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Correlated Coherent Errors in Stabilizer Codes: A General Cumulant Framework and Interference-Based Error Suppression

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Under correlated coherent Z noise, the logical channel induced by QEC depends on which stabilizer eigenspace is used as the codespace, and the right choice converts positive noise correlations from a liability into a resource.

desk verdict A genuinely new analytical framework for correlated coherent Z noise in stabilizer QEC; core derivations hold up, with the noiseless-gadget assumption the main caveat — but it is stated, acknowledged, and testable. read the letter →

arxiv 2607.22503 v1 pith:3C4LBLMF submitted 2026-07-24 quant-ph

Rohan N Rajmohan , Antoine Brillant , Peter Groszkowski , Alireza Seif , Jens Koch , Aashish Clerk This is my paper · ORCID
classification quant-ph
keywords stabilizercodescoherenterrorscorrelatednoiselogicalinfidelitycumulantexpansioneigenspacePROSEencodingPaulitwirling
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

This paper claims that, for correlated coherent Z errors, quantum error correction performance is controlled not only by the code but by which stabilizer eigenspace is designated as the logical codespace. It derives an exact expression for the logical channel after R QEC cycles and develops a cumulant expansion for the noise-averaged logical infidelity, non-perturbative in the noise strength and valid for arbitrary stabilizer codes. The central mechanism is interference among the amplitudes of different errors within the same syndrome set; the phase of that interference is set by the eigenvalues of the chosen codespace. This makes it possible to suppress errors by encoding in a protected stabilizer eigenspace, which the paper calls PROSE, and to combine PROSE with logical Pauli twirling to outperform standard suppression techniques. If correct, noise correlations—usually assumed harmful—can be used to push logical infidelity below the uncorrelated baseline.

What carries the argument

The central object is the single-cycle logical channel Λ(θ^(r)) = χII I + χZI,im H + χZZ Z̄L, where H = −i[Z̄L,·] is the generator of coherent logical errors and Z̄L is the logical Z operator. Its coefficients are built from coherent sums of the amplitudes of correctable and uncorrectable Z errors in each syndrome set, with phases (−1)^φ given by the eigenvalue of the relating Z stabilizer in the chosen codespace. These phases carry all the encoding-dependence: they determine whether error amplitudes interfere constructively or destructively. The exact R-cycle channel is the product Γ(R,θ) of complex factors constructed from χZZ and χZI,im, and the paper's cumulant expansion in the noise str

What would settle it

Measure the logical infidelity after 100 cycles of a distance-5 stabilizer code under controlled correlated Z rotations with fixed positive inter-qubit correlation, comparing the conventional +1 eigenspace with the PROSE eigenspace identified from a single-cycle scan. The theory predicts that the conventional eigenspace is the worst choice, that several PROSE eigenspaces are near-degenerate, and that PROSE reduces infidelity below the uncorrelated baseline. If instead the conventional eigenspace is not worst, or if PROSE does not go below the uncorrelated baseline, the central interference cla

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the exact logical channel after R QEC cycles under fixed correlated coherent Z noise is captured by Γ(R,θ) = ∏_r(1 − 2χZZ(θ^(r)) + 2iχZI,im(θ^(r))), with logical infidelity ϵ(R) = (1 − Re⟨Γ⟩)/3. The coefficients χZZ and χZI,im are coherent sums of error amplitudes within syndrome-specific sets, and their interference is controlled by the stabilizer eigenvalues of the chosen codespace. For Gaussian noise and odd Z-distance, χZZ scales as θ^(dZ+1) and χZI,im as θ^(dZ), universally across stabilizer codes; for even Z-distance, the hierarchy reverses, making inter-cycle correlations beneficial. Because the noise-averaged channel depends on

Load-bearing premise

The load-bearing premise is that the QEC gadget, comprising syndrome measurement and conditional recovery, is perfectly noiseless; if real measurement or correction noise is included, the interference structure that PROSE exploits could be polluted or renormalized, potentially changing the fidelity ordering of the protocols.

Editorial extensions

If this is right

  • For odd-Z-distance codes, long-ranged inter-cycle correlations are universally detrimental under the conventional +1 eigenspace encoding, while for even-Z-distance codes the same correlations are beneficial at leading order.
  • Selecting a protected stabilizer eigenspace (PROSE) can reduce logical infidelity below the uncorrelated-noise baseline when inter-qubit correlations are positive.
  • Logical Pauli twirling removes the coherent contribution to the second cumulant, turning inter-cycle correlations into a weak beneficial resource and making single-cycle identification of the PROSE valid over a wider parameter regime.
  • Combining PROSE with logical Pauli twirling matches or outperforms QEC-only, PROSE alone, logical twirling alone, logical dynamical decoupling, and physical Pauli twirling under weak stationary noise for arbitrary stabilizer codes.
  • For Gaussian noise with no inter-qubit correlations, the logical infidelity is independent of the encoding eigenspace to all orders; inter-qubit correlations are necessary for the PROSE advantage.

Reading between the lines

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

  • An implication left implicit is that the same eigenspace-interference mechanism should extend to coherent errors along multiple axes; the paper notes its cumulant framework readily generalizes to two orthogonal error axes, which would give a direct test of the mechanism's breadth.
  • A natural extension is to include finite syndrome-measurement noise and ask whether the optimal eigenspace shifts or the PROSE advantage degrades; the paper identifies imperfect QEC gadgets as its own acknowledged next step, but the quantitative consequences are not worked out.
  • Because the optimal eigenspace depends on the noise covariance, one could envision an adaptive calibration protocol that estimates the correlation matrix and switches the logical encoding to the PROSE for that operation; this follows from the single-cycle identification result but is not discussed in the paper.
  • The parity reversal suggests a code-design heuristic: under long-range correlated noise, moving from an odd to the next even Z-distance may be worth more than generic resource-count estimates suggest, though the paper does not perform a full cost-benefit analysis.
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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. This paper develops an analytical framework for the logical channel induced by repeated QEC cycles under correlated coherent Z rotations. For a single-logical-qubit stabilizer code and a maximum-likelihood decoder based on marginal Pauli error probabilities, it derives an exact expression for the R-cycle logical channel for a fixed noise realization (Eqs. 16 and 23, Apps. A-B), and a second-order cumulant expansion for the noise-averaged logical infidelity under Gaussian noise (Eqs. 33-39). The central observations are that the logical channel and infidelity depend on the stabilizer eigenspace chosen as the codespace, that this dependence can be exploited by PROSE encoding, and that combining PROSE with logical Pauli twirling outperforms standard suppression techniques in the weak-noise, stationary regime. The paper also reports a parity effect of the Z-distance and conditions under which positive correlations become a resource.

Significance. The strength of the paper is that the main results are parameter-free analytical statements derived from the covariance matrix, with detailed derivations in the appendices. The exact single-cycle and R-cycle channel formulas (Apps. A-B) and the proof that the χZI coefficient is imaginary (App. C) are careful. The paper audits its own approximations in App. G, comparing the exponential approximation and estimating the third cumulant, and the analytical estimates are checked against exact numerical averaging of Eq. (23) in several figures. If these results withstand scrutiny, the framework is a valuable tool for evaluating correlated coherent noise and provides a new, non-stochastic control knob (the codespace choice) for error suppression. The main caveats are the noiseless-QEC-gadget assumption and the single-logical-qubit scope, both of which should be stated as prominently in the abstract as they are in the main text.

major comments (2)
  1. [Abstract / Sec. III] The abstract and introduction claim the framework applies to "arbitrary stabilizer codes and correlation structures", but Sec. III explicitly restricts to Jn,1K codes; the scalar Γ expression (Eq. 24) and the infidelity formula (Eq. 23) rely on a single logical qubit with one logical Z operator. For multi-logical-qubit codes, the normalizer contains multiple logical Z operators and the simple product form does not follow. Please qualify the claims to single-logical-qubit stabilizer codes, or provide the multi-logical-qubit generalization.
  2. [Sec. V A 1 / Abstract] The paper states that the PROSE can be "efficiently identified in many relevant situations". The procedure described is a brute-force evaluation of the single-cycle infidelity for each of the 2^m Z-stabilizer eigenspaces (m the number of Z stabilizers). No polynomial-time algorithm or complexity bound is provided, and the optimization is a binary quadratic problem in general. Please either provide an algorithm exploiting the structure of the covariance, or soften the "efficiently" claim to "practically for small codes".
minor comments (5)
  1. [Sec. I / Sec. VII] The noiseless-QEC-gadget assumption is clearly stated and acknowledged as future work. Given that the practical comparisons in Sec. VI depend on this assumption, I suggest explicitly flagging in the abstract and conclusion that the PROSE+LT advantage is derived under ideal syndrome extraction and recovery.
  2. [App. H.4] The statement that logical spin-echo approaches the lower bound with corrections O(√δ_corr) is asserted without derivation. Please provide a proof or a reference, or present it as a numerical observation.
  3. [Abstract] The phrase "non-perturbative in the noise" is potentially misleading: Eq. (23) is exact/non-perturbative, but the cumulant expansion (Eq. 33) is an approximation valid for weak noise. Suggest rewording to distinguish the exact expression from the cumulant approximation.
  4. [Sec. V A 2] The "correlations as a resource" result is demonstrated with a specific surface-code example. The abstract's "can be a resource" is acceptable, but the section would benefit from stating more explicitly that this is a proof-of-principle, not a universally guaranteed property.
  5. [References] Reference [15] is cited as version 1; if a published version exists, please update the citation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: results are derived from the stated noise model and QEC assumptions; PROSE is a defined optimum, not a fitted prediction.

full rationale

The derivation chain is self-contained. Starting from the noise superoperator (Eqs. 1-6) and the maximum-likelihood decoding rule, the exact single-cycle channel (Eq. 16) is obtained by algebra in App. A; the eigenspace dependence follows from the explicit phase factors in Eqs. (19)-(22), not from an assumed conclusion. The R-cycle channel (Eqs. 23-24) is derived by induction in App. B. The cumulant expansion (Eqs. 33-39) is an approximation of the paper's own exact expression, and the numerics (noise-averaging Eq. 23) are checked against the analytical estimates as internal consistency, not as a fitted parameter called a prediction. No constants are fitted to data; the covariance matrix is an input, not inferred from outputs. The PROSE claim is definitionally the eigenspace minimizing the derived infidelity, but the nontrivial content - that a single-cycle comparison suffices in the relevant regime (Eq. 42), that this optimum can beat the uncorrelated baseline under correlated noise, and that LT+PROSE outperforms PT because min <= average (Eq. 54) - is derived. Prior experimental work (Ref. [16]) is cited as external support and as a special case the paper generalizes, not as load-bearing justification for the analytical results. The noiseless-gadget assumption is explicitly stated and acknowledged as a limitation in Sec. VII; this scopes the results but is not circular. There are no self-citation chains or uniqueness theorems invoked to force the conclusions.

Assumptions & free parameters 1 free parameters · 7 assumptions · 1 invented entities

The paper introduces no fitted constants and no new physical degrees of freedom. The load-bearing structure is: a clean model (Z-only noise, perfect gadget, fixed ML decoder), a Gaussian weak-noise regime for the cumulant machinery, and a set of explicit decoder-behavior assumptions that power the universal scaling claims. The only invented entity is the PROSE protocol, which carries a falsifiable prediction.

free parameters (1)
  • Noise-model parameters (σ², ϱ, τc/tcyc, ℓc) = σ²=10⁻³; τc/tcyc from 0.1 to 10⁷; ϱ∈[0,1]; ℓc=3 (Figs. 2-11)
    Hand-chosen for the illustrative numerics, not fitted to data and not needed for the derivations; the central scaling claims are parameter-free in these variables. Listed for ledger completeness.
assumptions (7)
  • domain assumption QEC gadget (syndrome measurement and conditional recovery) is noiseless
    Stated in Sec. I ('the QEC gadget—comprising the syndrome measurement and conditional recovery—is assumed to be noiseless'); Sec. VII lists relaxing it as future work. The single-cycle channel (Eq. 14) and codespace preservation behind the R-cycle product (App B) require it.
  • domain assumption Noise consists only of data-qubit Z rotations R_Zℓ(θ) with random angles; no X or Y errors
    Eq. (1) defines the model; the channel form (Eq. 16) has only I, H, Z̄ components because the noise is Z-type.
  • domain assumption Decoder is a maximum-likelihood decoder for uncorrelated stochastic Pauli Z noise with marginals ⟨sin²θ_ℓ⟩
    Sec. III A 1 fixes this decoding rule; the coefficients of the exact channel (Eq. 19) depend on which errors the decoder assigns to correctable/uncorrectable classes.
  • ad hoc to paper For odd dZ and small comparable per-qubit marginals, the decoder corrects all weight ≤ (dZ−1)/2 errors and the dominant uncorrectable errors have weight (dZ+1)/2
    Sec. III B 1 asserts this to obtain χZZ ~ θ^{dZ+1}, χZI,im ~ θ^{dZ}; it is assumed rather than proved for arbitrary codes, and the universal claims inherit it.
  • domain assumption Zero-mean multivariate Gaussian noise for the main analytical results
    Sec. III B 2 specializes the cumulants to Gaussian Σ; Wick's theorem is used in Eq. (39). Some claims are extended to symmetric non-Gaussian distributions (App E, Sec. VI B).
  • domain assumption Weak-noise regime with the stated validity conditions (σ≪1, Rc σ^{dZ−1}≲1, or Rc σ^{dZ+1}≪1 for LT)
    App G establishes when the second-order cumulant truncation and the exponential approximation (Eq. 29) hold; the PROSE efficiency claims (Secs. V A 1, VI A) rely on this regime.
  • standard math Standard stabilizer-code formalism (commutation structure, normalizer, CSS decomposition, Haar-averaged input fidelity)
    Used throughout; the infidelity metric (Eq. 10) averages over code states, giving the factor 1/3 in ϵ(R) = (1−Re⟨Γ⟩)/3.
invented entities (1)
  • PROSE (Protected Stabilizer Eigenspace) encoding independent evidence
    purpose: Error-suppression protocol: store the logical qubit in the stabilizer eigenspace that minimizes logical infidelity under correlated coherent noise.
    It predicts a measurable fidelity gain tied to a specific eigenspace choice and is falsifiable in any stabilizer-code experiment under coherent noise; a special case (Shor-code protected eigenspace) was demonstrated experimentally by Debroy et al. (Ref. [16]), which the paper builds on.

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Pith. "Pith review of Correlated Coherent Errors in Stabilizer Codes: A General Cumulant Framework and Interference-Based Error Suppression." pith.science (2026). https://pith.science/paper/3C4LBLMF

@misc{pith2026260722503,
  author       = {Pith},
  title        = {Pith review of: Correlated Coherent Errors in Stabilizer Codes: A General Cumulant Framework and Interference-Based Error Suppression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3C4LBLMF}},
  note         = {Machine review of arXiv:2607.22503}
}
abstract

Coherent errors in stabilizer codes are often correlated across qubits and QEC cycles. Having a general analytical treatment of such noise would thus be extremely valuable. We derive here the exact logical channel induced by repeated QEC cycles under correlated coherent $Z$ noise, and develop a broadly general cumulant-expansion framework that yields a tractable expression for the noise-averaged logical infidelity. Crucially, this expression is non-perturbative in the noise, and applies to arbitrary stabilizer codes and correlation structures. It reveals a feature with no analogue in standard stochastic Pauli error models: the induced channel depends on which stabilizer eigenspace is chosen as the codespace. Exploiting this, we introduce protected stabilizer eigenspace (PROSE) encoding, an error-suppression strategy that selects the optimal codespace. We show that this eigenspace can be efficiently identified in many relevant situations. Further, when combined with logical Pauli twirling, PROSE matches or outperforms standard error suppression techniques (dynamical decoupling, Pauli twirling of physical qubits). We also show that noise correlations, usually assumed to be harmful to QEC, can instead be a resource: with the right encoding, even positive correlations reduce the logical infidelity below the uncorrelated baseline. Our results offer a new, broadly applicable lens on correlated coherent noise in stabilizer codes.

Figures

Figures reproduced from arXiv: 2607.22503 by the authors.

Figure 1
Figure 1. Quantum-circuit representation of two consecutive QEC cycles for a stabilizer code. In each cycle r, errors on the n data qubits (red boxes) are modeled as coherent Z rotations, RˆZi (θ (r) i ) [Eq. (1)], where the stochastic rotation angles may be correlated both across qubits and across cycles (dashed red lines). The stabilizer generators gˆi are measured directly (yellow boxes), and the resulting syndrome ⃗m is d… view at source ↗
Figure 2
Figure 2. Infidelity of random odd-dZ CSS codes with the conventional syndrome-zero encoding under increasing strength of positive noise correlations. We plot the logical infidelity ϵ(R) after R = 100 cycles for three random J9, 1K CSS codes [see App. J] with dZ = 5, using the conventional syndrome-zero encoding and the noise model in Eq. (41), with noise strength σ 2 and inter-cycle correlation range τc/tcyc fixed while inte… view at source ↗
Figure 3
Figure 3. Impact of inter-cycle noise correlation range on infidelity ϵ(R) and sensitivity to the parity of the code’s Z distance. We plot ϵ(R) after R = 1000 cycles for the three-qubit (Rep-3; blue, circles) and four-qubit (Rep-4; red, squares) phase-flip repetition codes, under the noise model in Eq. (41), with noise strength σ 2 and inter-qubit correlation strength ϱ fixed. The inter-cycle correlation range τc/tcyc is vari… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: illustrates both how one identifies the PROSE, and the resulting performance gain it yields over 100 QEC cycles, using one of the dZ = 5 CSS codes and noise model from [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Impact of correlations with and without PROSE encoding. Left: A distance-3 rotated surface code, with X and Z stabilizer generators and qubits labeled. Right: Logical infidelity ϵ(R) after R = 100 QEC cycles for this code, under a simplified version of the noise model …
Figure 6
Figure 6. Figure 6: Fidelity improvement by applying logical Pauli twirling. We plot the logical infidelity ϵ(R) of the three-qubit phase-flip repetition code after R = 1000 cycles of QEC-only (black, circles) and QEC+LT (green, triangles), for the noise model in Eq. (41). Noise strength …
Figure 7
Figure 7. Figure 7: a presents this comparison for 100 cycles of a distance-3 rotated surface code [see [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: provides numerical evidence for this conclusion using a 3-qubit repetition code under Gaussian noise with covariance Σ (r,r′ ) ℓℓ′ = σ 2 exp  − |r − r ′ | tcyc τc  , (H9) fixing noise strength σ 2 and varying the inter-cycle correlation range set by τc/tcyc. For smal…
Figure 9
Figure 9. Figure 9: Near-equivalent performance of QEC+logical [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: Logical infidelity improvement under QEC+logical spin-echo relative to QEC-only for the three-qubit phase-flip repetition code and positive, long-ranged correlations, as the inter-cycle correlation range is varied. We plot the infidelity ϵ(R) of the three-qubit phase-…
Figure 11
Figure 11. Figure 11: Logical infidelity of an J9, 1K CSS code under QEC-only and QEC+PT, as the inter-qubit correlation strength is varied, showing the advantage of applying PT under positive noise correlations. We plot the infidelity ϵ(R) after R = 100 cycles, comparing QEC-only (black, …

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Works this paper leans on

57 extracted references · 1 canonical work pages

  1. [1]

    Efficiently identifying the PROSE Finding the PROSE may appear nontrivial: it will generically depend on the number of QEC cycles R, and having to evaluate each possible eigenspace over many-cycle evolution seems daunting. The problem simplifies, however, in the standard setting where the noise is stationary and RcσdZ −1 ≪ 1 (i.e., sufficiently weak noise...

  2. [2]

    Turning noise correlations into a resource Noise correlations, particularly positive ones, are generally considered detrimental to QEC [see e.g., our discussion in Sec. IV 2]. Fig. 4(b) however shows that with PROSE encoding, even positive correlations canreduce the logical infidelity below its value for uncorrelated noise of equal strength. This means th...

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    We briefly relate PROSE encoding to two closely related works

    Relation to previous work The idea to suppress logical infidelity under coherent errors by exploiting its encoding-dependence has only recently begun to be explored. We briefly relate PROSE encoding to two closely related works. Ref. [15] studies correlated coherent errors along one axis and comparable stochastic Pauli noise along an orthogonal axis, in a...

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    Here, ˆ¯XL denotes a representative of the logical X operator [ 37]

    Quantifying noise- and logical Pauli-averaged performance In the QEC+LT protocol, in each cycler, a logical Pauli gate ˆP (r) L ∈ {ˆI, ˆ¯XL, ˆ¯ZL, iˆ¯XL ˆ¯ZL}, chosen independently and uniformly at random, is inserted before the noisy evolution. Here, ˆ¯XL denotes a representative of the logical X operator [ 37]. Then, the same logical Pauli gate is appli...

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    For weak noise, the comparison simplifies by approximating ΓLT(R, ⃗θ) as in Eq

    Comparison with QEC-only and QEC+LDD Using ϵLT(R), we now compare QEC+LT to QEC-only and QEC+LDD. For weak noise, the comparison simplifies by approximating ΓLT(R, ⃗θ) as in Eq. (29), yielding: ϵLT(R)≈ 1− D exp −2 PR r=1 χZZ(⃗θ(r)) E 3 .(48) First, we compare with QEC-only. In principle, applying LT couldincreasethe logical infidelity, just as Pauli twirl...

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    (48) provides a convenient starting point to assess noise correlation effects on ϵLT(R)

    Impact of correlated coherent errors after applying LT Eq. (48) provides a convenient starting point to assess noise correlation effects on ϵLT(R). Approximating ϵLT(R) using a second-order cumulant expansion yields: ϵLT(R)≈ 1−exp κ1,LT(R) +κ 2,LT(R)/2 3 .(50) 10 1 100 101 102 103 104 Correlation time c/tcyc 2 × 10 3 3 × 10 3 4 × 10 3 Logical infidelity R...

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    Inter-cycle correlation effects are suppressed by LT; κ2,LT(R)/κ1,LT(R) is suppressed by σ2 relative to the QEC-only case

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    LT thus turns inter-cycle correlations into a resource, albeit a weak one

    These correlations are beneficial since κ2,LT(R) ≥ 0. LT thus turns inter-cycle correlations into a resource, albeit a weak one

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    Applying LT to an odd-dZ code reproduces the advantage of using an even- dZ code under long-ranged correlations

    With the conventional encoding and positive noise correlations, ϵLT(R) decreases monotonically in inter-cycle correlation strength or range, to leading- order in the noise strength. Applying LT to an odd-dZ code reproduces the advantage of using an even- dZ code under long-ran...

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    Note that in this regime, σ≪ 1 impliesR c ≫1

    Although unusual from the perspective of a naive cumulant hierarchy, this does not by itself signal a breakdown of the expansion: κcoh 2 (R) captures the contribution of coherent logical errors, which do not enter the first cumulant. Note that in this regime, σ≪ 1 impliesR c ≫...

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    The goal of LDD is to modulate the sign of the rotation about this axis, so that coherent logical errors from different cycles interfere destructively

    Protocol outline and computation ofϵ LDD(R) The residual coherent logical error after each QEC cycle lies along a fixed axis of the logical Bloch sphere, chosen here as the logical Z axis. The goal of LDD is to modulate the sign of the rotation about this axis, so that coheren...

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    (30) for a fixed noise realization provides intuition for this agreement

    Intuition for why QEC+LDD and QEC+L T agree in the weak-noise regime The approximation of the exact single-cycle channel in Eq. (30) for a fixed noise realization provides intuition for this agreement. It is given by: Λ(⃗θ(r)) =χII(⃗θ(r))I+χ ZI,im(⃗θ(r))H+χ ZZ(⃗θ(r)) ¯ZL (H10)...

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    Effect of noise correlations after applying LDD Now we turn to the effects of noise correlations on ϵLDD(R). For weak noise, ϵLDD(R) ≈ϵ LT(R), so noise correlations affect QEC+LDD similar to QEC+LT [see Sec. V B 3]. In brief: LDD suppresses the coherent logical errors, so inte...

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    These gates can be absorbed into state preparation and the correction steps, and thus carry no additional overhead

    Protocol outline and computation ofϵ PT(R) In the QEC+PT protocol, a random n-qubit Pauli gate ˆP (r) ∈ Pn is inserted before the noise layer in each cycle r and applied again after the correction step, with the correction step necessarily modified so that the QEC gadget itsel...

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    Comparison with conventional QEC-only , QEC+L T, and QEC+LDD With ϵPT(R) in hand, we now compare with QEC-only, QEC+LT, and QEC+LDD to determine when applying PT provides an advantage. Since ϵPT(R) is independent of the encoding eigenspace while the QEC-only infidelity is not,...

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Pith tools

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