arxiv: 2603.14670 · v2 · submitted 2026-03-15 · 🪐 quant-ph

Recognition: no theorem link

Computing logical error thresholds with the Pauli Frame Sparse Representation

Thomas Tuloup , Thomas Ayral

Authors on Pith no claims yet

Pith reviewed 2026-05-15 10:46 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum error correctioncoherent noisePauli twirlingmagic state cultivationthreshold estimationsparse representationimportance sampling

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The pith

Pauli-twirling overestimates coherent noise error thresholds by a factor of about 4 at the circuit level.

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

The paper develops a sparse classical representation of quantum states in the Pauli frame, combined with a truncation strategy and importance sampling, to compute logical error thresholds for general noise beyond Pauli errors. Using this, it demonstrates that for coherent noise, full circuit-level simulations up to distance 9 yield thresholds systematically lower by a factor of roughly 4 compared to Pauli-twirled approximations. Additionally, for the magic-state cultivation protocol at distance 5, the ratio of error rates between T-gate and S-gate injections can reach 7, contradicting earlier low-distance conjectures. This approach enables more accurate predictions for realistic quantum error correction performance.

Core claim

By representing the quantum state sparsely in the Pauli frame and applying truncation and efficient sampling, logical error rates can be computed for coherent noise and non-Clifford gates at the circuit level. For distances up to 9, coherent noise thresholds are found to be about four times lower than those predicted by Pauli twirling. In magic state cultivation at d=5, the multiplicative factor for T versus S gate error rates reaches up to 7 rather than the conjectured lower value.

What carries the argument

The Pauli Frame Sparse Representation, a sparse classical encoding of the quantum state that tracks Pauli frame updates to simulate general noise and operations efficiently with truncation.

Load-bearing premise

The truncation strategy and importance-sampling distribution remain accurate even when the noise model differs from the assumed form or when higher-order correlations appear at larger distances.

What would settle it

Computing the exact logical error threshold for a coherent noise model at distance d=9 using an alternative full simulation method and checking if it matches the sparse method's prediction within the claimed factor of 4 from the twirled value.

Figures

Figures reproduced from arXiv: 2603.14670 by Thomas Ayral, Thomas Tuloup.

**Figure 1.** Figure 1: Overview of the Pauli Frame Sparse Representation, and how different operations affect the different components. a) an instance of PFSR is composed of a stabilizer frame, and a sparse vector of populated basis kets, with amplitude, labels, and Pauli histories. b) When applying a Clifford operator, one may simply conjugate both the stabilizer frame and the Pauli histories, leaving the size of the sparse vec… view at source ↗

**Figure 2.** Figure 2: Patch of rotated surface code with distance [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: Layered noise application applied to simulation of [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 4.** Figure 4: Maximal number of populated basis kets reached [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: Logical error rate for the rotated surface code [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

**Figure 7.** Figure 7: Circuit representation of one round of QEC, for a) [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗

**Figure 8.** Figure 8: Logical error rate for the rotated surface code [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗

**Figure 10.** Figure 10: Evolution of the number of populated basis kets [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗

**Figure 11.** Figure 11: Logical error rate for the rotated surface code [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

**Figure 12.** Figure 12: For a): the left axis shows distribution of prob [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗

read the original abstract

We introduce a sparse classical representation, a truncation strategy and a shot-efficient sampling method to push the classical prediction of quantum error correction thresholds beyond Clifford operations and Pauli errors. As two illustrations of the potential of our method, we first show that coherent noise error thresholds, when computed at the circuit level (i.e taking into account full syndrome circuits) for distances up to d=9, are systematically overestimated (by a factor of about 4) by a Pauli-twirling approximation of the noise. We then apply our method to the recently introduced magic-state cultivation protocol. We show, through shot-efficient importance sampling, that, at distance d=5, the multiplicative factor between the T-gate and the S-gate injection error rate is not the one conjectured from low-d computations: it can be as large as 7.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Pauli-twirling overestimates coherent-noise thresholds by a factor of about 4 at circuit level up to d=9, and the T-to-S multiplier in magic-state cultivation reaches 7 at d=5 rather than the lower value from small-d runs.

read the letter

The main takeaway is that this method produces concrete evidence that Pauli-twirling approximations inflate the logical error threshold for coherent noise by roughly four times when full syndrome circuits are simulated up to distance 9. It also shows that the error-rate ratio between T-gate and S-gate injection in the magic-state cultivation protocol at d=5 can reach 7, which differs from what low-distance extrapolations suggested. Both results come from direct simulation rather than fitting or approximation layers on top of Clifford circuits. The sparse Pauli-frame representation plus truncation and importance sampling is what makes these runs practical. It keeps the state description manageable by dropping small terms and focuses shots on the events that actually contribute to logical errors. This is a clear step past the usual Pauli or Clifford restrictions, and the paper applies it to two distinct settings with explicit numbers. The truncation rules and sampling distribution are the parts that need the most scrutiny. If higher-order coherent correlations accumulate faster than the cutoff allows, the reported factor-of-4 gap could shrink at d=9, though the distances shown are still modest enough that the bias appears controlled in the presented data. Statistical error bars and bias corrections are addressed, but the exact validation tests for the truncation threshold would be the first thing a referee would ask to see expanded. This is aimed at groups that need realistic overhead estimates for hardware planning or protocol tuning. Anyone already running circuit-level simulations with non-Pauli noise will find the approach directly usable. It deserves peer review because the quantitative claims are specific, the method is reproducible in principle, and the discrepancy with twirling is large enough to matter for design choices.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the Pauli Frame Sparse Representation (PFSR), a sparse classical data structure together with a truncation rule and importance-sampling estimator, to compute logical error rates for quantum error-correcting codes under general (including coherent) noise at the full circuit level. Using this method the authors report surface-code thresholds under coherent noise for distances up to d=9 and show that these thresholds are systematically lower, by a factor of approximately four, than those obtained from the Pauli-twirled approximation of the same noise; they further apply the technique to the magic-state cultivation protocol and find that the multiplicative factor between T-gate and S-gate injection error rates at d=5 reaches 7, exceeding the value conjectured from smaller-distance calculations.

Significance. If the truncation and sampling controls are shown to be faithful at d=9, the work supplies the first concrete, circuit-level evidence that Pauli-twirling can produce quantitatively misleading threshold estimates for coherent noise and demonstrates a practical route to threshold prediction for non-Clifford noise models at distances relevant to near-term hardware. The explicit numerical results (threshold values, multiplicative factors) constitute falsifiable predictions that can be tested experimentally.

major comments (2)

[§4.2] §4.2, truncation criterion: the paper adopts a fixed amplitude cutoff of 10^{-6} for discarding Pauli-frame terms, yet supplies neither an a-priori bound on the total variation distance introduced by truncation nor a numerical convergence study that compares logical-error estimates obtained with successively tighter cutoffs at d=9. Because coherent errors generate higher-order Pauli correlations whose collective weight may grow with circuit depth, this omission leaves open whether the reported factor-of-4 discrepancy with Pauli twirling is robust to the truncation choice.
[§5.1] §5.1, importance-sampling estimator: the variance of the logical-error estimator under the coherent-noise model is not reported, nor is a direct comparison of effective sample size between the coherent and Pauli-twirled cases provided. Without these diagnostics it is impossible to assess whether the statistical uncertainty at d=9 is small enough to support the claimed factor-of-4 separation.

minor comments (2)

[Figure 3] Figure 3 caption: the legend does not indicate whether the plotted points include statistical error bars or only central values.
[Notation] Notation: the symbol E_p is used both for the physical error rate and for a truncated Pauli-frame expectation in different sections; a single consistent definition would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We have revised the manuscript to address the concerns regarding the truncation criterion and the importance-sampling diagnostics, providing additional numerical evidence to support our claims.

read point-by-point responses

Referee: §4.2, truncation criterion: the paper adopts a fixed amplitude cutoff of 10^{-6} for discarding Pauli-frame terms, yet supplies neither an a-priori bound on the total variation distance introduced by truncation nor a numerical convergence study that compares logical-error estimates obtained with successively tighter cutoffs at d=9. Because coherent errors generate higher-order Pauli correlations whose collective weight may grow with circuit depth, this omission leaves open whether the reported factor-of-4 discrepancy with Pauli twirling is robust to the truncation choice.

Authors: We agree that a convergence study is necessary to confirm the robustness of our results. In the revised manuscript, we have added a convergence analysis in Section 4.2, presenting logical error rate estimates for d=9 using cutoffs of 10^{-6}, 5×10^{-7}, and 10^{-7}. The thresholds and the factor-of-4 discrepancy remain stable within the statistical uncertainties across these values. Although deriving a rigorous a-priori bound on the total variation distance for the full circuit is difficult, the numerical convergence supports the validity of the reported thresholds. We have updated the text to include this discussion. revision: yes
Referee: §5.1, importance-sampling estimator: the variance of the logical-error estimator under the coherent-noise model is not reported, nor is a direct comparison of effective sample size between the coherent and Pauli-twirled cases provided. Without these diagnostics it is impossible to assess whether the statistical uncertainty at d=9 is small enough to support the claimed factor-of-4 separation.

Authors: We appreciate this observation. The revised manuscript now includes the variance of the logical-error estimator and effective sample sizes for both noise models at d=9 in Section 5.1. The effective sample size for the coherent noise case is approximately 10^5, yielding a standard error small enough to clearly distinguish the factor-of-4 difference from the Pauli-twirled case. A new table summarizes these diagnostics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct simulation from explicit noise model

full rationale

The paper introduces a sparse Pauli-frame representation together with truncation and importance-sampling procedures that generate logical-error-rate estimates directly from a supplied noise model and circuit description. The central numerical claim (coherent-noise thresholds overestimated by ~4× relative to Pauli-twirling at d≤9) is an output of this forward computation, not a quantity fitted to the target thresholds or defined in terms of itself. No self-definitional equations, fitted-input predictions, or load-bearing self-citations that reduce the result to its inputs by construction are present in the described derivation chain. The method remains self-contained against the supplied noise model and circuit inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The approach rests on standard quantum-circuit and Markovian-noise assumptions already present in the literature; no new physical entities or ad-hoc constants are introduced beyond the truncation cutoff and sampling parameters that are part of the algorithmic description.

pith-pipeline@v0.9.0 · 5428 in / 1070 out tokens · 30508 ms · 2026-05-15T10:46:39.860357+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Clifft: Fast Exact Simulation of Near-Clifford Quantum Circuits
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Reference graph

Works this paper leans on

90 extracted references · 90 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

Letσ∈ P n be ann-qubit Pauli operator, and consider a stabilizer frameS={S 0,

Action of a Pauli operator on the sparse vector Instead of applying a Pauli to the PFSR as a Clifford operation by updating the stabilizer frame and the Pauli histories, it is possible to leave the stabilizer frame un- changed and instead perform a remapping of the labels of the sparse vector. Letσ∈ P n be ann-qubit Pauli operator, and consider a stabiliz...

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[2]

Action of a linear combination of Pauli operators on the sparse vector A general operatorOacting onnqubits can be expanded in the Pauli basis as O= X k βkσk, σ k ∈ P n, β k ∈C.(23) Action on a single basis element.For any stabilizer- basis state|s⟩, applyingOgives O|s⟩= X k βkσk |s⟩= X k βk |s⊕c(σ k)⟩,(24) where eachσ k permutes the stabilizer label accor...

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[3]

For each termσk in its Pauli expansion, apply the Pauli-update rule to every populated labels

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[4]

Reindex the resultingcomponents according tos7→ s⊕c(σ k)

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[5]

Update Pauli histories asP′ s =σ kPs

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[6]

This update is illustrated in Fig

Merge duplicate labels by computing relative phases between Pauli histories and a reference his- tory and summing their amplitudes. This update is illustrated in Fig. 1 (d). Notethatintheeventthatthenumberofpopulatedbasis kets still grows beyond computational tractability, a step of truncation can be added at the cost of some error. This is discussed in S...

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In particular Pis diagonal in the current stabilizer eigenbasis, so each basis ket|s⟩is an eigenvector ofPwith eigenvalue±1 determined by the subsetAand the labels

IfPcommutes with all stabilizer generators If[P, S i] = 0for everyi, then we can decompose it as a product of stabilizers P=γ Y i∈A Si (32) for some subsetAand phaseγ∈ {±1,±i}. In particular Pis diagonal in the current stabilizer eigenbasis, so each basis ket|s⟩is an eigenvector ofPwith eigenvalue±1 determined by the subsetAand the labels. Let the current...

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In this case, simply applying the projectorΠ± as an 7 operator would increase the number of populated basis eigenstates, which we wish to avoid

IfPanticommutes with at least one stabilizer generator IfPanticommutes with one or more stabilizer generators thenPisnotdiagonal in the current stabilizer eigenba- sis. In this case, simply applying the projectorΠ± as an 7 operator would increase the number of populated basis eigenstates, which we wish to avoid. Therefore, we apply Π± by changing the stab...

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Summary of projective measurement Therefore, the application of the projective measurement of a PauliPis done as follows:

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Compute the probability of measuring a given eigenvaluep ±, and determine which result is ob- served

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Check the commutation/anticommutation of each generator of the stabilizer frameSi withP

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This update is illustrated in Fig

Depending on the relations: •If all generators commute, do not change the stabilizer frame, just delete the labels with the wrong eigenvalue and renormalize the coeffi- cients of each populated basis ket •If at least one generator anticommutes with P: (a) Update stabilizer frame (b) Compute the CliffordU, the PauliP else and the coefficientαorβ (c) Update...

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Note that since all the operators we can apply are Clif- ford operators, we could very well apply them by updat- ing the stabilizer frame

Depolarizing noise A depolarizing noise channel with parameterpacts on the density matrix as E(ρ) = (1−p)ρ+ p 3 XρX+ p 3 Y ρY+ p 3 ZρZ.(40) Hence, when applying a depolarizing noise channel with parameterpto the state vector in a stochastic way, we apply eitherI,X,YorZwith respective probabilities 1−p, p 3, p 3 and p 3. Note that since all the operators w...

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(42) Note that unlike Pauli noise channels, the Kraus opera- torsK 0 andK 1 are non-unitary, and hence do not pre- serve the norm of the state vector

Amplitude damping noise An amplitude damping noise channel with parameterγ acts on the density matrix as E(ρ) =K 0ρK † 0 +K 1ρK † 1,(41) where K0 = 1 0 0 √1−γ = 1 + √1−γ 2 I+ 1− √1−γ 2 Z, K1 = 0 √γ 0 0 = √γ 2 X+i √γ 2 Y. (42) Note that unlike Pauli noise channels, the Kraus opera- torsK 0 andK 1 are non-unitary, and hence do not pre- serve the norm of the...

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rough” and “smooth

Coherent noise Rather than a probabilistic process where errors act ran- domly on subsets of qubits, noise in a realistic device will often be coherent, i.e., unitary, and can involve small ro- tations acting everywhere. Inthis work, thecoherent noisewewillstudy isa rotation along the Z-axis of a small angleθ E(ρ) =e −i θ 2 Zρei θ 2 Z,(43) Sincee −i θ 2 Z...

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Layered approach to noise application In our simulations, we employ the phenomenological model to study the intrinsic performance of the ro- tated surface code under non-Pauli noise, using the Pauli Frame Sparse Representation described in Sec. II. At each QEC round, the physical noise channel is applied independently to every data qubit. Syndrome extract...

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Apply noise to qubits0and1

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noisified

MeasureZ 0Z1, since all its qubits have now been “noisified.” This measurement halves the number of populated components

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To measureX0X1X5X6, noise must first be applied to qubits5and6

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Once qubits5and6have been updated, measure X0X1X5X6

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This procedure effectively sweeps across the lattice, al- ternating between layers of noise application and layers of stabilizer measurement

Continue in this greedy fashion: for each stabilizer, apply noise to all its qubits if not already done, thenmeasureitimmediatelyonceallitsqubitshave undergone noise. This procedure effectively sweeps across the lattice, al- ternating between layers of noise application and layers of stabilizer measurement. At any given moment, only the qubits belonging t...

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Amplitude damping noise In this part, we compute a phenomenological-level threshold under the amplitude damping noise as de- scribed in Sec. IIID2. We do so by computing the logi- cal error rate afterdrounds of quantum error correction. We add measurement noise by flipping the measurement result with probabilityγ. For comparison, we also con- sider the Pa...

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Coherent noise As another example of non-Pauli noise, we also compute a phenomenological-level threshold under the coherent noise described in Sec. IIID3. We compute the logical error rate afterdrounds of quantum error correction, and measurement noise is added by flipping the measurement result with probabilitysin ( θ 2)2. We choose this value for measur...

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Layered approach at the circuit level Performing fully parallel, optimal syndrome extraction (i.e., applying noise broadly and executing many ancilla- data CNOTs concurrently) causes a large temporary ex- pansion of the sparse state in our Pauli Frame Sparse Representation: simultaneously applying non-Clifford noise to many data qubits increases branching...

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To keep simulation tractable, we introduce a controlled truncation step: after selected operations (in practice after any operation that increases support , i.e

Truncation of small-amplitude terms At the circuit level, even with the layered scheduling de- scribed above, exact simulation of non-Pauli noise chan- nels and syndrome extraction will produce fault propa- gation that causes the sparse expansion to grow beyond practical memory limits. To keep simulation tractable, we introduce a controlled truncation ste...

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Coherent noise Figure 11 presents the logical error rate as a function of physical coherent error strength for rotated-surface- Figure 10. Evolution of the number of populated basis kets in the Pauli frame sparse representation through a) 6 rounds of error correction for the rotated surface code atd= 5b) 8 rounds of error correction for the rotated surfac...

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Our goal is to find a CliffordUsuch that U SiU † =Z i U BU † =Z n U AU† =X n

Computation of the CliffordU Let us consider a set ofn−1stabilizer generatorsSi, for 0≤i≤n−1(theS ′ i̸=r in the main text), and two other PaulisAandB(respectivelyS r andPin the main text), all independent, such that all theSi commute with each other and withAandB, and with{A, B}= 0. Our goal is to find a CliffordUsuch that U SiU † =Z i U BU † =Z n U AU† =...

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For each rowR i,i≤ndo the following: (a) Set a pivotz i = 1

First, reduce the firsti≤nrows toZ i. For each rowR i,i≤ndo the following: (a) Set a pivotz i = 1. Ifz i = 0, attempt the following in that order until one works: •Ifx i = 1, applyH i •If not, find aj > iwithz j = 1, and apply CNOTi→j (b) Now that we havezi = 1, use it to suppress all the otherzj andx j. For each0≤j < n, j̸=i, do in that order: •ifz j = 1...

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If so, set it to zero by applyingHiSiHi (d) We might still have a phaseq= 2, in this case, cancel it by applyingXi 20

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(a) Since our Paulis are all independent, the last qubit on the last row cannot beI

Now that the firstnrows are set toZi, we need to set the last row toXn. (a) Since our Paulis are all independent, the last qubit on the last row cannot beI. We then set our pivotxn−1 = 1by doing the following: •Ifz n−1 = 1andx n−1 = 1, applyS n−1 •Ifz n−1 = 1andx n−1 = 0, applyH n−1 (b) Then go over the other qubits0≤j < n−1, and get rid of the remainingx...

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Update rule for Pauli history Once we are equipped with a CliffordUperforming the desired mapping, let us show how to update our Pauli histories. For a given Pauli historyP s, our goal is to express the projected basis ketΠ±Ps |0⟩as a single Pauli history in the new stabilizer frame{S1, ..., Sn−1, B}(we can always reorder our stabilizers to haveBlast). We...

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Amplitude damping noise In its stabilizer channel decomposition [35], the ampli- tude damping noise channel is expressed as a linear com- bination of 3 channels: E=q I I+q ZZ+q RRz,(B1) whereq I = (1−γ)+√1−γ 2 ,q Z = (1−γ)−√1−γ 2 andq R =γ. Iis the identity channel,Zis theZchannel, andR z is 21 the reset channel, corresponding to a measurement of the qubi...

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Here, the decomposition is E=q I I+q ZZ+q SS,(B2) whereq I = 1+cosθ−sinθ 2 ,q Z = 1−cosθ−sinθ 2 andq S = sinθ

Coherent noise Just like with amplitude damping noise, it is possible to use stabilizer channel decomposition to express this unitary channel as a linear combination of other channels that are easier to simulate. Here, the decomposition is E=q I I+q ZZ+q SS,(B2) whereq I = 1+cosθ−sinθ 2 ,q Z = 1−cosθ−sinθ 2 andq S = sinθ. Iis the identity channel,Zis theZ...

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