Pith. sign in

REVIEW 3 major objections 5 minor 122 references

Enhancing Decoding Performance using Efficient Error Learning

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read A quantum decoder that knows only about 1% of the measured Pauli error rates, with the rest supplied by a heuristic, suppresses logical errors about ten times better than a decoder that knows only the average noise strength.

desk verdict Novel top-K error-rate completion for ML decoding of concatenated codes, but the TVD table contradicts its own text—fix that before trusting the mechanism. read the letter →

arxiv 2507.08536 v1 pith:LSI5RJIN submitted 2025-07-11 quant-ph cs.ITmath.IT

Pavithran Iyer , Aditya Jain , Stephen D. Bartlett , Joseph Emerson This is my paper · ORCID
classification quant-phcs.ITmath.IT PACS 03.67.Pp
keywords quantumerrorcorrectionmaximumlikelihooddecodingcyclereconstructionPauliratesconcatenatedSteanecodenoisecharacterizationlogicalratefault-tolerantcomputing
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

Maximum-likelihood decoding of a quantum error-correcting code normally requires the full Pauli error distribution of the noise, which has exponentially many parameters. This paper argues that most of that information is unnecessary: a decoder fed only the $K$ largest Pauli error rates, obtained efficiently from cycle error reconstruction (CER), can perform nearly as well as fully noise-adapted decoding once a heuristic fills in the remaining rates. The heuristic, called Uncorrelated Split Search (USS), estimates each unmeasured multi-qubit error probability as a sum of products of probabilities of its lower-weight parts. For a level-2 concatenated Steane code, the authors report a typical tenfold reduction in logical error rate with about 163-164 Pauli error rates (roughly 1% of the full set) compared with a decoder that knows only the average process infidelity, across coherent and incoherent error models. If this holds, calibration data already obtainable at scale could lower the overhead of fault-tolerant quantum computation.

What carries the argument

The load-bearing object is the Uncorrelated Split Search (USS) algorithm, a greedy heuristic that completes the Pauli error distribution from partial data. For any Pauli error not in the measured set, USS sums, over all bipartitions of the qubits, the product of the probabilities of the two factors, recursing until the factors are found in the data; missing single-qubit probabilities are assigned by a depolarizing ansatz based on the measured identity rate. The completed distribution feeds a maximum-likelihood decoder for concatenated codes, which is efficient because noise is independent across code blocks and is decoded by message passing. The other half of the machinery is cycle error reconstruction (CER), which supplies the $K$ largest Pauli error rates with an experimental cost that scales with $K$.

What would settle it

Perform an exact or tomographically complete reconstruction of the Pauli error distribution of a small code block under a strongly correlated noise channel, then compare the logical error rate of the maximum-likelihood decoder using the true distribution with the decoder using the USS-completed distribution built from the $K$ largest rates. If the completed distribution does not reproduce the order-of-magnitude gain, or produces a loss, for a noise model whose high-weight error probabilities deviate strongly from the product ansatz, the central claim would be falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is that maximum-likelihood decoding of a concatenated quantum code can be made almost as good as fully noise-adapted decoding using a drastically incomplete description of the noise. Concretely, the authors show numerically that a level-2 concatenated Steane code, decoded with the 163 or 164 largest Pauli error rates supplied by cycle error reconstruction and with the remaining probabilities assigned by the USS heuristic, achieves a logical error rate roughly an order of magnitude below that of a decoder which knows only the average infidelity of the physical noise. This gain is demonstrated for random unitary errors from a coarse-grained one-dimensional Hamiltonian model, random unitary circuits, and generic non-catastrophic CPTP maps. The USS completion reduces the total variation distance between the decoder's error model and the true noise, indicating that the heuristic contributes substantially to the gain beyond the raw CER data.

Load-bearing premise

The gains rest on the USS heuristic's product-form assumption: every unmeasured multi-qubit Pauli error probability is well approximated by sums of products of its lower-weight factor probabilities, with missing single-qubit rates filled by a depolarizing ansatz; noise with correlations that violate this structure could make the completed distribution systematically wrong and erase the reported gains.

Editorial extensions

If this is right

  • A practical decoder for concatenated codes can be substantially improved without full process tomography: roughly 163-164 Pauli error rates, about 1% of the total, suffice for an order-of-magnitude gain over fidelity-only decoding.
  • The gain grows with the amount of calibration data and persists across coherent errors, random unitary circuits, and generic Markovian CPTP maps, so the benefit is not tied to one noise model.
  • When the largest rates are not available, the same USS completion works with $k$-body noise reconstruction data (all errors up to weight $k$), giving roughly 5x gains for unitary circuits and 10x for coarse-grained one-dimensional noise.
  • Because logical error rate is what determines the overhead of fault tolerance, CER-informed decoding can lower the resources needed to reach a target logical error rate.

Reading between the lines

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

  • If the same partial-characterization idea transfers to LDPC or surface codes with tensor-network or message-passing ML decoders, the practical impact could extend well beyond concatenated codes; the paper explicitly leaves this as future work.
  • The reported gain is a ratio relative to a fidelity-only decoder; the absolute benefit for a fault-tolerant architecture depends on where the physical noise sits relative to the code threshold, so the method may matter less in very high or very low noise regimes.
  • A direct hardware test would be to run CER on a 7-qubit block, decode with USS, and compare logical error rates against both fidelity-only and full-tomography decoders; the total variation distance numbers in the paper suggest the completed distribution is much closer to the true one, which is directly testable.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper proposes to improve maximum-likelihood (ML) decoding of concatenated Steane codes by feeding the decoder a partially characterized Pauli error distribution: the K largest Pauli error rates estimated by Cycle Error Reconstruction (CER), with the remaining rates completed by a greedy heuristic called Uncorrelated Split Search (USS). For level-2 concatenated Steane codes, the authors report numerical gains of roughly 10x in logical error rate relative to a decoder that only knows the average process infidelity, across ensembles of coherent CG1D unitary errors, random unitary circuits, and random non-catastrophic CPTP maps, using K=163 or 164 Pauli rates, about 1% of the 4^7 rates per block. They also attempt to isolate the contribution of USS through total-variation-distance (TVD) comparisons in Table I.

Significance. If the reported gains are robust, the paper makes a useful contribution by connecting efficient partial noise characterization (CER) to a concrete practical payoff in decoding concatenated codes, and it offers a computationally cheap heuristic for completing missing Pauli rates. The Monte Carlo methodology is a strength: the paper uses an importance-sampling estimator with a convergence check (Fig. 9), reports quartile spreads for the gain distribution, and covers several distinct noise families. The main weaknesses are that the quantitative evidence for the USS completion is internally inconsistent (Table I contradicts its own text) and that the USS heuristic has no accuracy or normalization guarantees, so the mechanism behind the headline 10x gain is not yet established. The paper does not ship code or data, which limits reproducibility.

major comments (3)
  1. [Section IV C, Table I] The text immediately above Table I states that 'TVD(χ(E), χK(E)) is approximately five to ten times larger than TVD(χ(E),χ̂K(E))', but every row of the table shows the opposite: the entries in the last column are about 5 to 7 times those in the second column. As presented, the USS-completed distribution is farther from the true Pauli rates, not closer, so the claim that USS 'plays a key role in achieving the observed performance gains' is unsupported. Please correct the table or the text, clarify whether the columns are mislabeled, and provide a direct comparison of decoding performance using χK (zeros for missing entries) versus χ̂K to separate the contribution of the K-largest data from the contribution of USS.
  2. [Algorithm 1 and Eq. (7)] The USS product ansatz has no stated accuracy guarantee and is not normalized. For the simple i.i.d. depolarizing case, a Pauli error of weight w and true probability p^w is assigned the sum over ordered bi-partitions of the support; when the lower-weight factors are taken from the data, this yields a value a_w p^w with a_w > 1 (already for w=3 the prefactor is 6 under the ordered-split convention of Eq. (7)). Because the completed distribution is not renormalized, the relative probabilities of logical cosets can be systematically distorted, and the paper gives no test (e.g., decoding with the true full distribution versus the USS completion) showing that this distortion is benign for the reported gain.
  3. [Section IV B and Algorithm 1 base case] The baseline decoder D1 is not precisely defined. It is described only as having access to the average fidelity, and the manuscript does not state whether D1 is the USS completion with K=0 (the base case of Algorithm 1) or another distribution. Moreover, the base-case formula in Eq. (8) and Algorithm 1 appears internally inconsistent and does not reduce to the standard fidelity-matched i.i.d. depolarizing model. Since the 10x gain is defined as a ratio relative to D1, the exact distribution used by D1 is load-bearing; please specify it and verify that it is the natural fidelity-only decoder.
minor comments (5)
  1. [Section IV B] The phrase '1% of the Pauli error rates in the system' should be clarified: the K=163/164 rates are 1% of the 4^7 rates for a single 7-qubit block, not for the full level-2 code with 49 physical qubits.
  2. [Appendix F, Fig. 9] The importance-sampling convergence check is shown only for the random-unitary model; a similar check for the CG1D and CPTP maps would strengthen confidence in the reported gains.
  3. [Table I] No error bars or confidence intervals are given for the TVD averages, despite the text noting that the values vary significantly across individual instances; please report the spread of the TVD values.
  4. [General] The paper does not include code or data; providing these would allow readers to reproduce the simulations and verify the TVD computation and the gain estimates.
  5. [Section IV A] The CG1D model samples local-interaction sizes from a Poisson distribution with mean two, but the text refers to a 'mean correlated length λ' and elsewhere sets λ=2; please define λ precisely and report its value for each figure.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 10X gain is a forward decoding simulation; USS is an explicit heuristic and its validation issue is a correctness concern, not a circular reduction.

full rationale

The paper's central claim is a numerical demonstration: for a chosen noise model E, it extracts the K largest Pauli error rates (simulating CER data), completes the distribution with the explicitly heuristic USS algorithm, runs a maximum-likelihood decoder on that completed distribution, and compares the resulting logical error rate with a decoder that uses only the average fidelity. Nothing in this chain fits a parameter to the logical error rate and then reports that rate as a predicted output. The USS completion is stated as a heuristic ansatz (Eq. 7 and Algorithm 1), not as a first-principles derivation; its accuracy is an empirical validation question. The baseline D1 is deliberately mismatched (depolarizing assumption), which may inflate the reported gain, but that is a baseline choice rather than circular reasoning. The self-citations to CER and cycle benchmarking provide experimental and theoretical tools that are independently demonstrated on multiple platforms; they are not used as the derivation of the claimed performance gain. The notable internal inconsistency in Table I - where the text claims TVD(chi(E), chi_K(E)) is five to ten times larger than TVD(chi(E), chi_hat_K(E)) but the table shows the opposite - is a correctness or validation defect, not a circular reduction of a prediction to its inputs. No load-bearing step is equivalent to its input by construction.

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

The paper's results rest on the USS product ansatz, the depolarizing base case, the perfect-QEC and block-i.i.d. assumptions, the use of Pauli-twirled probabilities for non-Pauli noise, and the external claim from Ref. [49] that top-K rates are efficiently learnable. No free parameters are fitted to the reported gains, but several numerical hyperparameters (correlation length lambda, importance sampling exponent alpha, single-qubit ansatz) are hand-chosen and affect the reported estimates.

free parameters (3)
  • Single-qubit ansatz form
    In Eq. (8), missing single-qubit error probabilities are set by a depolarizing ansatz with epsilon_0 = 1 - (1 - chi_{I,I})^{1/n}. The form is chosen by hand and is not data-fitted, but it controls the base case of USS.
  • Mean correlation length lambda = 2 (main figures)
    Error model generation samples local interaction sizes from a Poisson distribution with mean lambda; lambda=2 is used to introduce many-body terms. This hand-chosen value affects the error instances and the reported gains.
  • Importance sampling exponent alpha
    In Appendix F, Q(s)=Prob(s)^alpha is used to weight outlier syndromes in estimating the average logical error rate. The paper says alpha is selected per a threshold in Ref. [60] but does not report its value, and it affects all reported gains.
assumptions (6)
  • domain assumption Noise acts independently and identically on each of the n^{l-1} code blocks of the concatenated code.
    Section IV states this assumption to justify the message-passing ML decoder and the block-wise effective channel computation from Ref. [91].
  • domain assumption Syndrome measurements and recovery operations are executed perfectly.
    Section II C: "we assume that syndrome measurement and the application of the recovery operation prescribed by the decoder are executed perfectly." This simplifies the effective-channel analysis and excludes circuit-level noise.
  • domain assumption Decoding-relevant error probabilities can be obtained from the Pauli twirl of the full CPTP map.
    Appendix B states Prob(P)=chi_{P,P}(tau[E]) for non-Pauli errors, discarding off-diagonal chi-matrix elements. Coherent errors are thereby replaced by their twirled Pauli channel.
  • domain assumption The K largest Pauli error rates can be estimated efficiently with a number of experiments scaling linearly with K.
    Section III relies on Ref. [49] to justify that N_K is experimentally accessible; the paper does not prove this itself.
  • ad hoc to paper The USS product ansatz: probability of a composite error is the sum over bi-partitions of products of sub-error probabilities.
    Algorithm 1 and Eq. (7) define this heuristic; it is the core completion rule and is introduced ad hoc for this paper.
  • ad hoc to paper Depolarizing base-case ansatz for missing single-qubit errors.
    Eq. (8) assigns missing single-qubit error probabilities as if the channel were i.i.d. depolarizing with fidelity derived from chi_{I,I}; this is an ad hoc modeling choice.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Enhancing Decoding Performance using Efficient Error Learning." pith.science (2026). https://pith.science/paper/LSI5RJIN

@misc{pith2026250708536,
  author       = {Pith},
  title        = {Pith review of: Enhancing Decoding Performance using Efficient Error Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LSI5RJIN}},
  note         = {Machine review of arXiv:2507.08536}
}
abstract

Lowering the resource overhead needed to achieve fault-tolerant quantum computation is crucial to building scalable quantum computers. We show that adapting conventional maximum likelihood (ML) decoders to a small subset of efficiently learnable physical error characteristics can significantly improve the logical performance of a quantum error-correcting code. Specifically, we leverage error information obtained from efficient characterization methods based on Cycle Error Reconstruction (CER), which yields Pauli error rates on the $n$ qubits of an error-correcting code. Although the total number of Pauli error rates needed to describe a general noise process is exponentially large in $n$, we show that only a few of the largest few Pauli error rates are needed and that a heuristic technique can complete the Pauli error distribution for ML decoding from this restricted dataset. Using these techniques, we demonstrate significant performance improvements for decoding quantum codes under a variety of physically relevant error models. For instance, with CER data that constitute merely $1\%$ of the Pauli error rates in the system, we achieve a $10X$ gain in performance compared to the case where decoding is based solely on the fidelity of the underlying noise process. Our conclusions underscore the promise of recent error characterization methods for improving quantum error correction and lowering overheads.

Figures

Figures reproduced from arXiv: 2507.08536 by the authors.

Figure 1
Figure 1. FIG. 1. Figure showing a factor graph used for constructing [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Figures illustrating the performance enhancement in the QEC capabilities of the Steane code obtained by using partial [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The above figure shows a schematic for the CER [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Figure showing the encoding circuit of a concatenated [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Figure showing the operation of the message passing [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The above figure illustrates the performance improvements achieved by the ML decoder for concatenated Steane codes, [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The above figure illustrates the performance improvements achieved by the ML decoder for concatenated Steane codes, [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The figure above highlights the significance of calibration data from [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The figure demonstrates the rapid convergence of Monte Carlo estimates of the average logical error rate for the level [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

122 extracted references · 27 canonical work pages

  1. [1]

    Note that there are non-trivial Pauli errorsE̸=Iwhich have a trivial syndrome:s= 00

    Maximum Likelihood Decoding Before defining the decoding problem, it is worth ana- lyzing some structure of the Pauli group induced by the stabilizer group. Note that there are non-trivial Pauli errorsE̸=Iwhich have a trivial syndrome:s= 00. . .0; these errors belong toN(S): the normalizer of the sta- bilizer groupSwithin the Pauli groupP n. N(S) ={E∈ P n...

  2. [2]

    Firstly, it ne- cessitates the knowledge of all the 4 n Pauli error rates affecting the physical qubits of the code

    Efficient Decoding of Concatenated Codes While ML decoding is optimal [53, 58] for stabilizer codes, it comes with certain trade-offs. Firstly, it ne- cessitates the knowledge of all the 4 n Pauli error rates affecting the physical qubits of the code. Despite this practical roadblock, we will analyze the performance of ML decoding since it yields the best...

  3. [3]

    Quantum accuracy threshold for concatenated distance-3 codes.Quantum Information and Computa- tion, 6(2):97–165, 2006.doi:10.26421/QIC6.2-1

    Panos Aliferis, Daniel Gottesman, and John Preskill. Quantum accuracy threshold for concatenated distance-3 codes.Quantum Information and Computa- tion, 6(2):97–165, 2006.doi:10.26421/QIC6.2-1

    doi:10.26421/qic6.2-1 2006

  4. [4]

    Abanin, Laleh Aghababaie- Beni, Igor Aleiner, Trond I

    Rajeev Acharya, Dmitry A. Abanin, Laleh Aghababaie- Beni, Igor Aleiner, Trond I. Andersen, Markus Ans- mann, Frank Arute, Kunal Arya, Abraham As- faw, Nikita Astrakhantsev, Juan Atalaya, Ryan Bab- bush, Dave Bacon, Brian Ballard, Joseph C. Bardin, Johannes Bausch, Andreas Bengtsson, Alexander Bilmes, and Google Quantum AI et. al. Quan- tum error correctio...

    2025

  5. [5]

    Quantum error correction fails for hamiltonian models.Fluctuation and Noise Letters, 6(3):C23–C28, 2006.doi:10.1142/ S0219477506003343

    Robert Alicki. Quantum error correction fails for hamiltonian models.Fluctuation and Noise Letters, 6(3):C23–C28, 2006.doi:10.1142/ S0219477506003343

    2006

  6. [6]

    PhD thesis, University of Wa- terloo, 2023

    Stefanie Beale.Modeling and managing noise in quan- tum error correction. PhD thesis, University of Wa- terloo, 2023. URL:http://hdl.handle.net/10012/ 20047

    2023

  7. [7]

    Panos Aliferis and Barbara M. Terhal. Fault-tolerant quantum computation for local leakage faults.Quan- 17 (a) (b) FIG. 8. The figure above highlights the significance of calibration data fromk-body Noise Reconstruction (k−NR) in enhancing the performance of an ML decoder for concatenated Steane codes under two distinct models for coherent errors. In the ...

    2007

  8. [8]

    Surface code design for asymmetric error chan- nels.IET Quantum Communication, 3(3):174–183, 2022.doi:10.1049/qtc2.12042

    Utkarsh Azad, Aleksandra Lipi´ nska, Shilpa Mahato, Ri- jul Sachdeva, Debasmita Bhoumik, and Ritajit Majum- dar. Surface code design for asymmetric error chan- nels.IET Quantum Communication, 3(3):174–183, 2022.doi:10.1049/qtc2.12042

    work page doi:10.1049/qtc2.12042 2022

Show all 122 references
  1. [9]

    Wang, Sepehr Ebadi, Marcin Kalinowski, Alexander Keesling, Nishad Maskara, Hannes Pichler, Markus Greiner, Vladan Vuletia, and Mikhail D

    Dolev Bluvstein, Harry Levine, Giulia Semeghini, Tout T. Wang, Sepehr Ebadi, Marcin Kalinowski, Alexander Keesling, Nishad Maskara, Hannes Pichler, Markus Greiner, Vladan Vuletia, and Mikhail D. Lukin. A quantum processor based on coherent transport of en- tangled atom arrays....

    2022 doi

  2. [10]

    Stefanie J. Beale, Kristine Boone, Arnaud Carignan- Dugas, Anthony Chytros, Dar Dahlen, Hillary Dawkins, Joseph Emerson, Samuele Ferracin, Virginia Frey, Ian Hincks, David Hufnagel, Pavithran Iyer, Aditya Jain, Jason Kolbush, Egor Ospadov, Jos´ e Luis Pino, Ham- mam Qassim, Jo...

    2020 doi

  3. [11]

    Vaidya, and Prabha Man- dayam

    Debjyoti Biswas, Gaurav M. Vaidya, and Prabha Man- dayam. Noise-adapted recovery circuits for quantum er- ror correction.Physical Review Research, 6(4):043034, 2024.doi:10.1103/PhysRevResearch.6.043034

    2024 doi

  4. [12]

    Cross, Jay M

    Sergey Bravyi, Andrew W. Cross, Jay M. Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J. Yoder. High-threshold and low-overhead fault-tolerant quan- tum memory.Nature, 627(8005):778–782, 2024.doi: 10.1038/s41586-024-07107-7

    2024 doi

  5. [13]

    Pablo Bonilla Ataides, David K

    J. Pablo Bonilla Ataides, David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia, and Benjamin J. Brown. The xzzx surface code.Nature Communications, 12(1):2172, 2021.doi:10.1038/s41467-021-22274-1

    2021 doi

  6. [14]

    PhD thesis, University of Wa- terloo, 2021

    Kristine Boone.Concepts and Methods for Benchmark- ing Quantum Computers. PhD thesis, University of Wa- terloo, 2021. URL:https://uwspace.uwaterloo.ca/ items/72401eee-0981-4aa6-ac7b-ae3e873df07a

    2021

  7. [15]

    Simulation of rare events in quantum error correction.Phys

    Sergey Bravyi and Alexander Vargo. Simulation of rare events in quantum error correction.Phys. Rev. A, 88:062308, 2013.doi:10.1103/PhysRevA.88.062308

    2013 doi

  8. [16]

    Correcting coherent errors with sur- face codes.npj Quantum Information, 4(1):55, 2018

    Sergey Bravyi, Matthias Englbrecht, Robert Koenig, and Nolan Peard. Correcting coherent errors with sur- face codes.npj Quantum Information, 4(1):55, 2018. doi:10.1038/s41534-018-0106-y

    2018 doi

  9. [17]

    Efficient algorithms for maximum likelihood decoding in the surface code.Phys

    Sergey Bravyi, Martin Suchara, and Alexander Vargo. Efficient algorithms for maximum likelihood decoding in the surface code.Phys. Rev. A, 90:032326, 2014. doi:10.1103/PhysRevA.90.032326

    2014 doi

  10. [18]

    Beale, Samuele Ferracin, Joshua Skanes-Norman, Joseph Emerson, and Joel J

    Arnaud Carignan-Dugas, Dar Dahlen, Ian Hincks, Egor Ospadov, Stefanie J. Beale, Samuele Ferracin, Joshua Skanes-Norman, Joseph Emerson, and Joel J. Wall- man. The error reconstruction and compiled calibration of quantum computing cycles. 2023.doi:10.48550/ arXiv.2303.17714

    2023 doi arXiv

  11. [19]

    A class of quantum ldpc codes: construction and performances under iterative decoding

    Thomas Camara, Harold Ollivier, and Jean-Pierre Tillich. A class of quantum ldpc codes: construction and performances under iterative decoding. In2007 IEEE International Symposium on Information Theory, pages 811–815, 2007.doi:10.1109/ISIT.2007.4557324. 19

    2007

  12. [20]

    A polar decomposition for quantum channels (with applications to bounding error propa- gation in quantum circuits).Quantum, 3:173, 2019

    Arnaud Carignan-Dugas, Matthew Alexander, and Joseph Emerson. A polar decomposition for quantum channels (with applications to bounding error propa- gation in quantum circuits).Quantum, 3:173, 2019. doi:10.22331/q-2019-08-12-173

    2019 doi

  13. [21]

    Completely positive linear maps on complex matrices.Linear Algebra and its Applications, 10(3):285–290, 1975.doi:10.1016/0024-3795(75) 90075-0

    Man-Duen Choi. Completely positive linear maps on complex matrices.Linear Algebra and its Applications, 10(3):285–290, 1975.doi:10.1016/0024-3795(75) 90075-0

    1975 doi

  14. [22]

    Deep neural decoders for near term fault-tolerant experi- ments.Quantum Science and Technology, 3(4):044002, 2018.doi:10.1088/2058-9565/aad1f7

    Christopher Chamberland and Pooya Ronagh. Deep neural decoders for near term fault-tolerant experi- ments.Quantum Science and Technology, 3(4):044002, 2018.doi:10.1088/2058-9565/aad1f7

    2018 doi

  15. [23]

    Hard decoding algo- rithm for optimizing thresholds under general marko- vian noise.Phys

    Christopher Chamberland, Joel Wallman, Stefanie Beale, and Raymond Laflamme. Hard decoding algo- rithm for optimizing thresholds under general marko- vian noise.Phys. Rev. A, 95:042332, 2017.doi: 10.1103/PhysRevA.95.042332

    2017 doi

  16. [24]

    Christopher T. Chubb. General tensor network decod- ing of 2d pauli codes. 2021.doi:10.48550/arXiv.2101. 04125

    2021 doi

  17. [25]

    Quantum error correction in scrambling dynamics and measurement-induced phase transition

    Soonwon Choi, Yimu Bao, Xiao-Liang Qi, and Ehud Altman. Quantum error correction in scrambling dynamics and measurement-induced phase transition. Phys. Rev. Lett., 125:030505, 2020.doi:10.1103/ PhysRevLett.125.030505

    2020

  18. [26]

    Chuang and M

    Isaac L. Chuang and M. A. Nielsen. Prescription for ex- perimental determination of the dynamics of a quantum black box.Journal of Modern Optics, 44(11-12):2455– 2467, 1997.doi:10.1080/09500349708231894

    1997 doi

  19. [27]

    Cross.Fault-tolerant quantum com- puter architectures using hierarchies of quantum error- correcting codes

    Andrew W. Cross.Fault-tolerant quantum com- puter architectures using hierarchies of quantum error- correcting codes. PhD thesis, Dept. of Electrical Engi- neering and Computer Science, Massachusetts Institute of Technology., 2008. URL:http://hdl.handle.net/ 1721.1/44407

    2008

  20. [28]

    B. D. Clader, Colin J. Trout, Jeff P. Barnes, Kevin Schultz, Gregory Quiroz, and Paraj Titum. Impact of correlations and heavy tails on quantum error correc- tion.Phys. Rev. A, 103:052428, 2021.doi:10.1103/ PhysRevA.103.052428

    2021

  21. [29]

    Effective hamil- tonians for interacting superconducting qubits: local basis reduction and the schrieffer–wolff transformation

    Gioele Consani and Paul A Warburton. Effective hamil- tonians for interacting superconducting qubits: local basis reduction and the schrieffer–wolff transformation. New Journal of Physics, 22(5):053040, 2020.doi: 10.1088/1367-2630/ab83d1

    2020 doi

  22. [30]

    Darmawan and David Poulin

    Andrew S. Darmawan and David Poulin. Tensor- network simulations of the surface code under realis- tic noise.Phys. Rev. Lett., 119:040502, 2017.doi: 10.1103/PhysRevLett.119.040502

    2017 doi

  23. [31]

    Exact and approximate unitary 2-designs and their application to fidelity estimation

    Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A, 80:012304, Jul 2009.doi:10.1103/ PhysRevA.80.012304

    2009

  24. [32]

    Darmawan

    Andrew S. Darmawan. Optimal adaptation of surface- code decoders to local noise.arXiv preprint, 2024.doi: 10.48550/arXiv.2403.08706

    2024 doi arXiv

  25. [33]

    Scalable noise estimation with random unitary opera- tors.Journal of Optics B: Quantum and Semiclassical Optics, 7(10):S347, 2005.doi:10.1088/1464-4266/7/ 10/021

    Joseph Emerson, Robert Alicki, and Karol ˙Zyczkowski. Scalable noise estimation with random unitary opera- tors.Journal of Optics B: Quantum and Semiclassical Optics, 7(10):S347, 2005.doi:10.1088/1464-4266/7/ 10/021

    2005 doi

  26. [34]

    Darmawan and David Poulin

    Andrew S. Darmawan and David Poulin. Linear-time general decoding algorithm for the surface code.Phys. Rev. E, 97:051302, 2018.doi:10.1103/PhysRevE.97. 051302

    2018 doi

  27. [35]

    Crespo, and Josu Etxezarreta Martinez

    Antonio deMarti iOlius, Patricio Fuentes, Rom´ an Or´ us, Pedro M. Crespo, and Josu Etxezarreta Martinez. De- coding algorithms for surface codes.Quantum, 8:1498, 2024.doi:10.22331/q-2024-10-10-1498

    2024 doi

  28. [36]

    Constant overhead quantum fault tolerance with quantum expander codes.Commun

    Omar Fawzi, Antoine Grospellier, and Anthony Lev- errier. Constant overhead quantum fault tolerance with quantum expander codes.Commun. ACM, 64(1):106–114, 2020.doi:10.1145/3434163

    2020 doi

  29. [37]

    Cory, and Raymond Laflamme

    Joseph Emerson, Marcus Silva, Osama Moussa, Colm Ryan, Martin Laforest, Jonathan Baugh, David G. Cory, and Raymond Laflamme. Symmetrized char- acterization of noisy quantum processes.Science, 317(5846):1893–1896, 2007.doi:10.1126/science. 1145699

    2007 doi

  30. [38]

    Characterizing large-scale quantum com- puters via cycle benchmarking.Nature communications, 10(1):5347, 2019.doi:10.1038/s41467-019-13068-7

    Alexander Erhard, Joel J Wallman, Lukas Postler, Michael Meth, Roman Stricker, Esteban A Martinez, Philipp Schindler, Thomas Monz, Joseph Emerson, and Rainer Blatt. Characterizing large-scale quantum com- puters via cycle benchmarking.Nature communications, 10(1):5347, 2019.do...

    2019 doi

  31. [39]

    Austin G. Fowler. Coping with qubit leakage in topo- logical codes.Phys. Rev. A, 88:042308, 2013.doi: 10.1103/PhysRevA.88.042308

    2013 doi

  32. [40]

    Marciniak, Manuel Rispler, Robin Harper, Thomas Monz, Joseph Emerson, and Stephen D

    Nicholas Fazio, Robert Freund, Debankan Sannamoth, Alex Steiner, Christian D. Marciniak, Manuel Rispler, Robin Harper, Thomas Monz, Joseph Emerson, and Stephen D. Bartlett. Characterizing physical and logical errors in a transversal cnot via cycle error reconstruc- tion.arXiv ...

    2025 doi

  33. [41]

    Flammia and Joel J

    Steven T. Flammia and Joel J. Wallman. Efficient esti- mation of pauli channels.ACM Transactions on Quan- tum Computing, 1(1), 2020.doi:10.1145/3408039

    2020 doi

  34. [42]

    Fault-tolerant quantum computa- tion with constant overhead.Quantum Info

    Daniel Gottesman. Fault-tolerant quantum computa- tion with constant overhead.Quantum Info. Comput., 14(15–16):1338–1372, November 2014.doi:10.5555/ 2685179.2685184

    2014

  35. [43]

    Fowler, Matteo Mariantoni, John M

    Austin G. Fowler, Matteo Mariantoni, John M. Mar- tinis, and Andrew N. Cleland. Surface codes: To- wards practical large-scale quantum computation.Phys. Rev. A, 86:032324, 2012.doi:10.1103/PhysRevA.86. 032324

    2012 doi

  36. [44]

    Terno, and Christopher J

    Alexei Gilchrist, Daniel R. Terno, and Christopher J. Wood. Vectorization of quantum operations and its use.arXiv:0911.2539, 2009.doi:10.48550/arXiv. 0911.2539

    2009 doi arXiv

  37. [45]

    Mauricio Guti´ errez and Kenneth R. Brown. Compar- ison of a quantum error-correction threshold for exact 20 and approximate errors.Phys. Rev. A, 91:022335, 2015. doi:10.1103/PhysRevA.91.022335

    2015 doi

  38. [46]

    PhD thesis, California Institute of Technology, 1997.doi:doi:10.7907/rzr7-dt72

    Daniel Eric Gottesman.Stabilizer Codes and Quantum Error Correction. PhD thesis, California Institute of Technology, 1997.doi:doi:10.7907/rzr7-dt72

    1997 doi

  39. [47]

    Modeling coherent errors in quantum error correction.Quan- tum Science and Technology, 3(1):015007, 2017.doi: 10.1088/2058-9565/aa9a06

    Daniel Greenbaum and Zachary Dutton. Modeling coherent errors in quantum error correction.Quan- tum Science and Technology, 3(1):015007, 2017.doi: 10.1088/2058-9565/aa9a06

    2017 doi

  40. [48]

    Robin Harper and Steven T. Flammia. Learning cor- related noise in a 39-qubit quantum processor.PRX Quantum, 4:040311, 2023.doi:10.1103/PRXQuantum. 4.040311

    2023 doi

  41. [49]

    Mauricio Guti´ errez, Lukas Svec, Alexander Vargo, and Kenneth R. Brown. Approximation of realistic errors by clifford channels and pauli measurements.Phys. Rev. A, 87:030302, 2013.doi:10.1103/PhysRevA.87.030302

    2013 doi

  42. [50]

    Crosstalk diagnosis for the next generation of quantum processors.Quantum Views, 4:46, 2020.doi:10.22331/qv-2020-10-29-46

    Dominik Hangleiter. Crosstalk diagnosis for the next generation of quantum processors.Quantum Views, 4:46, 2020.doi:10.22331/qv-2020-10-29-46

    2020 doi

  43. [51]

    Bohdanowicz, Aleksander Kubica, Steven T

    Oscar Higgott, Thomas C. Bohdanowicz, Aleksander Kubica, Steven T. Flammia, and Earl T. Camp- bell. Improved decoding of circuit noise and frag- ile boundaries of tailored surface codes.Physical Re- view X, 13(3):031007, 2023.doi:10.1103/PhysRevX. 13.031007

    2023 doi

  44. [52]

    Robin Harper, Wenjun Yu, and Steven T. Flammia. Fast estimation of sparse quantum noise.PRX Quan- tum, 2:010322, 2021.doi:10.1103/PRXQuantum.2. 010322

    2021 doi

  45. [53]

    Naik, Alexis Morvan, Jean-Loup Ville, Bradley Mitchell, John Mark Kreikebaum, Marc Davis, Ethan Smith, Costin Iancu, Kevin P

    Akel Hashim, Ravi K. Naik, Alexis Morvan, Jean-Loup Ville, Bradley Mitchell, John Mark Kreikebaum, Marc Davis, Ethan Smith, Costin Iancu, Kevin P. O’Brien, Ian Hincks, Joel J. Wallman, Joseph Emerson, and Ir- fan Siddiqi. Randomized compiling for scalable quan- tum computing o...

    2021 doi

  46. [54]

    Doherty, and Steven Flammia

    Eric Huang, Andrew C. Doherty, and Steven Flammia. Performance of quantum error correction with coherent errors.Phys. Rev. A, 99:022313, 2019.doi:10.1103/ PhysRevA.99.022313

    2019

  47. [55]

    Hockings, Andrew C

    Evan T. Hockings, Andrew C. Doherty, and Robin Harper. Improving error suppression with noise-aware decoding.arXiv preprint, 2025.doi:10.48550/arXiv. 2502.21044

    2025 doi arXiv

  48. [56]

    Np-hardness of decoding quantum error-correction codes.Phys

    Min-Hsiu Hsieh and Francois Le Gall. Np-hardness of decoding quantum error-correction codes.Phys. Rev. A, 83:052331, 2011.doi:10.1103/PhysRevA.83.052331

    2011 doi

  49. [58]

    Shilin Huang, Michael Newman, and Kenneth R. Brown. Fault-tolerant weighted union-find decoding on the toric code.Phys. Rev. A, 102:012419, 2020. doi:10.1103/PhysRevA.102.012419

    2020 doi

  50. [59]

    Bartlett, and Joseph Emerson

    Pavithran Iyer, Aditya Jain, Stephen D. Bartlett, and Joseph Emerson. Efficient diagnostics for quantum error correction.Phys. Rev. Res., 4:043218, 2022.doi:10. 1103/PhysRevResearch.4.043218

    2022

  51. [60]

    Bartlett, and Joseph Emerson

    Aditya Jain, Pavithran Iyer, Stephen D. Bartlett, and Joseph Emerson. Improved quantum error correction with randomized compiling.Phys. Rev. Res., 5:033049, 2023.doi:10.1103/PhysRevResearch.5.033049

    2023 doi

  52. [61]

    Iyer and David Poulin

    Pavithran S. Iyer and David Poulin. Hardness of de- coding quantum stabilizer codes.IEEE Transactions on Information Theory, 61(9):5209–5223, 2015.doi: 10.1109/TIT.2015.2422294

    2015

  53. [62]

    A critical analysis of quantum error correction methods for real- istic noise processes

    Pavithran Iyer Sridharan.Une analyse critique de la correction d’erreurs quantiques pour du bruit r´ ealiste. PhD thesis, Universit´ e de Sherbrooke, Sherbrooke, Qu´ ebec, Canada, 2018. Also published as “A critical analysis of quantum error correction methods for real- istic ...

    2018

  54. [63]

    Quantum gate fidelity in terms of choi matrices.Journal of Physics A: Mathematical and Theoretical, 44(49):495303, 2011

    Nathaniel Johnston and David W Kribs. Quantum gate fidelity in terms of choi matrices.Journal of Physics A: Mathematical and Theoretical, 44(49):495303, 2011. doi:10.1088/1751-8113/44/49/495303

    2011 doi

  55. [64]

    Achieving fault toler- ance against amplitude-damping noise.Phys

    Akshaya Jayashankar, My Duy Hoang Long, Hui Khoon Ng, and Prabha Mandayam. Achieving fault toler- ance against amplitude-damping noise.Phys. Rev. Res., 4:023034, 2022.doi:10.1103/PhysRevResearch. 4.023034

    2022 doi

  56. [65]

    Us- ing concatenated quantum codes for universal fault- tolerant quantum gates.Phys

    Tomas Jochym-O’Connor and Raymond Laflamme. Us- ing concatenated quantum codes for universal fault- tolerant quantum gates.Phys. Rev. Lett., 112:010505, 2014.doi:10.1103/PhysRevLett.112.010505

    2014 doi

  57. [66]

    Character- izing crosstalk of superconducting transmon proces- sors.Phys

    Andreas Ketterer and Thomas Wellens. Character- izing crosstalk of superconducting transmon proces- sors.Phys. Rev. Appl., 20:034065, 2023.doi:10.1103/ PhysRevApplied.20.034065

    2023

  58. [67]

    Quantum maximum likelihood decoding for linear block codes

    Hyunwoo Jung, Jeonghwan Kang, and Jeongseok Ha. Quantum maximum likelihood decoding for linear block codes. InProceedings of the 11th International Con- ference on Information and Communication Technol- ogy Convergence (ICTC), pages 227–232, 2020.doi: 10.1109/ICTC49870.2020.9289350

    2020

  59. [68]

    Simulation and performance analysis of quantum error correction with a rotated surface code under a realistic noise model.Phys

    Mitsuki Katsuda, Kosuke Mitarai, and Keisuke Fujii. Simulation and performance analysis of quantum error correction with a rotated surface code under a realistic noise model.Phys. Rev. Res., 6:013024, 2024.doi: 10.1103/PhysRevResearch.6.013024

    2024 doi

  60. [69]

    Concatenated quantum codes.arXiv preprint quant-ph/9608012, 1996

    Emanuel Knill and Raymond Laflamme. Concatenated quantum codes.arXiv preprint quant-ph/9608012, 1996. doi:10.48550/arXiv.quant-ph/9608012

    1996 doi arXiv

  61. [70]

    Design of quantum error correcting code for biased error on heavy-hexagon structure.Quantum Infor- mation Processing, 22(6), June 2023.doi:10.1007/ s11128-023-03979-2

    Younghun Kim, Jeongsoo Kang, and Younghun Kwon. Design of quantum error correcting code for biased error on heavy-hexagon structure.Quantum Infor- mation Processing, 22(6), June 2023.doi:10.1007/ s11128-023-03979-2

    2023

  62. [72]

    A new way to use old codes

    Anirudh Krishna. A new way to use old codes. Nature Physics, 20(2):182–183, 2024.doi:10.1038/ s41567-023-02369-w

    2024

  63. [73]

    Kovalev and Leonid P

    Alexey A. Kovalev and Leonid P. Pryadko. Fault toler- ance of quantum low-density parity check codes with sublinear distance scaling.Phys. Rev. A, 87:020304, 2013.doi:10.1103/PhysRevA.87.020304

    2013 doi

  64. [74]

    Kraus, A

    K. Kraus, A. B¨ ohm, J.D. Dollard, and W.H. Wootters. States, Effects, and Operations: Fundamental Notions of Quantum Theory. Lecture Notes in Physics. Springer Berlin Heidelberg, 1983.doi:10.1007/3-540-12732-1

    1983 doi

  65. [75]

    Rectan- gular surface code under biased noise.Quantum In- formation Processing, 20(7):231, 2021.doi:10.1007/ s11128-021-03130-z

    Jonghyun Lee, Jooyoun Park, and Jun Heo. Rectan- gular surface code under biased noise.Quantum In- formation Processing, 20(7):231, 2021.doi:10.1007/ s11128-021-03130-z

    2021

  66. [76]

    Era- sure qubits: Overcoming theT 1 limit in superconduct- ing circuits.Phys

    Aleksander Kubica, Arbel Haim, Yotam Vaknin, Harry Levine, Fernando Brand˜ ao, and Alex Retzker. Era- sure qubits: Overcoming theT 1 limit in superconduct- ing circuits.Phys. Rev. X, 13:041022, 2023.doi: 10.1103/PhysRevX.13.041022

    2023 doi

  67. [77]

    Long, Andrew C

    Richard Kueng, David M. Long, Andrew C. Doherty, and Steven T. Flammia. Comparing experiments to the fault-tolerance threshold.Phys. Rev. Lett., 117:170502, 2016.doi:10.1103/PhysRevLett.117.170502. 21

    2016 doi

  68. [78]

    Theχ−matrix representation ofEshown in eq

    and satisfy X i K † i Ki =I. Theχ−matrix representation ofEshown in eq. (3) can be derived by expressing the Kraus operators in eq. (D1) as linear combinations of Pauli matrices. An important subclass of CPTP maps arenon- catastrophic channels, which are characterized byϵ(E)≥ ...

  69. [79]

    Ryan Levy, Di Luo, and Bryan K. Clark. Classical shadows for quantum process tomography on near-term quantum computers.Phys. Rev. Res., 6:013029, 2024. doi:10.1103/PhysRevResearch.6.013029

    2024 doi

  70. [80]

    Concate- nation Schemes for Topological Fault-tolerant Quan- tum Error Correction.Quantum, 7:1089, 2023.doi: 10.22331/q-2023-08-22-1089

    Zhaoyi Li, Isaac Kim, and Patrick Hayden. Concate- nation Schemes for Topological Fault-tolerant Quan- tum Error Correction.Quantum, 7:1089, 2023.doi: 10.22331/q-2023-08-22-1089

    2023 doi

  71. [81]

    Daniel A. Lidar. Lecture notes on the theory of open quantum systems.arXiv preprint, 2019.doi:10.48550/ arXiv.1902.00967

    2019 doi arXiv

  72. [82]

    Estimating the randomness of quantum circuit ensembles up to 50 qubits.npj Quantum Information, 8(1):137, 2022.doi:10.1038/s41534-022-00648-7

    Minzhao Liu, Junyu Liu, Yuri Alexeev, and Liang Jiang. Estimating the randomness of quantum circuit ensembles up to 50 qubits.npj Quantum Information, 8(1):137, 2022.doi:10.1038/s41534-022-00648-7

    2022 doi

  73. [83]

    Easwar Magesan, J. M. Gambetta, and Joseph Emer- son. Scalable and robust randomized benchmarking of quantum processes.Phys. Rev. Lett., 106:180504, May 2011.doi:10.1103/PhysRevLett.106.180504

    2011 doi

  74. [84]

    Gambetta, and Joseph Emer- son

    Easwar Magesan, Jay M. Gambetta, and Joseph Emer- son. Characterizing quantum gates via randomized benchmarking.Phys. Rev. A, 85:042311, 2012.doi: 10.1103/PhysRevA.85.042311

    2012 doi

  75. [85]

    Comparison of memory thresholds for planar qu- dit geometries.New Journal of Physics, 19(11):113022, 2017.doi:10.1088/1367-2630/aa939a

    Jacob Marks, Tomas Jochym-O’Connor, and Vlad Ghe- orghiu. Comparison of memory thresholds for planar qu- dit geometries.New Journal of Physics, 19(11):113022, 2017.doi:10.1088/1367-2630/aa939a

    2017 doi

  76. [86]

    Overcoming leakage in quantum error correc- tion.Nature Physics, 19(12):1780–1786, 2023.doi: 10.1038/s41567-023-02226-w

    Kevin C Miao, Matt McEwen, Juan Atalaya, Dvir Kafri, Leonid P Pryadko, Andreas Bengtsson, Alex Oprem- cak, Kevin J Satzinger, Zijun Chen, Paul V Klimov, et al. Overcoming leakage in quantum error correc- tion.Nature Physics, 19(12):1780–1786, 2023.doi: 10.1038/s41567-023-02226-w

    2023 doi

  77. [87]

    Arijit Mondal and Keshab K. Parhi. Quantum circuits for stabilizer error correcting codes: A tutorial.IEEE Circuits and Systems Magazine, 24(1):33–51, 2024.doi: 10.1109/MCAS.2024.3349668

    2024

  78. [88]

    On the ma- trix representation of quantum operations.quant- ph/0504091, 2005.doi:10.48550/arXiv.quant-ph/ 0504091

    Yoshihiro Nambu and Kazuo Nakamura. On the ma- trix representation of quantum operations.quant- ph/0504091, 2005.doi:10.48550/arXiv.quant-ph/ 0504091

    2005 doi

  79. [89]

    Nickerson and Benjamin J

    Naomi H. Nickerson and Benjamin J. Brown. Analysing correlated noise on the surface code using adaptive decoding algorithms.Quantum, 3:131, 2019.doi: 10.22331/q-2019-04-08-131

    2019 doi

  80. [90]

    Gate Set Tomography.Quantum, 5:557, 2021

    Erik Nielsen, John King Gamble, Kenneth Rudinger, Travis Scholten, Kevin Young, and Robin Blume- Kohout. Gate Set Tomography.Quantum, 5:557, 2021. doi:10.22331/q-2021-10-05-557

    2021 doi

  81. [91]

    PhD thesis, Uni- versity of Waterloo, January 2014

    Adam Paetznick.Resource optimization for fault- tolerant quantum computing. PhD thesis, Uni- versity of Waterloo, January 2014. PhD the- sis. URL:https://uwspace.uwaterloo.ca/items/ 3d9f1202-e0c2-474d-b9ec-c7b41a0d8ace

    2014

  82. [92]

    Pattison, Michael E

    Christopher A. Pattison, Michael E. Beverland, Mar- cus P. da Silva, and Nicolas Delfosse. Improved quantum error correction using soft information.arXiv preprint, 2021.doi:10.48550/arXiv.2107.13589

    2021 doi arXiv

  83. [93]

    Degenerate viterbi decoding.IEEE Transactions on Information The- ory, 59(6):3915–3921, 2013.doi:10.1109/TIT.2013

    Emilie Pelchat and David Poulin. Degenerate viterbi decoding.IEEE Transactions on Information The- ory, 59(6):3915–3921, 2013.doi:10.1109/TIT.2013. 2246815

    2013 doi

  84. [94]

    Optimal and efficient decoding of concatenated quantum block codes.Phys

    David Poulin. Optimal and efficient decoding of concatenated quantum block codes.Phys. Rev. A, 74:052333, 2006.doi:10.1103/PhysRevA.74.052333

    2006 doi

  85. [95]

    World Scientific, October 1998.doi:10.1142/9789812385253_0008

    John Preskill.Fault-Tolerant Quantum Computation, chapter 8, pages 213–269. World Scientific, October 1998.doi:10.1142/9789812385253_0008

    1998 doi

  86. [96]

    Benchmarking quantum computers.arXiv:2407.08828, 2024.doi:10.48550/ arXiv.2407.08828

    Timothy Proctor, Kevin Young, Andrew D Baczewski, and Robin Blume-Kohout. Benchmarking quantum computers.arXiv:2407.08828, 2024.doi:10.48550/ arXiv.2407.08828

    2024 doi

  87. [97]

    Leonid P. Pryadko. On maximum-likelihood decoding with circuit-level errors.Quantum, 4:304, 2020.doi: 10.22331/q-2020-08-06-304

    2020 doi

  88. [98]

    Gross, Alexan- der Grimm, Nicholas E

    Shruti Puri, Lucas St-Jean, Jonathan A. Gross, Alexan- der Grimm, Nicholas E. Frattini, Pavithran S. Iyer, Anirudh Krishna, Steven Touzard, Liang Jiang, Alexan- dre Blais, Steven T. Flammia, and S. M. Girvin. Bias- preserving gates with stabilized cat qubits.Science Ad- vances...

    2020 doi

  89. [99]

    Doherty, and Hideo Mabuchi

    Benjamin Rahn, Andrew C. Doherty, and Hideo Mabuchi. Exact performance of concatenated quan- tum codes.Phys. Rev. A, 66:032304, 2002.doi: 10.1103/PhysRevA.66.032304

    2002 doi

  90. [100]

    Karpuzcu

    Salonik Resch and Ulya R. Karpuzcu. Benchmarking quantum computers and the impact of quantum noise. ACM Computing Surveys, 54(7):1–31, 2021.doi:10. 1145/3464420

    2021

  91. [101]

    Towards practical quantum LDPC codes.Quantum Views, 5:63, 2021.doi:10.22331/ qv-2021-11-30-63

    Joschka Roffe. Towards practical quantum LDPC codes.Quantum Views, 5:63, 2021.doi:10.22331/ qv-2021-11-30-63

    2021

  92. [102]

    Hogle, Ravi K

    Kenneth Rudinger, Craig W. Hogle, Ravi K. Naik, Akel Hashim, Daniel Lobser, David I. Santiago, Matthew D. Grace, Erik Nielsen, Timothy Proctor, Stefan Seritan, Susan M. Clark, Robin Blume-Kohout, Irfan Siddiqi, and Kevin C. Young. Experimental characterization of crosstalk err...

    2021

  93. [104]

    Thomp- son, and Shruti Puri

    Kaavya Sahay, Junlan Jin, Jahan Claes, Jeff D. Thomp- son, and Shruti Puri. High-threshold codes for neutral- atom qubits with biased erasure errors.Phys. Rev. X, 13:041013, 2023.doi:10.1103/PhysRevX.13.041013

    2023 doi

  94. [105]

    Detecting crosstalk errors in quantum information pro- cessors.Quantum, 4:321, 2020.doi:10.22331/ q-2020-09-11-321

    Mohan Sarovar, Timothy Proctor, Kenneth Rudinger, Kevin Young, Erik Nielsen, and Robin Blume-Kohout. Detecting crosstalk errors in quantum information pro- cessors.Quantum, 4:321, 2020.doi:10.22331/ q-2020-09-11-321

    2020

  95. [106]

    Peter W. Shor. Fault-tolerant quantum computation. InProceedings of 37th Conference on Foundations of Computer Science (FOCS). IEEE, 1996.doi:10.1109/ SFCS.1996.548464

    1996

  96. [107]

    Spitz, Brian Tarasinski, Carlo W

    Stephen T. Spitz, Brian Tarasinski, Carlo W. J. Beenakker, and Thomas E. O’Brien. Adaptive weight estimator for quantum error correction in a time- dependent environment.Advanced Quantum Tech- nologies, 1(1):1800012, 2018.doi:10.1002/qute. 201800012

    2018 doi

  97. [108]

    Error regions in quantum state tomogra- phy: computational complexity caused by geometry of quantum states.New Journal of Physics, 19(9):093013, 2017.doi:10.1088/1367-2630/aa7ce9

    Daniel Suess, Lukasz Rudnicki, Thiago O maciel, and 22 David Gross. Error regions in quantum state tomogra- phy: computational complexity caused by geometry of quantum states.New Journal of Physics, 19(9):093013, 2017.doi:10.1088/1367-2630/aa7ce9

    2017 doi

  98. [109]

    Yu Tomita and Krysta M. Svore. Low-distance sur- face codes under realistic quantum noise.Phys. Rev. A, 90:062320, 2014.doi:10.1103/PhysRevA.90.062320

    2014 doi

  99. [110]

    Levenson-Falk, and Daniel A

    Vinay Tripathi, Huo Chen, Mostafa Khezri, Ka-Wa Yip, E.M. Levenson-Falk, and Daniel A. Lidar. Suppression of crosstalk in superconducting qubits using dynamical decoupling.Phys. Rev. Appl., 18:024068, 2022.doi: 10.1103/PhysRevApplied.18.024068

    2022 doi

  100. [111]

    Tuckett, Stephen D

    David K. Tuckett, Stephen D. Bartlett, and Steven T. Flammia. Ultrahigh error threshold for surface codes with biased noise.Phys. Rev. Lett., 120:050505, 2018. doi:10.1103/PhysRevLett.120.050505

    2018 doi

  101. [112]

    Hamiltonian learning for quantum error correction.Phys

    Agnes Valenti, Evert van Nieuwenburg, Sebastian Hu- ber, and Eliska Greplova. Hamiltonian learning for quantum error correction.Phys. Rev. Res., 1:033092, 2019.doi:10.1103/PhysRevResearch.1.033092

    2019 doi

  102. [113]

    Techniques for learning sparse Pauli-Lindblad noise models.Quantum, 8:1556, 2024.doi:10.22331/q-2024-12-10-1556

    Ewout van den Berg and Pawel Wocjan. Techniques for learning sparse Pauli-Lindblad noise models.Quantum, 8:1556, 2024.doi:10.22331/q-2024-12-10-1556

    2024 doi

  103. [114]

    Estimating the coherence of noise

    Joel Wallman, Chris Granade, Robin Harper, and Steven T Flammia. Estimating the coherence of noise. New Journal of Physics, 17(11):113020, 2015.doi: 10.1088/1367-2630/17/11/113020

    2015 doi

  104. [115]

    Wallman and Joseph Emerson

    Joel J. Wallman and Joseph Emerson. Noise tailor- ing for scalable quantum computation via randomized compiling.Phys. Rev. A, 94:052325, Nov 2016.doi: 10.1103/PhysRevA.94.052325

    2016 doi

  105. [116]

    Cambridge University Press, 2018.doi:10.1017/ 9781316848142

    John Watrous.The Theory of Quantum Informa- tion. Cambridge University Press, 2018.doi:10.1017/ 9781316848142

    2018

  106. [117]

    Weinstein, Timothy F

    Yaakov S. Weinstein, Timothy F. Havel, Joseph Emer- son, Nicolas Boulant, Marcos Saraceno, Seth Lloyd, and David G. Cory. Quantum process tomography of the quantum fourier transform.The Journal of Chemical Physics, 121(13):6117–6133, 10 2004.doi:10.1063/1. 1785151

    2004 doi

  107. [118]

    Wood, Jacob D

    Christopher J. Wood, Jacob D. Biamonte, and David G. Cory. Tensor networks and graphical calculus for open quantum systems.Quantum Info. Comput., 15(9– 10):759–811, 2015.doi:10.26421/QIC15.9-10-3

    2015 doi

  108. [119]

    Erasure conversion for fault-tolerant quan- tum computing in alkaline earth rydberg atom ar- rays.Nature communications, 13(1):4657, 2022.doi: 10.1038/s41467-022-32094-6

    Yue Wu, Shimon Kolkowitz, Shruti Puri, and Jeff D Thompson. Erasure conversion for fault-tolerant quan- tum computing in alkaline earth rydberg atom ar- rays.Nature communications, 13(1):4657, 2022.doi: 10.1038/s41467-022-32094-6

    2022 doi

  109. [120]

    Flammia, and Liang Jiang

    Qian Xu, Nam Mannucci, Alireza Seif, Aleksander Ku- bica, Steven T. Flammia, and Liang Jiang. Tailored xzzx codes for biased noise.Physical Review Research, 5(1):013035, 2023.doi:10.1103/PhysRevResearch.5. 013035

    2023 doi

  110. [121]

    Time-efficient constant-space-overhead fault-tolerant quantum com- putation.Nature Physics, 20(2):247–253, 2024.doi: 10.1038/s41567-023-02325-8

    Hayata Yamasaki and Masato Koashi. Time-efficient constant-space-overhead fault-tolerant quantum com- putation.Nature Physics, 20(2):247–253, 2024.doi: 10.1038/s41567-023-02325-8

    2024 doi

  111. [122]

    Maximum-likelihood decoding of quantum codes, 2023

    Theodore James Yoder. Maximum-likelihood decoding of quantum codes, 2023. U.S. Patent Filed by Inter- national Business Machines Corporation. URL:https: //patents.google.com/patent/US11736122B1/en

    2023

  112. [123]

    Concatenate codes, save qubits.npj Quan- tum Information, 11(1):88, 2025.doi:10.1038/ s41534-025-01035-8

    Satoshi Yoshida, Shiro Tamiya, and Hayata Ya- masaki. Concatenate codes, save qubits.npj Quan- tum Information, 11(1):88, 2025.doi:10.1038/ s41534-025-01035-8

    2025

  113. [124]

    A modified mwpm decoding algorithm for quantum surface codes over de- polarizing channels

    Yaping Yuan and Chung-Chin Lu. A modified mwpm decoding algorithm for quantum surface codes over de- polarizing channels. In2022 International Symposium on Information Theory and Its Applications (ISITA), pages 204–208, Ranzan, Japan, 2022. IEEE. URL: https://ieeexplore.ieee.o...

    2022

  114. [125]

    Quantum crosstalk analysis for simultaneous gate operations on superconducting qubits.PRX Quantum, 3:020301, 2022.doi:10.1103/PRXQuantum.3.020301

    Peng Zhao, Kehuan Linghu, Zhiyuan Li, Peng Xu, Ruixia Wang, Guangming Xue, Yirong Jin, and Haifeng Yu. Quantum crosstalk analysis for simultaneous gate operations on superconducting qubits.PRX Quantum, 3:020301, 2022.doi:10.1103/PRXQuantum.3.020301

    2022 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.