Pith. sign in

REVIEW 4 major objections 6 minor 4 cited by

Efficient simulation of Clifford circuits with small Markovian errors

T0 review · 4 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Clifford circuits with small coherent errors can be simulated classically in polynomial time.

desk verdict A genuinely new poly(n) strong-simulation algorithm for Clifford circuits with small coherent errors, but the flagship million-gate benchmark runs outside the stated small-error regime and no truncation bound is supplied. read the letter →

arxiv 2504.15128 v1 pith:UCVXVBWH submitted 2025-04-21 quant-ph

Ashe Miller , Corey Ostrove , Jordan Hines , Robin Blume-Kohout , Kevin Young , Timothy Proctor This is my paper · ORCID
classification quant-ph MSC 68Q1281P68
keywords CliffordcircuitsclassicalsimulationcoherenterrorsLindbladiannoiseelementaryerrorgeneratorssurfacecodesrandomizedbenchmarkingstabilizerstates
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 noisy Clifford circuits with small, sparse Markovian errors—including coherent (unitary) errors—can be classically simulated in polynomial time, a task previously out of reach whenever the noise is not purely stochastic. The algorithm writes each layer's error as a sparse Lindbladian in an elementary-error-generator basis, propagates all errors to the end of the circuit using Clifford conjugation, and then combines them with Baker-Campbell-Hausdorff and Taylor expansions. If correct, it provides an efficient strong simulator for this error class and makes concrete problems tractable, such as quantifying how coherent errors amplify or suppress syndrome flips in surface-code circuits with hundreds of qubits. Demonstrations include 100-qubit GHZ circuits with analytic cross-checks, 225-qubit random circuits with over a million gates, and syndrome extraction circuits for distance-3 through 11 rotated surface codes.

What carries the argument

The machinery is the elementary error generator (EEG) basis for n-qubit Lindbladians, together with three analytic capabilities it enables: exact symbolic commutation of any two basis elements, conjugation of any element by a Clifford superoperator to another signed element, and evaluation of expectation values after an element acts on a stabilizer state. These turn 'propagate all noise to the end, then expand' into a polynomial algorithm. Sparsity is the load-bearing structural fact: a noise model with $\kappa$ generators per layer stays $\kappa$-sparse under Clifford conjugation, and the BCH and Taylor steps produce only $O((d\kappa)^k)$ and $O((d\kappa)^{kl})$ terms, respectively, for expansion orders $k$ and $l$.

What would settle it

Run the algorithm with $k=1, l=2$ on the 100-qubit GHZ create-and-uncreate circuit while increasing the per-gate rotation angle $\theta$ so that the accumulated phase $\theta_{\mathrm{acc}}$ approaches order one, and compare the predicted success probability to the exact value $p_0 = \cos^2(\theta_{\mathrm{acc}}/2)$; a growing discrepancy would confirm that the expansion's small-total-error premise, not the sparsity representation, is the limiting precondition.

Watch

Extended reading notes

Core claim

The central discovery is that sparsity of an n-qubit error process is preserved when its generators are conjugated by Clifford gates, so a whole noisy Clifford circuit can be compressed into a single sparse Lindbladian before any perturbative approximation. Because the elementary error generators form a basis and transform under Clifford conjugation to signed basis elements, the error channels can be pushed to the end of the circuit exactly and efficiently. The remaining exponential cost is avoided by a kth-order BCH combination of the propagated Lindbladians and an lth-order Taylor expansion of the resulting channel, yielding a polynomial-term approximation to the circuit's error map. From that approximation, outcome probabilities, Pauli expectation values, and process fidelities are evaluated with stabilizer-state formulas. The paper reports simulations on up to 241 qubits and circuits of over a million gates.

Load-bearing premise

The algorithm's accuracy rests on the total error $\epsilon = \sum_{i,G\in S_i}|\epsilon_{i,G}|$ being small enough that low-order BCH and Taylor truncations dominate the error map.

Editorial extensions

If this is right

  • Syndrome extraction circuits for distance-3 through 11 rotated surface codes can be simulated with coherent gate errors, revealing up to roughly a 3-fold enhancement and a 0.33-fold suppression of marginal qubit error rates relative to stochastic noise of equal gate fidelity.
  • Deep random 225-qubit Clifford circuits of depth up to 8196 show process infidelities that grow by up to about 1000-fold as scrambling power decreases, and remain roughly 10-fold above the perfect-scrambling limit even at the highest simulated Hadamard density.
  • The algorithm computes analytic sensitivity matrices, such as S_omega for syndrome bit-flip rates, giving leading-order dependence of observables on all coherent error parameters simultaneously and with little additional computational cost.
  • The method extends beyond the demonstrated examples to non-Markovian noise by promoting error rates to time-dependent stochastic processes, and to learning sparse Lindblad noise models from experimental data.
  • The algorithm also applies to randomized benchmarking circuits, as demonstrated on binary randomized benchmarking circuits with mixed coherent, incoherent, and non-unital errors, where second-order Taylor terms correct a systematic overestimate at larger depths.

Reading between the lines

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

  • A direct stress test suggested by the paper's own validity condition would be to fix k=1, l=2 and increase the total error epsilon to order one on a shallow Clifford circuit where exact simulation remains feasible, checking where the approximate outcome probabilities diverge from exact ones.
  • Because the algorithm returns entire sensitivity matrices, one could close the loop on calibration: use coherent amplification factors as an objective to tune single-qubit rotation angles so that errors cancel within a surface-code syndrome extraction cycle, an optimization the paper suggests but does not run.
  • The poly(n) guarantee holds for fixed expansion orders k and l; a useful benchmark is to measure how quickly approximation error degrades as circuit depth grows at fixed per-gate error rate, since total error typically grows with depth.
  • The approach is likely to combine with gate-set-tomography-style fitting, since the same sparse Lindbladian representation is the natural parameterization for fitting noise models to experimental data; the paper notes this possibility but stops short of demonstrating it.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper introduces an efficient classical algorithm for approximate strong simulation of Clifford circuits subject to sparse, small Markovian noise, including coherent errors. The algorithm propagates each layer's error generator to the end of the circuit using Clifford-conjugation rules for the elementary error generator (EEG) basis, combines the propagated generators with a k-th order Baker-Campbell-Hausdorff (BCH) expansion, and evaluates outcome probabilities or Pauli expectation values using an l-th order Taylor expansion. The authors claim polynomial runtime for fixed k and l when sparsity and depth are polynomial in n, and they demonstrate the method on three examples: a 100-qubit GHZ preparation/un-creation circuit validated against an analytic solution; 225-qubit random Clifford circuits up to depth 8196 (over 1.8 million gates); and syndrome extraction circuits for rotated surface codes of distance 3 to 11 (up to 241 qubits). The surface-code study reports that coherent errors can increase or decrease marginal error rates by factors of about 3 and 0.33, respectively, relative to stochastic models with the same gate fidelity.

Significance. If the approximation error is controlled, the paper provides a substantial extension of efficient noisy-Clifford simulation beyond stochastic Pauli noise to general small sparse Lindbladian noise, including coherent errors. The supplemental derivations of EEG commutation relations and of measurement/expectation-value formulae are self-contained, and the GHZ and binary randomized benchmarking (BiRB) validations provide meaningful checks; the BiRB comparison against exact simulation in particular demonstrates that the algorithm can capture the energy distribution, not just its mean, for 10 qubits. The surface-code results on coherent amplification and suppression are physically interesting and, to the best of my knowledge, go beyond what existing specialized methods (e.g., tensor networks or z-error-only models) can efficiently access at this scale. The open-source implementation in pyGSTi with the use of stim is a further practical strength. However, the central demonstration on deep random circuits appears to operate outside the stated small-total-error regime, so the claimed performance there is not yet supported by the paper's own validity condition.

major comments (4)
  1. [Simulation algorithm and Fig. 3d] The stress-test concern is well-founded: the deep random-circuit benchmark violates the stated expansion-validity condition. The text says the expansions are valid when the total error ε = Σ_{i,G∈S_i}|ε_{i,G}| is small. For the random circuits, n=225, d=8196, and each qubit receives a coherent rotation θ=1e-5 per layer, giving ε ≈ n d θ ≈ 18.4, which is not O(1). At ξ=0 the exact analytic infidelity is 1−cos^{450}(dθ/2) ≈ 0.31, so the plotted range is not confined to a small-error regime. No alternative effective small parameter is identified, and no convergence check (e.g., comparing k=1 vs. k=2 or l=1 vs. l=2) is reported at depth 8196. Since the BiRB validation in the supplement (Supplemental Figs. 5–6) shows that at depth 16 the first-order Taylor expansion systematically overestimates energies and second-order is needed, the depth-8196 curves in Fig. 3d require either a rigorous error bound, a demonstration of an effective small parameter in this regime, or additional convergence checks at representative depths.
  2. [Methods: Computing a circuit's process fidelity] The computation of process fidelity in the random-circuit example is not fully specified. The Methods state that the process fidelity is well-approximated by Eq. (28), and the main text says that at ξ=0 the algorithm's prediction agrees closely with the exact formula cos^{2n}(dθ/2). However, for the stated parameters, the leading-order formula (28) with per-qubit cumulative rotation dθ gives a process infidelity of order n(dθ)^2 ≈ 1.5, whereas the exact infidelity is approximately 0.31. The manuscript should explain how the process fidelity is actually evaluated in this example (e.g., whether the exact exponential of the commuting first-order generator is used, and how this is done efficiently) and why Eq. (28) is applicable, or state which alternative formula is used.
  3. [Example applications: Coherent error scrambling in random circuits] The expansion orders (k,l) are not stated for the random-circuit example. The GHZ example explicitly sets k=1, l=2, but the random-circuit section does not specify the orders. Because the truncation error depends strongly on the Taylor order at large depth, as shown by the BiRB validation, the authors should state k and l for each example and provide evidence that the retained orders suffice, for instance by reporting the difference between l=1 and l=2 (or k=1 and k=2) for a subset of the plotted parameters at depth 8196.
  4. [Methods: Perturbative expansions] The validity condition for the perturbative expansions is given without a precise error bound. The text defines ε twice with different meanings: within the BCH term-size statement ε bounds the maximum per-gate rate, while the validity condition uses the summed rates Σ|ε_{i,G}|. These are different quantities, and the accuracy of the approximation depends on both the sizes of the individual rates and the commutation structure of the propagated generators. A precise statement of the approximation error, such as an operator-norm bound on the difference between E_c and the BCH/Taylor approximation in terms of the summed rates, the expansion orders, and the number of non-commuting terms, would make the algorithm's guarantees testable. This is particularly important because the surface-code example uses θ=0.001 across many gates, so its summed error may also be borderline.
minor comments (6)
  1. [Notation (Eq. 30 and surrounding text)] The symbol k is used for the BCH order in the algorithm description but is reused for the Taylor order in Eq. (30) and the following paragraph. Using distinct symbols (e.g., k_B and k_T) would eliminate ambiguity.
  2. [Introduction vs. Example applications] The Introduction and the random-circuit section give different depth values: the Introduction says depth up to 8192, while the Example applications section says 8196 layers (and Fig. 3c caption says 8192). These should be harmonized.
  3. [Example applications: GHZ setup] The sentence 'To simulate these circuits, we set k = 1 and l = 2 in our algorithm' appears immediately after the GHZ description, but it could be misread as applying to the subsequent random-circuit example. Rephrase to make clear that these orders are for the GHZ example.
  4. [Code and Data Availability] The statement that Python notebooks 'will be released shortly' is weaker than the usual availability expectation and does not provide a persistent repository or version. Providing a DOI or a stable public repository link would improve reproducibility.
  5. [Throughout the examples] The convention for θ differs between examples: in the random-circuit section θ appears to be the rotation angle, while in the surface-code section θ corresponds to twice the rotation angle (as stated in the text). Making this convention explicit at each use is important because it directly affects the total-error estimates and the comparison to the analytic formula.
  6. [Methods: Random circuits simulation] The paper does not report the actual runtime of the three example simulations beyond saying they are feasible on a laptop. A brief table of runtimes and the parameters (n, d, k, l, κ) would help readers assess the practical scalability of the method.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the simulated quantities are outputs of the stated BCH/Taylor expansions for specified noise models, and the EEG-basis self-citation is independent, not target-equivalent.

full rationale

The derivation chain is self-contained: the algorithm's predictions are computed from the stated sparse Lindblad noise model by (i) exactly propagating elementary error generators under Clifford layers using the conjugation formula (Eq. 14), (ii) BCH-combining the propagated generators (Eqs. 25-26), and (iii) Taylor-expanding the combined error map (Eq. 27) with analytic matrix-element formulas (Eqs. 32-35 and the supplemental derivations). The reported quantities--GHZ success probability, random-circuit process infidelity, surface-code coherent amplification factors, and sensitivity matrices S_omega and v_omega--are outputs of these expansions for specified model parameters, not fitted inputs renamed as predictions. The analytic GHZ check and the xi=0 random-circuit check compare against independent closed-form expressions, and no parameter is calibrated to the quantity it later predicts. The only overlapping-author citation is the elementary-error-generator basis of Ref. [21], which is an externally published taxonomy supplying a representation rather than any target prediction; per the stated independence rules, that citation does not raise the circularity score. The deep random-circuit example's large summed error rate is a validity concern about whether the expansion is controlled, not a circularity, because no equation or definition forces the simulation result to equal the model input.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The algorithm's central claim rests on the EEG basis from prior work (Ref. [21], with overlapping authorship), the smallness and sparsity of the noise, and standard BCH/Taylor and stabilizer machinery. No new physical entities are introduced. The free parameters listed are demonstration choices and truncation orders, not fitted data; the headline coherent amplification factors depend on the specific hand-chosen θ vector.

free parameters (4)
  • Surface code Hamiltonian rates θ (18-element vector) = θZ = θIX = -θZI = 0.001, all others 0
    Chosen by hand for the coherent amplification demonstration; the headline 3.33x and 0.33x factors depend on this specific choice, and the paper shows sensitivity to θ in Fig. 1e.
  • Random circuit per-qubit rotation angle θ = 1e-5
    Chosen for the 225-qubit depth-8192 random circuit simulations; the paper does not fit it to data, but the validity of the perturbative expansion depends on its smallness relative to circuit depth.
  • GHZ validation rotation angle θ = 1e-4
    Chosen for the accuracy check against the analytic cos^2 solution.
  • Expansion orders k (BCH) and l (Taylor) = k=1, l=2 stated for GHZ; surface code section states second-order terms were computed but does not state the BCH order
    The polynomial runtime is only guaranteed for fixed k and l, and the numerical accuracy of all reported quantities depends on these truncation choices.
assumptions (5)
  • domain assumption The elementary error generators (EEGs) of Ref. [21] form a basis for trace-preserving superoperators, and any sufficiently small CPTP map can be written (or well approximated) as exp of a sum of EEGs with real rates.
    Invoked in Methods Eq. (9)-(11); the algorithm's sparse representation of noise depends on this published construction, which is from a paper with overlapping authorship.
  • standard math The Baker-Campbell-Hausdorff formula and Taylor expansion of the error superoperator converge and give controlled error in the small-error regime.
    Used in Algorithm step 2; the paper asserts validity when total error is small but gives no explicit convergence bound.
  • standard math Clifford unitaries and stabilizer states can be represented and manipulated efficiently (Aaronson-Gottesman, Gidney's stim).
    Used throughout for conjugation, symplectic updates, and evaluation of the alpha and beta coefficients in the supplement.
  • domain assumption The noise models considered are sparse in the EEG basis, with κ small (e.g., growing at most linearly with n for local noise).
    Stated as a requirement in the abstract and Methods; efficiency and the O((dκ)^k) term count depend on it.
  • domain assumption The total accumulated error in the simulated circuits is small enough that low-order truncations dominate.
    The paper states expansions are valid when total error ε is small, but does not quantify ε for the depth-8192 random circuit example, where a naive sum of per-qubit-per-layer rates is about 18.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Efficient simulation of Clifford circuits with small Markovian errors." pith.science (2026). https://pith.science/paper/UCVXVBWH

@misc{pith2026250415128,
  author       = {Pith},
  title        = {Pith review of: Efficient simulation of Clifford circuits with small Markovian errors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UCVXVBWH}},
  note         = {Machine review of arXiv:2504.15128}
}
abstract

Classical simulation of noisy quantum circuits is essential for understanding quantum computing experiments. It enables scalable error characterization, analysis of how noise impacts quantum algorithms, and optimized implementations of quantum error correction. However, most existing efficient simulation techniques can only simulate the effects of stochastic (incoherent) noise. The lack of efficient ways to simulate coherent errors, which are common and significant in contemporary quantum computing systems, has frustrated research. We remedy this gap by introducing an efficient algorithm for approximate simulation of Clifford circuits with arbitrary small errors (including coherent errors) that can be described by sparse $n$-qubit Lindbladians. We use this algorithm to study the impact of coherent errors on syndrome extract circuits for distance-3, 5, 7, 9, and 11 rotated surface codes, and on deep random 225-qubit circuits containing over a million gates.

Figures

Figures reproduced from arXiv: 2504.15128 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. a, creates a 100-qubit GHZ state and then uncreates it. If no errors occur, measuring each of the 100 qubits will yield 0, so the probability p0 of this outcome is a simple measure for circuit success. We simulated a noise model in which each gate (including single-qubit idles) is followed by a coherent error that rotates the target qubit around the z-axis by either −θ, for the first η qubits, or +θ, for the rest. T… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Each BiRB circuit consists of (1) sampling a random nonidentity Pauli operator, [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of the exactly computed BiRB circuit energies with those computed using error generator propagation using the first [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Absolute error between approximately and exactly computed BiRB circuit energies, [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

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

  1. Theory and Architecture of Syndrome-Resolved Logical Gates

    quant-ph 2025-10 conditional novelty 7.0 of 10

    A new theorem gives weak (probabilistic) transversal implementations of arbitrary multi-qubit Pauli rotations for CSS codes, enabling an in-place rotation architecture with large projected resource savings.

  2. SymFT: Universal Fault-Tolerant Quantum Circuit Simulation via Symbolic Clifford--Pauli Frames and Stabilizer Coordinates

    quant-ph 2026-07 conditional novelty 6.5 of 10

    SymFT reaches state-of-the-art exact sampling of Clifford-dominated FT circuits by combining symbolic Clifford–Pauli frames with planned dense stabilizer-coordinate updates.

  3. Plaquette: A hardware-aware design platform for fault-tolerant quantum computers

    quant-ph 2026-07 conditional novelty 6.0 of 10

    Plaquette compiles realistic quantum hardware noise models into multiple sampler representations, showing that Pauli-twirled approximations can misestimate logical error rates by an order of magnitude compared to leak...

  4. Continuous Noise Model for Quantum Circuits

    quant-ph 2026-04 unverdicted novelty 6.0 of 10

    On small stabilizer-code and Grover circuits, coherent Gaussian noise and Pauli noise rank differently under entropy-matched comparison, and a simplified variance-propagation model matches full simulation only in unco...

Reference graph

Works this paper leans on

78 extracted references · 54 canonical work pages · cited by 4 Pith papers

  1. [1]

    Takahashi, Y

    Y . Takahashi, Y . Takeuchi, and S. Tani,45th International Sym- posium on Mathematical Foundations of Computer Science (MFCS 2020) (Schloss Dagstuhl - Leibniz-Zentrum f ¨ur Infor- matik, 2020) pp. 83:1–83:13

    work page 2020

  2. [2]

    Gidney, Stim: a fast stabilizer circuit simulator, arXiv:2103.02202 (2021)

    C. Gidney, Stim: a fast stabilizer circuit simulator, arXiv:2103.02202 (2021)

    arXiv 2021

  3. [3]

    Aaronson and D

    S. Aaronson and D. Gottesman, Improved simulation of stabi- lizer circuits, Phys. Rev. A 70, 052328 (2004)

    2004

  4. [4]

    Delfosse and A

    N. Delfosse and A. Paetznick, Simulation of noisy cli fford cir- cuits without fault propagation, arXiv:2309.15345 (2023)

    arXiv 2023

  5. [5]

    Gosset, D

    D. Gosset, D. Grier, A. Kerzner, and L. Schae ffer, Fast simula- tion of planar clifford circuits, Quantum 8, 1251 (2024)

    work page 2024

  6. [6]

    Fontana, M

    E. Fontana, M. S. Rudolph, R. Duncan, I. Rungger, and C. Cˆırstoiu, Classical simulations of noisy variational quantum circuits, arXiv:2306.05400 (2023)

    arXiv 2023

  7. [7]

    Gonz ´alez-Garc´ıa, J

    G. Gonz ´alez-Garc´ıa, J. I. Cirac, and R. Trivedi, Pauli path simulations of noisy quantum circuits beyond average case, arXiv:2407.16068 (2024)

    arXiv 2024

  8. [8]

    Xun and D

    G. Xun and D. Luming, E fficient classical simulation of noisy quantum computation, arXiv:1810.03176 (2018)

    arXiv 2018

Show all 78 references
  1. [9]

    Bouwmeester, J.-W

    D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, Experimental quantum teleportation, Nature 390, 575 (1997). 9

    1997

  2. [10]

    Google Quantum AI and Collaborators, Quantum error correc- tion below the surface code threshold, Nature 638, 920 (2025)

    2025

  3. [11]

    Magesan, J

    E. Magesan, J. M. Gambetta, and J. Emerson, Scalable and ro- bust randomized benchmarking of quantum processes, Phys. Rev. Lett. 106, 180504 (2011)

    2011

  4. [12]

    T. J. Proctor, A. Carignan-Dugas, K. Rudinger, E. Nielsen, R. Blume-Kohout, and K. Young, Direct randomized bench- marking for multiqubit devices, Phys. Rev. Lett. 123, 030503 (2019)

    2019

  5. [13]

    Hines, D

    J. Hines, D. Hothem, R. Blume-Kohout, B. Whaley, and T. Proctor, Fully scalable randomized benchmarking without motion reversal, PRX quantum 5, 030334 (2024)

    2024

  6. [14]

    Blume-Kohout, J

    R. Blume-Kohout, J. K. Gamble, E. Nielsen, K. Rudinger, J. Mizrahi, K. Fortier, and P. Maunz, Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography, Nat. Commun. 8, 14485 (2017)

    2017

  7. [15]

    M. T. Madzik, S. Asaad, A. Youssry, B. Joecker, K. M. Rudinger, E. Nielsen, K. C. Young, T. J. Proctor, A. D. Baczewski, A. Laucht, V . Schmitt, F. E. Hudson, K. M. Itoh, A. M. Jakob, B. C. Johnson, D. N. Jamieson, A. S. Dzurak, C. Ferrie, R. Blume-Kohout, and A. Morello, Prec...

    2022

  8. [16]

    H. J. Briegel, D. E. Browne, W. D ¨ur, R. Raussendorf, and M. Van den Nest, Measurement-based quantum computation, Nat. Phys. 5, 19 (2009)

    2009

  9. [17]

    J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Optimal frequency measurements with maximally correlated states, Phys. Rev. A 54, R4649 (1996)

    1996

  10. [18]

    Z. Bao, S. Xu, Z. Song, K. Wang, L. Xiang, Z. Zhu, J. Chen, F. Jin, X. Zhu, Y . Gao, Y . Wu, C. Zhang, N. Wang, Y . Zou, Z. Tan, A. Zhang, Z. Cui, F. Shen, J. Zhong, T. Li, J. Deng, X. Zhang, H. Dong, P. Zhang, Y .-R. Liu, L. Zhao, J. Hao, H. Li, Z. Wang, C. Song, Q. Guo, B. H...

    2024

  11. [19]

    T. J. Proctor, P. A. Knott, and J. A. Dunningham, Multiparam- eter estimation in networked quantum sensors, Phys. Rev. Lett. 120, 080501 (2018)

    2018

  12. [20]

    Hashim, L

    A. Hashim, L. B. Nguyen, N. Goss, B. Marinelli, R. K. Naik, T. Chistolini, J. Hines, J. P. Marceaux, Y . Kim, P. Gokhale, T. Tomesh, S. Chen, L. Jiang, S. Ferracin, K. Rudinger, T. Proc- tor, K. C. Young, R. Blume-Kohout, and I. Siddiqi, A practical introduction to benchmarkin...

    2024

  13. [21]

    Blume-Kohout, M

    R. Blume-Kohout, M. P. da Silva, E. Nielsen, T. Proctor, K. Rudinger, M. Sarovar, and K. Young, A taxonomy of small markovian errors, PRX Quantum 3, 020335 (2022)

    2022

  14. [22]

    R. S. Bennink, E. M. Ferragut, T. S. Humble, J. A. Laska, J. J. Nutaro, M. G. Pleszkoch, and R. C. Pooser, Unbiased simulation of near-clifford quantum circuits, Phys. Rev. A 95, 10.1103/physreva.95.062337 (2017)

    2017 doi

  15. [23]

    Y .-T. Chen, C. Farquhar, and R. M. Parrish, Low-rank density- matrix evolution for noisy quantum circuits, Npj Quantum Inf. 7, 1 (2021)

    2021

  16. [24]

    Pednault, J

    E. Pednault, J. A. Gunnels, G. Nannicini, L. Horesh, T. Mager- lein, E. Solomonik, E. W. Draeger, E. T. Holland, and R. Wis- nieff, Pareto-efficient quantum circuit simulation using tensor contraction deferral, arXiv:1710.05867 (2017)

    2017 arXiv

  17. [25]

    P. Rall, D. Liang, J. Cook, and W. Kretschmer, Simulation of qubit quantum circuits via pauli propagation, Physical Review A 99, 062337 (2019)

    2019

  18. [26]

    Angrisani, A

    A. Angrisani, A. A. Mele, M. S. Rudolph, M. Cerezo, and Z. Holmes, Simulating quantum circuits with arbitrary local noise using pauli propagation, arXiv:2501.13101 (2025)

    2025 arXiv

  19. [27]

    G. Nebe, E. M. Rains, and N. J. A. Sloane,Self-Dual Codes and Invariant Theory, 2006th ed., Algorithms and Computation in Mathematics (Springer, Berlin, Germany, 2006)

    2006

  20. [28]

    Bravyi and D

    S. Bravyi and D. Gosset, Improved classical simulation of quan- tum circuits dominated by cli fford gates, Phys. Rev. Lett. 116, 250501 (2016)

    2016

  21. [29]

    Huang and P

    Y . Huang and P. Love, Approximate stabilizer rank and im- proved weak simulation of cli fford-dominated circuits for qu- dits, Phys. Rev. A 99, 052307 (2019)

    2019

  22. [30]

    Kocia, Improved strong simulation of universal quantum cir- cuits, arXiv:2012.11739 (2020)

    L. Kocia, Improved strong simulation of universal quantum cir- cuits, arXiv:2012.11739 (2020)

    2020 arXiv

  23. [31]

    K. N. Smith, M. A. Perlin, P. Gokhale, P. Frederick, D. Owusu- Antwi, R. Rines, V . Omole, and F. Chong, Cli fford-based cir- cuit cutting for quantum simulation, in Proceedings of the 50th Annual International Symposium on Computer Architec- ture (ACM, New York, NY , USA, 202...

    2023

  24. [32]

    G. Park, H. Kwon, and H. Jeong, Extending classically simu- latable bounds of cli fford circuits with nonstabilizer states via framed wigner functions, Phys. Rev. Lett. 133, 220601 (2024)

    2024

  25. [33]

    Bu and D

    K. Bu and D. E. Koh, E fficient classical simulation of cli fford circuits with nonstabilizer input states, Phys. Rev. Lett. 123, 170502 (2019)

    2019

  26. [34]

    Katsuda, K

    M. Katsuda, K. Mitarai, and K. Fujii, Simulation and perfor- mance analysis of quantum error correction with a rotated sur- face code under a realistic noise model, Phys. Rev. Res. 6, 013024 (2024)

    2024

  27. [35]

    Bravyi, M

    S. Bravyi, M. Englbrecht, R. K ¨onig, and N. Peard, Correct- ing coherent errors with surface codes, Npj Quantum Inf. 4, 1 (2018)

    2018

  28. [36]

    Venn and B

    F. Venn and B. B ´eri, Error-correction and noise-decoherence thresholds for coherent errors in planar-graph surface codes, Phys. Rev. Res. 2, 043412 (2020)

    2020

  29. [37]

    B. Pato, J. W. Staples, and K. R. Brown, Logical coherence in two-dimensional compass codes, Phys. Rev. A 111, 032424 (2025)

    2025

  30. [38]

    F. Venn, J. Behrends, and B. B ´eri, Coherent-error threshold for surface codes from majorana delocalization, Phys. Rev. Lett. 131, 060603 (2023)

    2023

  31. [39]

    Behrends and B

    J. Behrends and B. B´eri, The surface code under genericx-error channels: Statistical mechanics, error thresholds, and errorfield double phenomenology, arXiv:2412.21055 (2024)

    2024

  32. [40]

    Behrends and B

    J. Behrends and B. B ´eri, Statistical mechanical mapping and maximum-likelihood thresholds for the surface code under generic single-qubit coherent errors, arXiv:2410.22436 (2024)

    2024

  33. [41]

    Marton and J

    A. Marton and J. K. Asboth, Coherent errors and readout errors in the surface code, Quantum 7, 1116 (2023)

    2023

  34. [42]

    Pataki, A

    D. Pataki, A. Marton, J. K. Asboth, and A. Palyi, Coherent er- rors in stabilizer codes caused by quasistatic phase damping, Phys. Rev. A 110, 012417 (2024)

    2024

  35. [43]

    Suzuki, K

    Y . Suzuki, K. Fujii, and M. Koashi, E fficient simulation of quantum error correction under coherent error based on the nonunitary free-fermionic formalism, Phys. Rev. Lett. 119, 190503 (2017)

    2017

  36. [44]

    Fontana, N

    E. Fontana, N. Fitzpatrick, D. M. Ramo, R. Duncan, and I. Rungger, Evaluating the noise resilience of variational quan- tum algorithms, Phys. Rev. A 104, 022403 (2021)

    2021

  37. [45]

    C. T. Hann, G. Lee, S. M. Girvin, and L. Jiang, Resilience of quantum random access memory to generic noise, PRX Quan- tum 2, 020311 (2021)

    2021

  38. [46]

    Malekakhlagh, A

    M. Malekakhlagh, A. Seif, D. Puzzuoli, L. C. G. Govia, and E. van den Berg, E fficient lindblad synthesis for noise model 10 construction, arXiv:2502.03462 (2025)

    2025

  39. [47]

    Cincio, K

    L. Cincio, K. Rudinger, M. Sarovar, and P. J. Coles, Machine learning of noise-resilient quantum circuits, PRX Quantum 2, 10.1103/prxquantum.2.010324 (2021)

    2021 doi

  40. [48]

    Ferracin, A

    S. Ferracin, A. Hashim, J.-L. Ville, R. Naik, A. Carignan- Dugas, H. Qassim, A. Morvan, D. I. Santiago, I. Siddiqi, and J. J. Wallman, Efficiently improving the performance of noisy quantum computers, Quantum 8, 1410 (2024)

    2024

  41. [49]

    P. D. Nation, H. Kang, N. Sundaresan, and J. M. Gambetta, Scalable mitigation of measurement errors on quantum comput- ers, PRX Quantum 2, 10.1103/prxquantum.2.040326 (2021)

    2021 doi

  42. [50]

    Proctor, K

    T. Proctor, K. Rudinger, K. Young, E. Nielsen, and R. Blume- Kohout, Measuring the capabilities of quantum computers, Nat. Phys. 18, 75 (2021)

    2021

  43. [51]

    D. M. Debroy, M. Li, M. Newman, and K. R. Brown, Stabi- lizer slicing: Coherent error cancellations in low-density parity- check stabilizer codes, Phys. Rev. Lett. 121, 250502 (2018)

    2018

  44. [52]

    X.-C. Wu, S. Di, E. M. Dasgupta, F. Cappello, H. Finkel, Y . Alexeev, and F. T. Chong, Full-state quantum circuit simula- tion by using data compression, in Proceedings of the Interna- tional Conference for High Performance Computing, Network- ing, Storage and Analysis (ACM, N...

    2019

  45. [53]

    Boixo, S

    S. Boixo, S. V . Isakov, V . N. Smelyanskiy, and H. Neven, Sim- ulation of low-depth quantum circuits as complex undirected graphical models, arXiv:1712.05384 (2017)

    2017 arXiv

  46. [54]

    C. Guo, Y . Liu, M. Xiong, S. Xue, X. Fu, A. Huang, X. Qiang, P. Xu, J. Liu, S. Zheng, H.-L. Huang, M. Deng, D. Poletti, W.-S. Bao, and J. Wu, General-purpose quantum circuit sim- ulator with projected entangled-pair states and the quantum supremacy frontier, Phys. Rev. Lett.1...

    2019

  47. [55]

    Z.-Y . Chen, Q. Zhou, C. Xue, X. Yang, G.-C. Guo, and G.-P. Guo, 64-qubit quantum circuit simulation, Sci. Bull. (Beijing) 63, 964 (2018)

    2018

  48. [56]

    J. Chen, F. Zhang, C. Huang, M. Newman, and Y . Shi, Classical simulation of intermediate-size quantum circuits, arXiv:1805.01450 (2018)

    2018 arXiv

  49. [57]

    A. S. Darmawan and D. Poulin, Tensor-network simulations of the surface code under realistic noise, Phys. Rev. Lett. 119, 040502 (2017)

    2017

  50. [58]

    Bouwmeester, J.-W

    D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Observation of three-photon greenberger-horne- zeilinger entanglement, Phys. Rev. Lett. 82, 1345 (1999)

    1999

  51. [59]

    Blanes and F

    S. Blanes and F. Casas, On the convergence and optimization of the baker–campbell–hausdorff formula, Linear Algebra and its Applications 378, 135 (2004)

    2004

  52. [60]

    Dankert, R

    C. Dankert, R. Cleve, J. Emerson, and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Phys. Rev. A 80, 012304 (2009)

    2009

  53. [61]

    Gross, K

    D. Gross, K. Audenaert, and J. Eisert, Evenly distributed uni- taries: On the structure of unitary designs, J. Math. Phys. 48, 052104 (2007)

    2007

  54. [62]

    Boixo, S

    S. Boixo, S. V . Isakov, V . N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven, Char- acterizing quantum supremacy in near-term devices, Nat. Phys. 14, 595 (2018)

    2018

  55. [63]

    Arute et al., Quantum supremacy using a programmable su- perconducting processor, Nature 574, 505 (2019)

    F. Arute et al., Quantum supremacy using a programmable su- perconducting processor, Nature 574, 505 (2019)

    2019

  56. [64]

    S. J. Beale, J. J. Wallman, M. Guti ´errez, K. R. Brown, and R. Laflamme, Quantum error correction decoheres noise, Phys. Rev. Lett. 121, 190501 (2018)

    2018

  57. [65]

    J. K. Iverson and J. Preskill, Coherence in logical quantum channels, New J. Phys. 22, 073066 (2020)

    2020

  58. [66]

    Harper and S

    R. Harper and S. T. Flammia, Learning correlated noise in a 39-qubit quantum processor, PRX quantum 4, 040311 (2023)

    2023

  59. [67]

    Proctor, K

    T. Proctor, K. Young, A. D. Baczewski, and R. Blume-Kohout, Benchmarking quantum computers, Nat. Rev. Phys. 7, 105 (2025)

    2025

  60. [68]

    Sarovar, T

    M. Sarovar, T. Proctor, K. Rudinger, K. Young, E. Nielsen, and R. Blume-Kohout, Detecting crosstalk errors in quantum infor- mation processors, Quantum 4, 321 (2020)

    2020

  61. [69]

    Harper, S

    R. Harper, S. T. Flammia, and J. J. Wallman, E fficient learning of quantum noise, Nat. Phys. 16, 1184 (2020)

    2020

  62. [70]

    Harper, W

    R. Harper, W. Yu, and S. T. Flammia, Fast estimation of sparse quantum noise, PRX Quantum 2, 10.1103 /prxquan- tum.2.010322 (2021)

    2021

  63. [71]

    S. T. Flammia, Averaged circuit eigenvalue sampling, arXiv:2108.05803 (2021)

    2021 arXiv

  64. [72]

    Hines and T

    J. Hines and T. Proctor, Pauli noise learning for mid- circuit measurements, Phys. Rev. Lett. 134, 10.1103 /phys- revlett.134.020602 (2025)

    2025

  65. [73]

    Nielsen, K

    E. Nielsen, K. Rudinger, T. Proctor, A. Russo, K. Young, and R. Blume-Kohout, Probing quantum processor performance with pyGSTi, Quantum Sci. Technol. 5, 044002 (2020)

    2020

  66. [74]

    H. Ball, T. M. Stace, S. T. Flammia, and M. J. Biercuk, E ffect of noise correlations on randomized benchmarking, Phys. Rev. A 93, 022303 (2016). 11 SUPPLEMENTAL MATERIAL I. ADDITIONAL APPLICATIONS In this section we will provide an additional example of the application of our...

    2016

  67. [75]

    Randomly sampling a nonidentity Pauli operator, s, and performing a layer of single-qubit gates which maps the input state to a +1 eigenstate of s

  68. [76]

    Propagation through the randomly sampled Cli fford layers has the effect of mapping the originally sampled Pauli operator s to some new Pauli operator s′

    d layers of Clifford gates sampled according to some distribution Ω. Propagation through the randomly sampled Cli fford layers has the effect of mapping the originally sampled Pauli operator s to some new Pauli operator s′

  69. [77]

    Given a BiRB circuit, we’ll refer to the expectation value of the final Pauli observable⟨sc⟩ as the energy for that circuit

    A final layer of single-qubit gates mapping s′ to tensor-product of Z and I Pauli operators, sc. Given a BiRB circuit, we’ll refer to the expectation value of the final Pauli observable⟨sc⟩ as the energy for that circuit. Using our algorithm one can compute approximations to t...

  70. [78]

    (64)-(67) forα(x, G,ψ )

    Derivations We now derive Eqs. (64)-(67) forα(x, G,ψ ). We begin with Eq. (64), where G = S P. In this case α(x, S P,ψ ) = 2ζ(ψ)Tr(|x⟩⟨x|S P[|ψ⟩⟨ψ|]) = 2ζ(ψ) (⟨x|P|ψ⟩⟨ψ|P|x⟩−⟨ x|ψ⟩⟨ψ|x⟩) (68) = Φψ,x(P, P)− Φψ,x(I, I). (69) Next, we derive Eq. (65), which is a formula forα(x, H...

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.