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arxiv: 2604.26008 · v1 · submitted 2026-04-28 · 🪐 quant-ph

Recognition: unknown

Continuous Noise Model for Quantum Circuits

Yunos El Kaderi , Andreas Honecker , Iryna Andriyanova
Authors on Pith no claims yet

Pith reviewed 2026-05-07 16:35 UTC · model grok-4.3

classification 🪐 quant-ph
keywords continuous coherent noisequantum error correctionPauli noise comparisonvon Mises-Fisher distributionbinary entropy matchingClifford circuit propagationlogical error ratescoherent error accumulation
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The pith

Continuous coherent noise degrades logical performance more strongly than matched Pauli noise in quantum error-correction circuits.

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

The paper develops a model of continuous coherent noise where small gate errors accumulate as random rotations drawn from a von Mises-Fisher distribution on the Bloch sphere, reducing to an isotropic Gaussian in the small-angle limit. It compares this model directly to the standard discrete Pauli noise model by aligning both to produce the same binary entropy at readout. The comparison is tested on error-correction circuits for the [[5,1,3]] and [[7,1,3]] codes as well as Grover search circuits of varying size. The central result is that the continuous model produces stronger degradation of logical performance than the Pauli model at equivalent uncertainty. An approximate analytical propagation method for tracking error distributions through Clifford circuits is also derived and validated against full simulations.

Core claim

We introduce a continuous coherent noise model based on von Mises-Fisher distributed rotations and show, via a binary-entropy matching scheme that holds readout uncertainty fixed, that this continuous model degrades logical performance in the [[5,1,3]] and [[7,1,3]] error-correcting codes more than an equivalent Pauli noise model; the same distinction appears in Grover search circuits, while an approximate analytical method tracks coherent error propagation through Clifford circuits without full Monte Carlo sampling.

What carries the argument

The binary-entropy matching scheme that aligns continuous coherent noise channels to discrete Pauli channels at fixed readout uncertainty, isolating the structural effect of noise type.

If this is right

  • The approximate analytical propagation method reduces simulation cost for Clifford circuits while preserving accuracy within its identified regime of validity.
  • Continuous coherent noise produces measurably higher logical error rates than Pauli noise once readout entropy is held constant.
  • Simplified error-propagation models succeed for bare Clifford circuits but break down under error correction, requiring full sampling in the latter case.
  • The performance gap between continuous and Pauli noise appears consistently across both error-correction and search circuits of different qubit counts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Simulations that rely only on Pauli noise may systematically underestimate logical error rates when real hardware contains coherent rotation errors.
  • The matching scheme could be reused to compare other structured noise models fairly without relying on circuit-specific metrics.
  • Extending the continuous model beyond Clifford gates would test whether the degradation advantage persists in non-Clifford settings.
  • Error-correction thresholds derived from Pauli-only simulations may need downward revision once continuous coherent noise is included.

Load-bearing premise

The binary-entropy matching scheme at readout isolates the structural difference between continuous and discrete noise without introducing bias from the specific choice of metric or from circuit-dependent effects.

What would settle it

A side-by-side measurement of logical error rates under controlled small coherent rotations versus randomized Pauli errors in the [[5,1,3]] code, with the two noise strengths tuned so that readout binary entropy is identical.

Figures

Figures reproduced from arXiv: 2604.26008 by Andreas Honecker, Iryna Andriyanova, Yunos El Kaderi.

Figure 1
Figure 1. Figure 1: FIG. 1. Encode one logical qubit of any code, apply series of view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Circuits for the [[5 view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Circuits for the [[7 view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. A general Grover’s search circuit [ view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Same as Fig view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Error probability of Grover search circuits for view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. [[5 view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Mean (top) and variance (bottom) of the ratio in view at source ↗
read the original abstract

Quantum noise is a central challenge in quantum computing across many applications. Extensive work has examined how qubits couple to their environment, leading to decoherence and relaxation, which is irreversible. Current studies focus on coherent gate errors caused by control misalignment, which accumulate with circuit depth but can, in principle, be corrected. This work studies a continuous coherent noise model for quantum circuits and compares it with a discrete Pauli model. The focus is on small coherent gate errors that build up across circuit depth. These errors are modeled as random rotations on the Bloch sphere using a von Mises-Fisher distribution. In the small-angle limit, the model reduces to an isotropic Gaussian distribution. We test the model on quantum error-correction circuits based on the [[5,1,3]] and [[7,1,3]] codes. A variant of Grover's search circuit with different qubit counts is also examined. To enable fair comparison, we introduce a model-independent matching scheme. Pauli and continuous noise channels are aligned using the binary entropy at readout. This isolates the effect of noise structure at fixed uncertainty. An approximate analytical method for coherent error propagation is also developed. The method tracks error distributions through Clifford circuits without full Monte Carlo sampling. It reduces simulation cost while preserving accuracy for circuit-level error estimates. The approximation is validated against brute-force simulations, identifying its regime of validity with Clifford circuits and limits under error correction. Our results show that continuous coherent noise can degrade logical performance more strongly than Pauli noise. They also clarify when simplified propagation models succeed and when they break down.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a continuous coherent noise model for quantum circuits in which gate errors are modeled as random rotations on the Bloch sphere drawn from a von Mises-Fisher distribution (reducing to isotropic Gaussian for small angles). It develops an approximate analytical propagation method that tracks error distributions through Clifford circuits without full Monte Carlo sampling, validates the approximation against brute-force simulations, and identifies its regime of validity. To compare with discrete Pauli noise, the authors introduce a model-independent matching scheme that aligns the two channels by equating binary entropy at readout. The model is tested on the [[5,1,3]] and [[7,1,3]] error-correcting codes as well as variants of Grover's search algorithm. The central result is that, under this matching, continuous coherent noise degrades logical performance more strongly than Pauli noise.

Significance. If the binary-entropy matching isolates structural differences without metric-induced bias, the work provides useful insight into how coherent error accumulation affects logical qubits differently from stochastic Pauli errors, with direct implications for error-correction thresholds and circuit-level noise budgeting. The approximate propagation method offers a practical reduction in simulation cost for Clifford circuits while preserving accuracy in the identified regime. Strengths include explicit validation against independent Monte Carlo runs and clear statements of the approximation's limits under error correction.

major comments (2)
  1. [Matching scheme] Matching scheme (Section describing the binary-entropy alignment at readout): the central claim that continuous coherent noise degrades logical performance more strongly rests on this post-circuit scalar matching. Because coherent rotations accumulate deterministically (with possible constructive interference) through Clifford gates while Pauli channels are stochastic, equating only final readout entropy may not equalize effective logical error rates; any residual difference could therefore reflect propagation structure rather than noise type alone. A direct comparison of logical error rates or intermediate matching points would be needed to confirm the claim is not partly an artifact of the chosen metric.
  2. [Approximate propagation method] Approximate analytical propagation (Section on the method and its validation): while the approximation is cross-checked against brute-force Monte Carlo and its validity regime for Clifford circuits is identified, the manuscript notes limits under error correction without providing quantitative error bounds or explicit failure cases for the [[5,1,3]] and [[7,1,3]] codes. This weakens in the reported logical-performance comparisons when the approximation is applied inside error-corrected circuits.
minor comments (2)
  1. [Methods] Notation for the von Mises-Fisher concentration parameter and its relation to the small-angle Gaussian limit should be introduced with an explicit equation early in the methods to improve readability.
  2. [Results] Several figures comparing logical error rates would benefit from explicit error bars or confidence intervals derived from the Monte Carlo runs to allow readers to assess the statistical significance of the reported performance gaps.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and recommendation for major revision. We address each point below, indicating revisions that will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Matching scheme] Matching scheme (Section describing the binary-entropy alignment at readout): the central claim that continuous coherent noise degrades logical performance more strongly rests on this post-circuit scalar matching. Because coherent rotations accumulate deterministically (with possible constructive interference) through Clifford gates while Pauli channels are stochastic, equating only final readout entropy may not equalize effective logical error rates; any residual difference could therefore reflect propagation structure rather than noise type alone. A direct comparison of logical error rates or intermediate matching points would be needed to confirm the claim is not partly an artifact of the chosen metric.

    Authors: The binary-entropy matching at readout is chosen as a model-independent scalar that equates the uncertainty in final measurement outcomes, thereby isolating the structural effects of noise propagation (deterministic coherent accumulation versus stochastic Pauli errors) on logical performance. The deterministic buildup is an intrinsic feature of the continuous coherent model, so observed differences under this alignment reflect that distinction rather than an artifact. To address the concern directly, we will add explicit comparisons of logical error rates and intermediate propagation points in the revised manuscript. revision: yes

  2. Referee: [Approximate propagation method] Approximate analytical propagation (Section on the method and its validation): while the approximation is cross-checked against brute-force Monte Carlo and its validity regime for Clifford circuits is identified, the manuscript notes limits under error correction without providing quantitative error bounds or explicit failure cases for the [[5,1,3]] and [[7,1,3]] codes. This weakens in the reported logical-performance comparisons when the approximation is applied inside error-corrected circuits.

    Authors: The manuscript validates the approximation via Monte Carlo for Clifford circuits and states its regime of validity, including noted limits under error correction. We agree that quantitative error bounds and explicit failure cases for the [[5,1,3]] and [[7,1,3]] codes would increase confidence in the logical comparisons. In the revision we will supply these bounds and highlight specific failure cases where the approximation deviates. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper defines a continuous noise model via von Mises-Fisher rotations (reducing to Gaussian), introduces an approximate propagation method for Clifford circuits, validates it directly against Monte Carlo sampling, and normalizes Pauli vs. continuous models via an external binary-entropy metric at readout before comparing logical degradation. None of these steps reduce a claimed result to its inputs by construction; the matching normalizes one observable to compare a distinct observable (logical performance), the approximation is independently checked, and no load-bearing self-citation or ansatz smuggling is present. The central claim therefore rests on explicit simulation cross-checks rather than definitional equivalence.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that small coherent errors are well-described by von Mises-Fisher rotations and on the matching procedure that equates binary entropy; no new physical entities are postulated and the free parameters are the distribution concentration and the matched noise strength.

free parameters (2)
  • von Mises-Fisher concentration parameter
    Controls the angular spread of the coherent rotations and is set to the small-angle regime.
  • noise strength matched by binary entropy
    The continuous and Pauli channels are aligned by equating readout binary entropy, which effectively sets the overall error scale for comparison.
axioms (2)
  • domain assumption Small coherent gate errors can be represented as random rotations distributed according to the von Mises-Fisher distribution on the Bloch sphere
    Invoked in the model definition for continuous noise.
  • domain assumption Error distributions can be tracked analytically through Clifford circuits without full sampling
    Basis for the approximate propagation method.

pith-pipeline@v0.9.0 · 5578 in / 1461 out tokens · 73851 ms · 2026-05-07T16:35:42.707244+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

68 extracted references · 8 canonical work pages · 1 internal anchor

  1. [1]

    Only single-qubit gates induce coherent noise

  2. [2]

    CNOT gates are noiseless but propagate existing errors

  3. [3]

    Algorithm1Circuit simulation with coherent Gaussian noise

    The noise is isotropic, withσ θ =σ ϕ =σ. Algorithm1Circuit simulation with coherent Gaussian noise. Choose circuitC, noise grid Σ ={σ}, number of circuit instancesN c, andSimulator forσ∈Σdo Initializeresults←∅ fori= 1, . . . , N c do SetC ′ ←C for allsingle-qubit gateGinC ′ do Sample (θ, ϕ)∼ N(0, σ 2)▷Eq. (6) InsertU err(θ, ϕ, ϕ) afterG ▷Eq. (2) end for E...

  4. [4]

    2b and 3b, respectively

    Implementations of the logical Hadamard for both codes are shown in Figs. 2b and 3b, respectively. For the [[5,1,3]] code, the logical Hadamard requires a combination of physical Hadamard gates and SWAP operations, making it non-transversal. For the [[7,1,3]] code, the logical Hadamard consists solely of physical Hadamard gates, and is therefore transvers...

  5. [5]

    Each ofX,Y, andZoccurs with probabilityp/3

    Pauli Noise (Reference Model) We consider a symmetric Pauli channel with total error ratep. Each ofX,Y, andZoccurs with probabilityp/3. The gate acts ideally with probability 1−p. AZ-basis measurement yields a BSC with bit-flip (bf) probability pPauli bf = 2p 3 ,(12) sinceXandYflip the outcome. The corresponding bi- nary entropy [55] is HPauli(p) =h 2 2p ...

  6. [6]

    Letλdenote the angle between the intended and actual axes

    Continuous rotations: von Mises–Fisher (vMF) Coherent control errors are modeled as random tilts of the rotation axis. Letλdenote the angle between the intended and actual axes. AZ-basis bit flip occurs with probability sin 2(λ/2) = 1 2(1−cosλ). The effective flip rate follows from averaging this quantity over the noise distribution. For axis fluctuations...

  7. [7]

    Montanaro, Quantum algorithms: an overview, npj Quantum Inf.2, 15023 (2016)

    A. Montanaro, Quantum algorithms: an overview, npj Quantum Inf.2, 15023 (2016)

    2016

  8. [8]

    Bharti, A

    K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys.94, 015004 (2022)

    2022

  9. [9]

    Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Hug- gins, Y. Li, J. R. McClean, and T. E. O’Brien, Quantum error mitigation, Rev. Mod. Phys.95, 045005 (2023)

    2023

  10. [10]

    S. P. Tera, R. Chinthaginjala, X. Zhao, and M. Hamdi, Advancing quantum machine learning from conceptual design to practical use, Expert Systems with Applications 320, 132127 (2026)

    2026

  11. [11]

    B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys.87, 307 (2015)

    2015

  12. [12]

    Tripathi, D

    V. Tripathi, D. Kowsari, K. Saurav, H. Zhang, E. M. Levenson-Falk, and D. A. Lidar, Benchmarking quantum gates and circuits, Chemical Reviews125, 5745 (2025)

    2025

  13. [13]

    Lescanne, M

    R. Lescanne, M. Villiers, T. Peronnin, A. Sarlette, M. Delbecq, B. Huard, T. Kontos, M. Mirrahimi, and Z. Leghtas, Exponential suppression of bit-flips in a qubit encoded in an oscillator, Nat. Phys.16, 509 (2020)

    2020

  14. [14]

    Acharya, I

    R. Acharya, I. Aleiner, R. Allen, T. I. Andersen, M. Ans- mann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, R. Bab- bush, D. Bacon, J. C. Bardin, J. Basso, A. Bengtsson, S. Boixo,et al.(Google Quantum AI), Suppressing quan- tum errors by scaling a surface code logical qubit, Nature 614, 676 (2023)

    2023

  15. [15]

    Bravyi, A

    S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low- overhead fault-tolerant quantum memory, Nature627, 778 (2024)

    2024

  16. [16]

    Acharya, D

    R. Acharya, D. A. Abanin, L. Aghababaie-Beni, I. Aleiner, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, N. Astrakhantsev, J. Atalaya, R. Babbush, D. Bacon, B. Ballard, J. C. Bardin,et al. (Google Quantum AI and Collaborators), Quantum er- ror correction below the surface code threshold, Nature 638, 920 (2025)

    2025

  17. [17]

    D. Ruiz, J. Guillaud, A. Leverrier, M. Mirrahimi, and C. Vuillot, LDPC-cat codes for low-overhead quantum computing in 2D, Nat. Commun.16, 1040 (2025)

    2025

  18. [18]

    Rousseau, D

    R. Rousseau, D. Ruiz, E. Albertinale, P. d’Avezac, D. Banys, U. Blandin, N. Bourdaud, G. Campanaro, G. Cardoso, N. Cottet, C. Cullip, S. Del´ eglise, L. De- vanz, A. Devulder, A. Essig, P. F´ evrier, A. Gicquel,´Elie Gouzien, A. Gras, J. Guillaud, E. G¨ um¨ u¸ s, M. Hall´ en, A. Jacob, P. Magnard, A. Marquet, S. Miklass, T. Per- onnin, S. Polis, F. Rautsc...

    work page arXiv 2025

  19. [19]

    The Heisenberg Representation of Quantum Computers

    D. Gottesman, The Heisenberg representation of quan- tum computers (1998), arXiv:quant-ph/9807006 [quant- ph]

    work page internal anchor Pith review arXiv 1998

  20. [20]

    Aaronson and D

    S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A70, 052328 (2004)

    2004

  21. [21]

    R. A. Fisher, Dispersion on a sphere, Proc. R. Soc. A 217, 295 (1953)

    1953

  22. [22]

    Sheldon, L

    S. Sheldon, L. S. Bishop, E. Magesan, S. Filipp, J. M. Chow, and J. M. Gambetta, Characterizing errors on qubit operations via iterative randomized benchmarking, Phys. Rev. A93, 012301 (2016)

    2016

  23. [23]

    Kaufmann, I

    N. Kaufmann, I. Rojkov, and F. Reiter, Characterization of coherent errors in gate layers with robustness to Pauli noise, Phys. Rev. Appl.23, 034014 (2025)

    2025

  24. [24]

    Hanks, S

    M. Hanks, S. Lee, and M. Kim, Noise-tailored construc- tions for spin Wigner function kernels, Adv. Phys. Res. 3, 2300124 (2024)

    2024

  25. [25]

    Schultz, Orientation statistics and quantum informa- tion (2016), arXiv:1609.08074 [quant-ph]

    K. Schultz, Orientation statistics and quantum informa- tion (2016), arXiv:1609.08074 [quant-ph]

    work page arXiv 2016

  26. [26]

    Wudarski, J

    F. Wudarski, J. Marshall, A. Petukhov, and E. Rieffel, Augmented fidelities for single-qubit gates, Phys. Rev. A 102, 052612 (2020)

    2020

  27. [27]

    H. Peng, S. Koudia, and S. Chatzinotas, Security analysis of MDI-QKD in turbulent free-space polariza- tion channels-a composite channel framework (2025), arXiv:2509.02087 [quant-ph]

    work page arXiv 2025

  28. [28]

    Maiti, K

    S. Maiti, K. Sen, and U. Sen, Quantum phase estimation in presence of glassy disorder (2022), arXiv:2112.04411 [quant-ph]

    work page arXiv 2022

  29. [29]

    Huynh, G

    T. Huynh, G. An, M. Kim, Y.-S. Jeon, and J. Lee, A vari- ational quantum algorithm by Bayesian inference with von Mises-Fisher distribution (2024), arXiv:2410.03130 [quant-ph]

    work page arXiv 2024

  30. [30]

    J. T. Thomas, M. Lababidi, and M. Tian, Robustness of single-qubit geometric gate against systematic error, Phys. Rev. A84, 042335 (2011)

    2011

  31. [31]

    Ragazzi, S

    G. Ragazzi, S. Cavazzoni, P. Bordone, and M. G. A. Paris, Generalized phase estimation in noisy quantum gates, Phys. Rev. A110, 052425 (2024)

    2024

  32. [32]

    J. P. Barnes, C. J. Trout, D. Lucarelli, and B. D. Clader, Quantum error-correction failure distributions: Compar- ison of coherent and stochastic error models, Phys. Rev. A95, 062338 (2017)

    2017

  33. [33]

    Huang, A

    E. Huang, A. C. Doherty, and S. Flammia, Performance of quantum error correction with coherent errors, Phys. Rev. A99, 022313 (2019)

    2019

  34. [34]

    Marshall and D

    J. Marshall and D. Kafri, Incoherent approximation of leakage in quantum error correction, Phys. Rev. Appl. 23, 054025 (2025)

    2025

  35. [35]

    K. Noh, C. Chamberland, and F. G. Brand˜ ao, Low- overhead fault-tolerant quantum error correction with the surface-GKP code, PRX Quantum3, 010315 (2022)

    2022

  36. [36]

    M´ arton and J

    ´A. M´ arton and J. K. Asb´ oth, Coherent errors and readout errors in the surface code, Quantum7, 1116 (2023)

    2023

  37. [37]

    Bravyi, M

    S. Bravyi, M. Englbrecht, R. K¨ onig, and N. Peard, Cor- recting coherent errors with surface codes, npj Quantum 13 Inf.4, 55 (2018)

    2018

  38. [38]

    Laflamme, C

    R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Perfect quantum error correcting code, Phys. Rev. Lett. 77, 198 (1996)

    1996

  39. [39]

    A. M. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett.77, 793 (1996)

    1996

  40. [40]

    Guti´ errez, C

    M. Guti´ errez, C. Smith, L. Lulushi, S. Janardan, and K. R. Brown, Errors and pseudothresholds for incoherent and coherent noise, Phys. Rev. A94, 042338 (2016)

    2016

  41. [41]

    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

  42. [42]

    Z. Cai, X. Xu, and S. C. Benjamin, Mitigating coherent noise using Pauli conjugation, npj Quantum Inf.6, 17 (2020)

    2020

  43. [43]

    Gottesman,Stabilizer Codes and Quantum Error Cor- rection, Ph.D

    D. Gottesman,Stabilizer Codes and Quantum Error Cor- rection, Ph.D. thesis, California Institute of Technology (1997), available from ProQuest Dissertations & Theses Global (Order No. 9800357, Document No. 304364982)

    1997

  44. [44]

    Z. P. Bradshaw, J. J. Dale, and E. N. Evans, Introduction to quantum error correction with stabilizer codes, Ann. Phys. (N. Y.)487, 170353 (2026)

    2026

  45. [45]

    T. J. Yoder, R. Takagi, and I. L. Chuang, Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes, Phys. Rev. X6, 031039 (2016)

    2016

  46. [46]

    Q. Zhao, Q. Li, H. Mao, X. Wen, Q. Han, and M. Li, Fault-tolerant quantum error correction code preparation in UBQC, Quantum Inf. Process.19, 236 (2020)

    2020

  47. [47]

    M. A. Nielsen and I. L. Chuang,Quantum Computa- tion and Quantum Information: 10th Anniversary Edi- tion(Cambridge University Press, 2010)

    2010

  48. [48]

    L. K. Grover, A fast quantum mechanical algorithm for database search, inProceedings of the Twenty-Eighth An- nual ACM Symposium on Theory of Computing, STOC ’96 (Association for Computing Machinery, New York, NY, USA, 1996) p. 212–219

    1996

  49. [49]

    L. K. Grover, Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett.79, 325 (1997)

    1997

  50. [50]

    El Kaderi, A

    Y. El Kaderi, A. Honecker, and I. Andriyanova, Perfor- mance of uncoded implementation of Grover’s algorithm on today’s quantum processors, in2023 IEEE Informa- tion Theory Workshop (ITW)(2023) pp. 209–214

    2023

  51. [51]

    Kumar, B

    D. Kumar, B. Raj, and G. Singh, Comparative analysis for effect of noise in Grover’s quantum search algorithm, Research Square (2025), preprint, Version 1

    2025

  52. [52]

    M. A. Ijaz and M. Faryad, Noise analysis of Grover and phase estimation algorithms implemented as quan- tum singular value transformations for a small number of noisy qubits, Sci. Rep.13, 20144 (2023)

    2023

  53. [53]

    Bravyi, D

    S. Bravyi, D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard, Simulation of quantum circuits by low- rank stabilizer decompositions, Quantum3, 181 (2019)

    2019

  54. [54]

    M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Annu. Rev. Condens. Matter Phys.14, 335 (2023)

    2023

  55. [55]

    Sauli \`e re, G

    A. Sauliere, G. Lami, P. Ribeiro, A. De Luca, and J. De Nardis, Error-correction transitions in finite-depth quan- tum channels (2026), arXiv:2603.20369 [quant-ph]

    work page arXiv 2026

  56. [56]

    Brown and O

    W. Brown and O. Fawzi, Decoupling with random quan- tum circuits, Commun. Math. Phys.340, 867 (2015)

    2015

  57. [57]

    M. J. Gullans, S. Krastanov, D. A. Huse, L. Jiang, and S. T. Flammia, Quantum coding with low-depth random circuits, Phys. Rev. X11, 031066 (2021)

    2021

  58. [58]

    A. S. Darmawan, Y. Nakata, S. Tamiya, and H. Ya- masaki, Low-depth random Clifford circuits for quan- tum coding against Pauli noise using a tensor-network decoder, Phys. Rev. Res.6, 023055 (2024)

    2024

  59. [59]

    Miller, C

    A. Miller, C. Ostrove, J. Hines, R. Blume-Kohout, K. Young, and T. Proctor, Efficient simulation of Clifford circuits with small Markovian errors (2025), arXiv:2504.15128 [quant-ph]

    work page arXiv 2025

  60. [60]

    J.-S. Kim, L. S. Bishop, A. D. C´ orcoles, S. Merkel, J. A. Smolin, and S. Sheldon, Hardware-efficient random cir- cuits to classify noise in a multiqubit system, Phys. Rev. A104, 022609 (2021)

    2021

  61. [61]

    Richardson and R

    T. Richardson and R. Urbanke,Modern Coding Theory (Cambridge University Press, 2008)

    2008

  62. [62]

    A. M. Dalzell, N. Hunter-Jones, and F. G. S. L. Brand˜ ao, Random quantum circuits transform local noise into global white noise, Commun. Math. Phys.405, 78 (2024)

    2024

  63. [63]

    Deshpande, P

    A. Deshpande, P. Niroula, O. Shtanko, A. V. Gorshkov, B. Fefferman, and M. J. Gullans, Tight bounds on the convergence of noisy random circuits to the uniform dis- tribution, PRX Quantum3, 040329 (2022)

    2022

  64. [64]

    Foldager and B

    J. Foldager and B. Koczor, Can shallow quantum circuits scramble local noise into global white noise?, J. Phys. A: Math. Theor.57, 015306 (2023)

    2023

  65. [65]

    Haferkamp, P

    J. Haferkamp, P. Faist, N. B. T. Kothakonda, J. Eisert, and N. Yunger Halpern, Linear growth of quantum cir- cuit complexity, Nat. Phys.18, 528 (2022)

    2022

  66. [66]

    Suzuki, J

    R. Suzuki, J. Haferkamp, J. Eisert, and P. Faist, Quan- tum complexity phase transitions in monitored random circuits, Quantum9, 1627 (2025)

    2025

  67. [67]

    Tsubouchi, Y

    K. Tsubouchi, Y. Mitsuhashi, K. Sharma, and N. Yosh- ioka, Symmetric Clifford twirling for cost-optimal quan- tum error mitigation in early FTQC regime, npj Quan- tum Inf.11, 104 (2025)

    2025

  68. [68]

    Fefferman, S

    B. Fefferman, S. Ghosh, M. Gullans, K. Kuroiwa, and K. Sharma, Effect of nonunital noise on random-circuit sampling, PRX Quantum5, 030317 (2024)

    2024