a public code repository.

Significance. If the quantitative claims hold, the paper resolves an important open question: it shows that the no-go theorem on bias-preserving CNOT gates does not preclude leveraging noise bias for two-level qubit QEC at the circuit level, provided native bias-preserving CZ gates are available. The introduction of the HBD noise model is a useful contribution, and the numerical study is extensive, with large Monte Carlo shot counts and a public code repository. The qualitative insight that residual CNOT bias helps is likely robust. However, the quantitative headline results depend on the exact residual-CNOT Pauli channel, which is not specified in the manuscript, limiting the ability to verify or reproduce the reported thresholds and footprint reductions.

major comments (2)
  1. [§III B, Appendix E] The manuscript never specifies the two-qubit Pauli channel used for the residually biased CNOT gates in the circuit-level simulations. Section III B states that the tuple (η_sys, η_CNOT) uses the values obtained in Section II, but a bias ratio η_CNOT ≈ 5 does not uniquely determine the 15 non-identity two-qubit error probabilities; two channels with the same ratio can differ in the split among ZI, IZ, ZZ and in the non-Z rates. Since the minimum-weight perfect-matching decoder edge weights depend on the full error distribution, the threshold improvement from 0.93% (Figure 2a) to 1.27% (Table III) and the footprint reductions in Tables IV and V are directly dependent on this unspecified channel. Please provide the exact error probabilities (or a formula derived from the Lindblad simulation) for all Pauli terms in the CNOT channel, including the per-η_sys values, so that the simulations are reproducible.
  2. [§II, Figure 1] The definition of the residual bias η_CNOT for a two-qubit channel is not given. It is unclear whether η_CNOT is the ratio of the total probability of errors with a Z on either qubit to the total of all other non-identity errors, or some other aggregation. Since the values of η_CNOT obtained from Figure 1 are used as input to the circuit-level simulations (Section III B), an unambiguous definition is required to interpret the mapping from η_sys to the CNOT noise channel.
minor comments (5)
  1. [Abstract, §III B 1, Figure 3] The abstract quotes a 90% threshold improvement and the text of §III B 1 gives 91%, while the caption of Figure 3 says 93%. These numbers should be reconciled to a single consistent value derived from Table III.
  2. [§IV] The statement that silicon spin qubits 'could operate well below threshold right now' overstates the present situation: Table I shows that for silicon spin qubits only the weak-exchange limit yields a bias-preserving CZ gate, and the required fidelity has not been experimentally demonstrated. The claim should be qualified accordingly.
  3. [§II, Figure 1] The values of η_CNOT corresponding to each η_sys are only shown in a plot (Figure 1); a table of these values (or an equation for the fitting curve) would make the simulation inputs explicit and facilitate reproduction.
  4. [Appendix C and bibliography] There are typographical errors in the appendix titles and bibliography, e.g., 'Liebfried' for 'Leibfried' in Appendix C and 'tehcnologies' in the title of Appendix C.
  5. [§III B] The paper would benefit from error bars or confidence intervals on the threshold estimates, or at least a plot of the raw logical error rates for the relevant distances, to support the statistical significance of the claimed threshold differences.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the residual CNOT bias is a computed output of a Lindblad model, the HBD thresholds are independent Monte Carlo results, and the few self-citations are peripheral.

full rationale

The claimed derivation chain is: compute the residual CNOT bias from the Lindblad equation with the Hamiltonian in Eq. (1) and dissipator rates set to give η_sys and roughly 99.7% fidelity (Appendix B); define the HBD circuit-level noise model with bias-preserving CZ gates and depolarizing or residually biased CNOTs (Appendix E); simulate the XZZX code with Stim and decode with MWPM/pymatching (Appendix G), comparing against the standard depolarizing model. At no point is a parameter fitted to the threshold numbers; the residual bias η_CNOT ≈ 5 is an output of the master-equation simulation, and the 1.27% threshold is a numerical consequence of the stated noise model. The self-citations ([2], [6], [17], [49], [52]) are background references or optional decoder suggestions, not load-bearing: the no-go theorem [25] and the gate Hamiltonian [27] are external, and the central numerical claims are benchmarked against the standard depolarizing model and are backed by the released Stim-based code. The incomplete specification of the full 15-term CNOT Pauli channel in Section IIIB is a reproducibility or correctness concern, not circularity, since no threshold value is used to define that channel. Therefore no circular step is established.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central quantitative results rest on the HBD circuit-level noise model (Appendix E), the Lindblad-based residual-bias computation (Appendix B), and the availability of bias-preserving native CZ gates (Table I). These are all assumptions or modeling choices; the paper is transparent about most of them.

free parameters (3)
  • CNOT gate fidelity target = 0.997
    In Appendix B, dissipator rates are set so lambda_ZI+lambda_IZ+lambda_ZZ=0.002, yielding a gate fidelity of about 99.7%; the residual bias curve depends on this hand-chosen target.
  • System bias eta_sys = swept from 1/2 to 10^4
    Input bias parameter of the noise model and Lindblad dissipators; it is swept, not fitted, but the model outputs (eta_CNOT, thresholds) are functions of it.
  • Physical error rate p = 0.003 and 0.001 for footprints; swept for thresholds
    Per-operation error probability in the HBD model, assumed equal for all operations; values chosen to match state-of-the-art experiments, not fitted to the target results.
assumptions (6)
  • domain assumption Each circuit operation is modeled as an ideal gate followed by a Pauli error channel with total error rate p (Pauli twirling).
    Used throughout Appendix E; standard in QEC circuit-level simulations, justified by Pauli twirling.
  • domain assumption The Lindblad equation with Pauli dissipators (Eq. B1) accurately represents the gate noise for computing residual bias.
    Appendix B; Markovian noise with X, Y, Z dissipators on each qubit is assumed; the CNOT residual bias eta_CNOT about 5 follows from this choice.
  • domain assumption CZ gates commute with the dominant Z dephasing and can be natively implemented bias-preserving in a broad class of platforms.
    Section II.C and Table I; this is the key physical assumption, without which the threshold improvement disappears (Figure 2b).
  • ad hoc to paper All circuit operations (gates, idling, SPAM) fail with the same probability p in the HBD model.
    Appendix E; the authors explicitly choose equal error rates as a generic model, while noting real hardware has non-identical rates.
  • ad hoc to paper SPAM errors are unbiased flips with probability p.
    Appendix E; the paper contrasts this with other biased circuit-level models that bias SPAM probabilities; the physics of SPAM bias is left to future work.
  • ad hoc to paper The residual CNOT bias values from Section II are representative of CNOT gates used in the circuit-level simulations.
    Section III.B; the exact CNOT channel probabilities are not specified in the text, but the tuple (eta_sys, eta_CNOT) from Figure 1 is used.

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Pith. "Pith review of Leveraging biased noise for more efficient quantum error correction at the circuit-level with two-level qubits." pith.science (2026). https://pith.science/paper/GWPQGJKV

@misc{pith2026250517718,
  author       = {Pith},
  title        = {Pith review of: Leveraging biased noise for more efficient quantum error correction at the circuit-level with two-level qubits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GWPQGJKV}},
  note         = {Machine review of arXiv:2505.17718}
}
abstract

Tailoring quantum error correction codes (QECC) to biased noise has demonstrated significant benefits. However, most of the prior research on this topic has focused on code capacity noise models. Furthermore, a no-go theorem prevents the construction of CNOT gates for two-level qubits in a bias preserving manner which may, in principle, imply that noise bias cannot be leveraged in such systems. In this work, we show that a residual bias up to $\eta\sim$5 can be maintained in CNOT gates under certain conditions. Moreover, we employ controlled-phase (CZ) gates in syndrome extraction circuits and show how to natively implement these in a bias-preserving manner for a broad class of qubit platforms. This motivates the introduction of what we call a hybrid biased-depolarizing (HBD) circuit-level noise model which captures these features. We numerically study the performance of the XZZX surface code and observe that bias-preserving CZ gates are critical for leveraging biased noise. Accounting for the residual bias present in the CNOT gates, we observe an increase in the code threshold up to a $1.27\%$ physical error rate, representing a $90\%$ improvement. Additionally, we find that the required qubit footprint can be reduced by up to a $75\%$ at relevant physical error rates.

Figures

Figures reproduced from arXiv: 2505.17718 by the authors.

Figure 1
Figure 1. Remainder bias towards dephasing errors when im [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Threshold of the rotated XZZX code as a function of the system bias for the HBD circuit-level noise model with and [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Threshold of the rotated XZZX code as a function [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Layout of the rotated XZZX surface code. White [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: An XZZX check measurement. The check qubit is [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

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Forward citations

Cited by 1 Pith paper

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