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Decoding across transversal Clifford gates in the surface code

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper shows that decoding across fast transversal Clifford gates in the unrotated surface code can be done with minimum-weight perfect matching on per-observable subgraphs, preserving the code-distance guarantee of d/2.

desk verdict A genuinely new per-observable matching decoder for fast transversal Clifford gates, with an honest but incomplete treatment of the one property the main theorem depends on. read the letter →

arxiv 2505.13599 v4 pith:OVMAEYV2 submitted 2025-05-19 quant-ph

Marc Serra-Peralta , Mackenzie H. Shaw , Barbara M. Terhal This is my paper · ORCID
classification quant-ph
keywords surfacecodetransversalCliffordgateslogicalobservablematchingminimum-weightperfectwindoweddecodingfragileobservableshypergraphfast
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

The paper shows that a surface-code decoder can keep up with fast transversal Clifford gates and T-gate injections without losing the code's distance, even though the raw decoding problem contains hyperedges that ordinary matching cannot handle. Its logical observable matching (lom) decoder runs a separate minimum-weight perfect matching instance for each reliable logical measurement, on a matchable subgraph built by projecting the decoding hypergraph onto the region where errors can flip that measurement. For circuits made of H, S, and CNOT gates with one QEC round per gate, the paper proves that any basic error of weight less than d/2 is corrected. Numerical benchmarks under phenomenological and circuit-level depolarizing noise show thresholds close to memory experiments, for both repeated and arbitrary two-qubit Clifford circuits. Windowed versions of the decoder are also proposed, with trade-offs between computational efficiency, reset speed, and fault-tolerance that are only conjecturally resolved.

What carries the argument

The load-bearing object is the decoding subgraph G_O for an observable O, defined by taking the observing edge set H_O — the basic errors that flip O, which lie on one spatial boundary — and including all detectors of the same Pauli type at the same logical circuit locations, then projecting every hyperedge of the full decoding hypergraph onto those vertices. For one gate per QEC round in the {H,S,CNOT} set, this projection contains only edges, so minimum-weight perfect matching applies. The proof of Theorem 1 then follows the familiar surface-code distance argument: if the combined error-and-correction string had odd overlap with H_O, it would have to connect the two spatial boundaries and hence contain at least d edges, which contradicts the bound w(error) < d/2 together with minimality of the correction. The pre-gate detector frame keeps detecting regions local in space-time, which is what makes the observing edge set a small, boundary-local set of edges.

What would settle it

Search exhaustively over small unrotated surface codes and circuits with two H, S, or CNOT gates between QEC rounds for a reliable observable O whose projected subgraph G_O contains a weight-4 hyperedge under the basic error model; if one exists, the graph assumption fails and the matching-based d/2 guarantee collapses in that setting.

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Extended reading notes

Core claim

The central claim is that decoding across arbitrary sequences of fast transversal Clifford gates in the unrotated surface code can be reduced to independent minimum-weight matching problems, one per logical observable, without sacrificing the d/2 correction radius. For any reliable observable, the single-lom decoder projects the full decoding hypergraph onto a subgraph G_O that contains the observing edge set of that observable and only same-time, same-Pauli-type detectors; for the {H,S,CNOT} gate set with one gate per QEC round this projection is a graph, not a hypergraph. Theorem 1 then proves that the decoder correctly predicts whether an error of weight below d/2 flips the observable, using the standard surface-code argument that a mistaken logical flip would require a correction path of weight at least d. Fragile observables are never decoded directly: their outcomes are sampled randomly and combined with later observables so that only reliable products are decoded. The paper also shows that naive hyperedge-splitting and hierarchical matching decoders have distanceless failure patterns, and that windowed variants need additional short-cut edges and synchronized resets or measurements to avoid sublinear-weight logical failures.

Load-bearing premise

The d/2 guarantee relies on the decoding subgraph G_O for every observable O being a true graph with no hyperedges for circuits of {H,S,CNOT} gates with one gate per QEC round, a property the paper supports by inspection and an intuitive argument rather than a full proof.

Editorial extensions

If this is right

  • The lom decoder sustains full code distance while running one QEC round per transversal gate, so fast logical gates do not force a Θ(d) slow-down for decoding.
  • Naive splitting of weight-3 hyperedges and hierarchical matching decoders are not fault-tolerant in this setting; the lom decoder avoids their constant-weight and distance-independent logical failure patterns.
  • Numerically, thresholds under phenomenological and circuit-level depolarizing noise are close to those of memory experiments for repeated and random two-qubit Clifford circuits, suggesting that transversal gates need not degrade logical performance.
  • The basic windowed-lom decoder is computationally efficient under slow resets, while the two-step variant handles fast resets but may be inefficient; both require synchronization and short-cut edges to conjecturally correct all errors of weight below d/2.

Reading between the lines

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

  • The matchable-subgraph projection is a general decoding technique: since the paper notes it works with any graph-based decoder, it likely extends to Union-Find and to rotated or color codes with appropriate boundary structures.
  • The 'independent observers' framing suggests that any windowed decoder whose windows overlap in space-time can suffer from time-like snakes; decoders with non-overlapping commit regions avoid this failure mode by construction.
  • A testable extension is to benchmark the windowed-lom decoder with short-cut edges under circuit-level noise and compare thresholds to the non-windowed lom decoder, which would show whether the conjectured fault-tolerance is practically relevant.
  • The sublinear-weight failure examples imply that distance alone is not a reliable proxy for windowed-decoder performance; circuit-by-circuit validation may be necessary for fast-logic decoding strategies.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes a minimum-weight-perfect-matching (MWPM) based decoder, called logical observable matching (lom), for decoding the unrotated surface code when transversal Clifford gates are applied with a single QEC round between gates. The central construction is to project the full decoding hypergraph onto a subgraph G_O associated with each reliable logical observable O, then run independent MWPM instances on these subgraphs. The paper claims that, under a basic independent X/Z error model, the lom decoder corrects any error of weight less than d/2 for arbitrary circuits built from fast transversal Clifford gates and T-gate injections. It also introduces windowed variants, analyzes failures of hierarchical and splitting-hyperedge decoders, and presents extensive numerical benchmarks under phenomenological and circuit-level depolarizing noise, including comparisons with minimum-weight decoding.

Significance. If the central claim holds, this is an important step toward practical decoding of fast transversal logic: it replaces minimum-weight hypergraph decoding by standard MWPM on carefully chosen subgraphs, and it provides a concrete route to decoding with one QEC round per transversal gate. The paper is strong in several respects: the proofs of Lemma 1 and Theorem 1 are clearly structured, the numerical study is extensive with confidence intervals, thresholds, and comparisons to minimum-weight decoding, and the authors are transparent about the conjectural status of the windowed decoder variants and about the circuit-level distance-reducing errors for the repeated-S experiment. The main weakness is that the load-bearing structural property that each projected subgraph G_O is a graph is not rigorously proved; the current Appendix E gives an intuitive parity argument rather than a formal proof. This gap, together with the informal treatment of fragile observables in full circuits, means the d/2 guarantee is conditional in a way that should be resolved before the central claim is accepted as proven.

major comments (2)
  1. The claim that E_O = {h ∩ V_O : h ∈ H, h ∩ V_O ≠ ∅} is a set of edges for every reliable observable O and every circuit built from {H,S,CNOT} with one gate per round is load-bearing for Theorem 1, but it is not proven. The main text calls this 'straightforward... by visual inspection' and refers to the Supplemental Material, yet Appendix E explicitly presents only an 'intuitive, general, argument.' The argument is not logically sufficient: the fact that a space-time stabilizer has even overlap with the observing region H_O does not imply that each individual weight-3 hyperedge h satisfies |h ∩ V_O| even, since two hyperedges with odd overlap could cancel in the total stabilizer. Please provide a rigorous proof, e.g. by enumerating all hyperedge types in the basic error model (weight-1, weight-2, and weight-3 time-like hyperedges from I, H, S, and CNOT in the pre-gate frame) and verifying for each observable type that |h ∩ V_O| ≤ 2. Until this is supplied, the statement that the single-lom decoder runs MWPM on a matching instance, and hence the d/2 guarantee, is conditional.
  2. Theorem 1 is stated and proved only for a single reliable observable. The extension to an arbitrary full circuit, including fragile observables, conditioning measurements, and T-gate injections, is described algorithmically but not formalized. In particular, the procedure of randomly assigning an outcome to a fragile observable, then decoding a reliable product observable and inferring the original outcome, requires a proof that no basic error of weight < d/2 can flip the final corrected logical outcome. The discussion of replacing a fragile-conditioned S gate by a Pauli gate tracked in software is an argument sketch rather than a lemma. Please state and prove a theorem for the full lom decoder on arbitrary circuits, or explicitly mark this extension as a conjecture. This matters because the Introduction's central claim is about 'arbitrary circuits' and not only about a single reliable observable.
minor comments (5)
  1. The text says the numerical computation of |e_min| verifies that the decoder is 'circuit-distance preserving,' but then notes that for the repeated-S experiment in the X-basis under circuit-level noise, |e_min| = 2,4 for d = 3,5, i.e. d-1. This is acknowledged, but it would be clearer to state in the Introduction or Abstract that the d/2 guarantee applies to the basic error model and that circuit-level noise can reduce the effective distance by one in specific cases.
  2. The caption states that the decoded Z-measurement observable 'happens to be fragile,' which may confuse readers because Section III.B2 says fragile observables need not be decoded. Consider choosing a reliable observable in the figure or explaining explicitly why the figure is pedagogical despite the observable being fragile.
  3. The year in reference [63] appears as '20245'; this should be '2025'.
  4. The efficiency argument for the basic windowed-lom decoder uses the function f(t) quantifying operator spreading, but f(t) is only defined later in Appendix B2. Define f(t) at first use in Section V.B1 or add a forward reference.
  5. The discussion notes that more than one S and/or CNOT gate per QEC round can produce weight-4 hyperedges in G_O, which prevents reduction to matching. This is an important scope limitation of the lom decoder and should be mentioned prominently, perhaps in the abstract or introduction, so that readers do not assume the decoder works for arbitrary gate densities.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lom decoder's guarantees are derived from the code's distance and the decoding-subgraph structure, not from fitted inputs, renamed predictions, or load-bearing self-citations.

full rationale

The derivation chain is self-contained. Theorem 1's d/2 guarantee is a standard distance argument: for a reliable observable, any error-plus-correction string that flips the observable must connect the two spatial boundaries, hence has weight at least d, while minimum-weight correction gives w(e XOR c) < d. The only structural input is the claim that G_O is a graph, which the paper supports in Appendix E by an even-overlap argument involving space-time stabilizers and the observing edge set; this is a lemma to be proved, not an assumption of the target result. Even if that argument is informal, that would be a proof-completeness or correctness risk, not circularity. The numerical thresholds and logical-error rates are Monte Carlo measurements, not fitted parameters presented as predictions; the repeated-S hook-error reduction (Appendix F) is explicitly reported as a decoder weakness rather than being adjusted away. Self-citations are confined to software repositories and a standard review reference on decoder backlog, and none is load-bearing for the central claim. No uniqueness theorem from the authors is invoked, and no known empirical pattern is renamed as a new principle. The central result therefore does not reduce to its inputs by construction.

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

The LOM decoder imports the fault-tolerance of fast transversal gates from Ref. [10] and assumes the geometric matchability of each observable's subgraph; no free parameters are fitted, and no new physical entities are introduced.

assumptions (3)
  • domain assumption One QEC round after each transversal gate is sufficient for fault-tolerant fast logic (with |T> injection).
    The entire setup of decoding across gates with constant rounds relies on the result of Ref. [10]; this paper does not reprove it.
  • ad hoc to paper For any observable O, the projected decoding subgraph G_O is a graph for {H,S,CNOT} circuits with one gate per round in the pre-gate frame.
    Central to the LOM construction; supported by an informal Appendix E argument, not a complete proof.
  • domain assumption Magic states |T> are supplied fault-tolerantly with known stabilizer eigenvalues and error rate below ~p^{d/2}.
    Assumed in Section II.B item 2 as a black box.

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Pith. "Pith review of Decoding across transversal Clifford gates in the surface code." pith.science (2026). https://pith.science/paper/OVMAEYV2

@misc{pith2026250513599,
  author       = {Pith},
  title        = {Pith review of: Decoding across transversal Clifford gates in the surface code},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVMAEYV2}},
  note         = {Machine review of arXiv:2505.13599}
}
abstract

Transversal logical gates offer the opportunity for fast and low-noise logic, particularly when interspersed by a single round of parity check measurements of the underlying code. Using such circuits for the surface code requires decoding across logical gates, complicating the decoding task. We show how one can decode across an arbitrary sequence of transversal gates for the unrotated surface code, using a fast "logical observable" minimum-weight-perfect-matching (MWPM) based decoder, and benchmark its performance in Clifford circuits under circuit-level noise. We propose windowed logical observable matching decoders to address the problem of fully efficient decoding: our basic windowed decoder is computationally efficient under the restriction of quiescent (slow) resets. Our 'advanced' two-step windowed decoder can be computationally inefficient but allows fast resets. For both windowed decoders we identify errors which scale sublinearly in $d$ - depending on the structure of the circuit - which can lead to logical failure, and we propose methods to adapt the decoding to remove such failures. Our work highlights the complexity and interest in efficient decoding of fast logic for the surface code.

Figures

Figures reproduced from arXiv: 2505.13599 by the authors.

Figure 1
Figure 1. FIG. 1. (a) The [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. At each circuit location (t, j), the space-like hy￾peredges form an identical subgraph consisting of a dis￾connected X- and Z-component corresponding to the X￾and Z-detectors. Some space-like hyperedges have weight one, i.e. they are connected to only one vertex, and these we say are connected to the boundary. When decoding with a minimum-weight decoder, one would add a bound￾ary vertex vbdy to the graph and connect… view at source ↗
Figure 3
Figure 3. FIG. 3. Hyperedge decomposition and error combination that explains the bad performance of a “splitting-hyperedge” matching [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (29 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The decoding hypergraph and subgraph [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. An example of a weight-5 error in the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. A simple example of a fragile observable. (a) A [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Examples in the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. An example of successful correction by a hierarchi [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. An example of a failure of hierarchical matching due to the presence of a loop in the first track. (a) The circuit consists [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Logical circuits used to benchmark the decoder in the [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Logical error probability of the ‘splitting-hyperedge’ [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Logical error probability of the [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Focusing on single logical observables for the two-qubit repeated-gate experiments under phenomenological noise. As [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Performance of the [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Logical error probability of the [PITH_FULL_IMAGE:figures/full_fig_p023_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Performance comparison between [PITH_FULL_IMAGE:figures/full_fig_p023_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. The operation of a sliding window matching decoder [PITH_FULL_IMAGE:figures/full_fig_p025_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. (a) A visual representation of the windowed- [PITH_FULL_IMAGE:figures/full_fig_p026_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. (a) An example of a fragile time-boundary in the for [PITH_FULL_IMAGE:figures/full_fig_p029_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. An example of a failure of hierarchical matching [PITH_FULL_IMAGE:figures/full_fig_p031_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. A circuit [PITH_FULL_IMAGE:figures/full_fig_p032_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. A simplified representation of the decoding hyper [PITH_FULL_IMAGE:figures/full_fig_p032_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23. The first window of decoding the error from Figs. 21 and 22. We are only interested in four of the iterations of the [PITH_FULL_IMAGE:figures/full_fig_p033_23.png]
Figure 22
Figure 22. Figure 22: Fig. 23 shows how the proliferation of defects [PITH_FULL_IMAGE:figures/full_fig_p033_22.png]
Figure 24
Figure 24. Figure 24: FIG. 24. The second window of decoding the errors from Figs. 21 and 22 along with the left-over defects from the first window [PITH_FULL_IMAGE:figures/full_fig_p034_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. A circuit [PITH_FULL_IMAGE:figures/full_fig_p035_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26. The first window of decoding the error from Fig. 25. Each [PITH_FULL_IMAGE:figures/full_fig_p038_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27. The second window of decoding the left-over defects from the errors and corrections in Figs. 25 and 26. We are only [PITH_FULL_IMAGE:figures/full_fig_p039_27.png]
Figure 28
Figure 28. Figure 28: FIG. 28. An example of a space-time stabilizer (purple) that has even overlap with an observing hyperedge region (red and [PITH_FULL_IMAGE:figures/full_fig_p043_28.png]
Figure 29
Figure 29. Figure 29: FIG. 29. Example of a weight-4 error which is undetectable when decoding the sequence of repeated [PITH_FULL_IMAGE:figures/full_fig_p044_29.png]
Figure 30
Figure 30. Figure 30: FIG. 30. Physical circuits for the [PITH_FULL_IMAGE:figures/full_fig_p045_30.png]
Figure 31
Figure 31. Figure 31: FIG. 31. Extended version of Fig. 12(f)–(j) showing the scaling of the logical error probability of the [PITH_FULL_IMAGE:figures/full_fig_p047_31.png]

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Forward citations

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Reference graph

Works this paper leans on

87 extracted references · 41 canonical work pages · cited by 4 Pith papers

  1. [1]

    The issue is with the second-step single-loms that are obtained by the back-propagation of a multi-logical-qubit Pauli operator

    Efficiency of the two-step windowed-lomdecoder Now that we have described the action of the two-step windowed-lomdecoder, we explain why the decoder is not efficient in general. The issue is with the second-step single-loms that are obtained by the back-propagation of a multi-logical-qubit Pauli operator. As foreshadowed in Section IIIB2, there is no gene...

  2. [2]

    It is true that time- like loops may be present in each single-lomtrack,and it is also true that edges from the single-lomare used to infer artificial defects in the next window

    Time-like loops First, we explain that the windowed-lomdecoder does notfail due to the presence of time-like loops, unlike the hierarchical decoder Section IIIC1. It is true that time- like loops may be present in each single-lomtrack,and it is also true that edges from the single-lomare used to infer artificial defects in the next window. However, by def...

  3. [3]

    Fragile time-boundaries In contrast, the presence of fragile time-boundaries can cause an issue for the windowed-lomdecoder because they can cause the time-like edges to be incorrectly in- ferred, as shown in Fig. 19. Because we propagate oper- ators forwardsandbackwards in the windowed-lomde- coder, this problem can even arise in the basic windowed- lomd...

  4. [4]

    far away

    Time-like snakes The final issue that we discuss is the circuit-dependent occurrence of time-likesnakes. Intuitively, the issue arises because a low-weight spatial error in one window can be decoded multiple times by different single-loms, each of which leaves behind a pair of defects for the fol- lowing window to decode—we call theseleft-overdefects as t...

    2022

  5. [5]

    In particular, in Figs

    Simple example of a time-like snake We begin by showing our relatively simple example of how a time-like “snake” can allow a low-weight error to lead to a logical decoding error. In particular, in Figs. 21 to 24 we show how an error of weight2d/5 + 6can lead to a logical error in the windowed-lomdecoder. In this example there are multiple minimum-weight c...

  6. [6]

    proliferation

    the “proliferation” of defects: multiple single-loms in the first windowseethe same set of∆yspatial 33 FIG. 23. The first window of decoding the error from Figs. 21 and 22. We are only interested in four of the iterations of the single-lom(abbreviatedslomin the figure), corresponding to the Zoperators on logical qubits (i) 1, (ii) 3, (iii) 5 and (iv) 7 in...

  7. [7]

    the logical structure of the circuit in the second window implies that the pro- liferation of defects is seen by a next single-lomde- coder

    a time-like “snake”, i.e. the logical structure of the circuit in the second window implies that the pro- liferation of defects is seen by a next single-lomde- coder. This decoder has the opportunity to match these defects in novel ways, in particular partially using time-like edges (measurement errors). Specif- ically, we require that any time-like corre...

  8. [8]

    descending staircase

    a “descending staircase” pattern of defects in the second window, i.e. some of the artificial defects are left behind at different spatial coordinates, causing a pattern of defects where the bad correction is al- ways as short as the shortest good correction. We show the structure of the circuit in Fig. 21 and the placement of the errors in the decoding h...

    work page

Show all 87 references
  1. [9]

    descending staircase

    An asymptotically worse example To construct an example where the weight of the error which causes a logical error scales slower thanΘ(d), we use the more elaborate circuit and set of errors shown in Figs. 25 to 27, which represent a generalization and modification of the smal...

  2. [10]

    descending staircase

    Short-cut edges Weprovideaprecisedefinitionoftheshort-cutedgesin either the basic or two-step windowed-lomdecoders for which we introduce the following notation. We write the full decoding hypergraph asG= (V,H), and the decod- ing subgraph for a single-lomasG= (V,E). Moreover,...

  3. [11]

    1 ford= 3, with spatial coordinates(x,y)withx,y∈1 2 Z and0≤x,y≤d−1

    Qubit coordinates and fold-transversal gates We label each qubit in the unrotated surface code, see Fig. 1 ford= 3, with spatial coordinates(x,y)withx,y∈1 2 Z and0≤x,y≤d−1. Data qubits havex+y∈Z, ancilla qubits measuringX-stabilizers havex∈Zandy∈Z+ 1 2, andZ-ancilla qubits hav...

  4. [12]

    For a memory experiment without logical gates, all frames are equivalent and the detectors are defined in the standard way, i.e

    The pre-gate frame In this section, we discuss various “frame” definitions of detectors in the presence of logical gates, with a particular focus on thepre-gateframe that we use throughout the main text. For a memory experiment without logical gates, all frames are equivalent ...

  5. [13]

    one measurementm(t−1/2,j,x,y)at timet, and

  6. [14]

    co-moving

    any number of measurements at timet+1/2that guarantee that the detecting region only spans two consecutive QEC rounds. The detailed definition of the detectors is given in Table II. In contrast, the post-gate frame is defined such that every detectord(t,j,x,y)only contains a s...

  7. [15]

    only forXbasis

    Distance-reducing hook errors for thelomdecoder In this section we discuss a circuit-level error of weightd−1which can occur in the repeated-Sexperiment and which is undetectable by thelomdecoder, possibly affecting the numerical results in Section IV.D.1 from the main text. F...

  8. [16]

    B. Zeng, A. Cross, and I. L. Chuang, Transversality versus universality for additive quantum codes, IEEE Transactions on Information Theory57, 6272 (2011)

    2011

  9. [17]

    Eastin and E

    B. Eastin and E. Knill, Restrictions on transversal encoded quantum gate sets, Physical Review Letters102(2009)

    2009

  10. [18]

    Bombin and M

    H. Bombin and M. A. Martin-Delgado, Topological quantum distillation, Physical Review Letters97(2006)

    2006

  11. [19]

    Kubica, B

    A. Kubica, B. Yoshida, and F. Pastawski, Unfolding the color code, New Journal of Physics17, 083026 (2015)

    2015

  12. [20]

    J. E. Moussa, Transversal Clifford gates on folded surface codes, Physical Review A94, 042316 (2016)

    2016

  13. [21]

    N. P. Breuckmann and S. Burton, Fold-transversal Clifford gates for quantum codes, Quantum8, 1372 (2024)

    2024

  14. [22]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, X. Gao, P. Sales Rodriguez, T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V. Vuletić, and M. D. Lukin,...

    2023

  15. [23]

    Specs of Quantinuum Model H2 Chip (2025)

    2025

  16. [24]

    M. Cain, C. Zhao, H. Zhou, N. Meister, J. P. B. Ataides, A. Jaffe, D. Bluvstein, and M. D. Lukin, Correlated decoding of logical algorithms with transversal gates (2024), arXiv:2403.03272 [quant-ph]

    2024 arXiv

  17. [25]

    H. Zhou, C. Zhao, M. Cain, D. Bluvstein, N. Maskara, C. Duckering, H.-Y. Hu, S.-T. Wang, A. Kubica, and M. D. Lukin, Low-overhead transversal fault tolerance for universal quantum computation, Nature , 1 (2025)

    2025

  18. [26]

    Zhang, Z.-Y

    J. Zhang, Z.-Y. Chen, J.-N. Li, T.-H. Wei, H.-Y. Liu, X.-N. Zhuang, Q.-S. Li, Y.-C. Wu, and G.-P. Guo, Scalable constant- time logical gates for large-scale quantum computation using window-based correlated decoding (2025), arXiv:2410.16963 [quant-ph]. 48

    2025 arXiv

  19. [27]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics43, 4452–4505 (2002)

    2002

  20. [28]

    E. T. Campbell, A theory of single-shot error correction for adversarial noise, Quantum Science and Technology4, 025006 (2019)

    2019

  21. [29]

    Balasubramanian, M

    S. Balasubramanian, M. Davydova, and E. Lake, A local automaton for the 2d toric code (2024), arXiv:2412.19803 [quant- ph]

    2024 arXiv

  22. [30]

    McEwen, D

    M. McEwen, D. Bacon, and C. Gidney, Relaxing hardware requirements for surface code circuits using time-dynamics, Quantum7, 1172 (2023)

    2023

  23. [31]

    Sundaresan, T

    N. Sundaresan, T. J. Yoder, Y. Kim, M. Li, E. H. Chen, G. Harper, T. Thorbeck, A. W. Cross, A. D. Córcoles, and M. Takita, Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders, Nature Communications14(2023)

    2023

  24. [32]

    Sahay, Y

    K. Sahay, Y. Lin, S. Huang, K. R. Brown, and S. Puri, Error correction of transversal CNOT gates for scalable surface code computation (2024), arXiv:2408.01393 [quant-ph]

    2024

  25. [33]

    K. H. Wan, M. Webber, A. G. Fowler, and W. K. Hensinger, An iterative transversal CNOT decoder (2024), arXiv:2407.20976 [quant-ph]

    2024 arXiv

  26. [34]

    Guernut and C

    A. Guernut and C. Vuillot, Fault-tolerant constant-depth Clifford gates on toric codes (2024), arXiv:2411.18287 [quant-ph]

    2024 arXiv

  27. [35]

    Chen, M.-C

    Z.-H. Chen, M.-C. Chen, C.-Y. Lu, and J.-W. Pan, Transversal logical Clifford gates on rotated surface codes with reconfigurable neutral atom arrays (2024), arXiv:2412.01391 [quant-ph]

    2024 arXiv

  28. [36]

    M. E. Beverland, A. Kubica, and K. M. Svore, Cost of universality: A comparative study of the overhead of state distillation and code switching with color codes, PRX Quantum2, 020341 (2021)

    2021

  29. [37]

    Delfosse and N

    N. Delfosse and N. H. Nickerson, Almost-linear time decoding algorithm for topological codes, Quantum5, 595 (2021)

    2021

  30. [38]

    Delfosse, A

    N. Delfosse, A. Paetznick, J. Haah, and M. B. Hastings, Splitting decoders for correcting hypergraph faults (2023), arXiv:2309.15354 [quant-ph]

    2023 arXiv

  31. [39]

    Gidney, Stim: a fast stabilizer circuit simulator, Quantum5, 497 (2021)

    C. Gidney, Stim: a fast stabilizer circuit simulator, Quantum5, 497 (2021)

    2021

  32. [41]

    Caune, L

    L. Caune, L. Skoric, N. S. Blunt, A. Ruban, J. McDaniel, J. A. Valery, A. D. Patterson, A. V. Gramolin, J. Majaniemi, K. M. Barnes,et al., Demonstrating real-time and low-latency quantum error correction with superconducting qubits, arXiv preprint arXiv:2410.05202 (2024)

    2024 arXiv

  33. [42]

    Google Quantum AI and Collaborators, Quantum error correction below the surface code threshold, Nature (Dec 9, 2024)

    2024

  34. [43]

    Tomita and K

    Y. Tomita and K. M. Svore, Low-distance surface codes under realistic quantum noise, Physical Review A90(2014)

    2014

  35. [44]

    T. E. O’Brien, B. Tarasinski, and L. DiCarlo, Density-matrix simulation of small surface codes under current and projected experimental noise, npj Quantum Information3, 10.1038/s41534-017-0039-x (2017)

    2017 doi

  36. [45]

    Skoric, D

    L. Skoric, D. E. Browne, K. M. Barnes, N. I. Gillespie, and E. T. Campbell, Parallel window decoding enables scalable fault tolerant quantum computation, Nature Communications14, 7040 (2023)

    2023

  37. [46]

    X. Tan, F. Zhang, R. Chao, Y. Shi, and J. Chen, Scalable surface-code decoders with parallelization in time, PRX Quantum 4, 040344 (2023)

    2023

  38. [47]

    Viszlai, J

    J. Viszlai, J. D. Chadwick, S. Joshi, G. S. Ravi, Y. Li, and F. T. Chong, Predictive window decoding for fault-tolerant quantum programs (2024), arXiv:2412.05115 [quant-ph]

    2024 arXiv

  39. [48]

    M. Cain, D. Bluvstein, C. Zhao, S. Gu, N. Maskara, M. Kalinowski, A. A. Geim, A. Kubica, M. D. Lukin, and H. Zhou, Fast correlated decoding of transversal logical algorithms (2025), arXiv:2505.13587 [quant-ph]

    2025 arXiv

  40. [49]

    Bacon, S

    D. Bacon, S. T. Flammia, A. W. Harrow, and J. Shi, Sparse quantum codes from quantum circuits, IEEE Transactions on Information Theory63, 2464 (2017)

    2017

  41. [50]

    Delfosse and A

    N. Delfosse and A. Paetznick, Spacetime codes of Clifford circuits (2023), arXiv:2304.05943 [quant-ph]

    2023 arXiv

  42. [51]

    Gottesman, Opportunities and challenges in fault-tolerant quantum computation (2022), arXiv:2210.15844 [quant-ph]

    D. Gottesman, Opportunities and challenges in fault-tolerant quantum computation (2022), arXiv:2210.15844 [quant-ph]

    2022 arXiv

  43. [52]

    See Supplemental Material

  44. [53]

    Chamberland and E

    C. Chamberland and E. T. Campbell, Universal quantum computing with twist-free and temporally encoded lattice surgery, PRX Quantum3, 010331 (2022)

    2022

  45. [54]

    A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Physical Review A86(2012)

    2012

  46. [55]

    Higgott and C

    O. Higgott and C. Gidney, Sparse Blossom: correcting a million errors per core second with minimum-weight matching, Quantum9, 1600 (2025)

    2025

  47. [56]

    Serra-Peralta, surface-sim on GitHub (2024)

    M. Serra-Peralta, surface-sim on GitHub (2024)

    2024

  48. [57]

    Serra-Peralta, lomatching on GitHub (2024)

    M. Serra-Peralta, lomatching on GitHub (2024)

    2024

  49. [58]

    Versluis, S

    R. Versluis, S. Poletto, N. Khammassi, B. Tarasinski, N. Haider, D. J. Michalak, A. Bruno, K. Bertels, and L. DiCarlo, Scalable quantum circuit and control for a superconducting surface code, Phys. Rev. Appl.8, 034021 (2017)

    2017

  50. [59]

    Wintersperger, F

    K. Wintersperger, F. Dommert, T. Ehmer, A. Hoursanov, J. Klepsch, W. Mauerer, G. Reuber, T. Strohm, M. Yin, and S. Luber, Neutral atom quantum computing hardware: performance and end-user perspective, EPJ Quantum Technology 10, 32 (2023)

    2023

  51. [60]

    Krinner, N

    S. Krinner, N. Lacroix, A. Remm, A. Di Paolo, E. Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Herrmann, G. J. Norris, C. K. Andersen, M. Müller, A. Blais, C. Eichler, and A. Wallraff, Realizing repeated quantum error correction in a distance-three surface code, Nat...

    2022

  52. [61]

    Combes, C

    J. Combes, C. Granade, C. Ferrie, and S. T. Flammia, Logical randomized benchmarking (2017), arXiv:1702.03688 [quant- ph]. 49

    2017 arXiv

  53. [62]

    Lacroixet al., Scaling and logic in the color code on a superconducting quantum processor (2024), arXiv:2412.14256 [quant-ph]

    N. Lacroixet al., Scaling and logic in the color code on a superconducting quantum processor (2024), arXiv:2412.14256 [quant-ph]

    2024

  54. [63]

    A. R. Calderbank, E. M. Rains, P. M. Shor, and N. J. Sloane, Quantum error correction via codes over GF(4), IEEE Transactions on Information Theory44, 1369 (1998)

    1998

  55. [64]

    Grier and L

    D. Grier and L. Schaeffer, The classification of Clifford gates over qubits, Quantum6, 734 (2022)

    2022

  56. [65]

    M. L. Turner, E. T. Campbell, O. Crawford, N. I. Gillespie, and J. Camps, Scalable decoding protocols for fast transversal logic in the surface code (2025), arXiv:2505.23567 [quant-ph]

    2025 arXiv

  57. [66]

    Gidney, M

    C. Gidney, M. Newman, A. Fowler, and M. Broughton, A fault-tolerant honeycomb memory, Quantum5, 605 (2021)

    2021

  58. [67]

    S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, et al., High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature622, 268 (2023)

    2023

  59. [68]

    P. S. Rodriguez, J. M. Robinson, P. N. Jepsen, Z. He, C. Duckering, C. Zhao, K.-H. Wu, J. Campo, K. Bagnall, M. Kwon, et al., Experimental demonstration of logical magic state distillation, arXiv preprint arXiv:2412.15165 (2024)

    2024

  60. [69]

    T. M. Graham, L. Phuttitarn, R. Chinnarasu, Y. Song, C. Poole, K. Jooya, J. Scott, A. Scott, P. Eichler, and M. Saffman, Midcircuit measurements on a single-species neutral alkali atom quantum processor, Physical Review X13, 041051 (2023)

    2023

  61. [70]

    Deist, Y.-H

    E. Deist, Y.-H. Lu, J. Ho, M. K. Pasha, J. Zeiher, Z. Yan, and D. M. Stamper-Kurn, Mid-circuit cavity measurement in a neutral atom array, Physical Review Letters129, 203602 (2022)

    2022

  62. [71]

    Serra-Peralta, qec-util on GitHub (2024)

    M. Serra-Peralta, qec-util on GitHub (2024)

    2024

  63. [72]

    E. B. Wilson, Probable inference, the law of succession, and statistical inference, Journal of the American Statistical Association22, 209 (1927)

    1927

  64. [73]

    Heußen, D

    S. Heußen, D. Winter, M. Rispler, and M. Müller, Dynamical subset sampling of quantum error-correcting protocols, Physical Review Research6, 013177 (2024)

    2024

  65. [74]

    L. D. Brown, T. T. Cai, and A. DasGupta, Interval estimation for a binomial proportion, Statistical science16, 101 (2001)

    2001

  66. [75]

    Hillmann, G

    T. Hillmann, G. Dauphinais, I. Tzitrin, and M. Vasmer, Single-shot and measurement-based quantum error correction via fault complexes, arXiv preprint arXiv:2410.12963 (2024)

    2024

  67. [76]

    Wang,Surface code error correction during logical operations, Master’s thesis, TU Delft (2023)

    Z. Wang,Surface code error correction during logical operations, Master’s thesis, TU Delft (2023)

    2023

  68. [77]

    A. R. O’Rourke and S. Devitt, Compare the pair: Rotated vs. unrotated surface codes at equal logical error rates (2024), arXiv:2409.14765 [quant-ph]

    2024 arXiv

  69. [78]

    Serra-Peralta, mle-decoder on GitHub (20245)

    M. Serra-Peralta, mle-decoder on GitHub (20245)

  70. [79]

    A. J. Landahl, J. T. Anderson, and P. R. Rice, Fault-tolerant quantum computing with color codes (2011), arXiv:1108.5738 [quant-ph]

    2011 arXiv

  71. [80]

    Gurobi Optimization, Gurobi Optimizer Reference Manual (2024)

    2024

  72. [81]

    Nahum, S

    A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Physical Review X8, 10.1103/phys- revx.8.021014 (2018)

    2018 doi

  73. [82]

    B. M. Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics87, 307–346 (2015)

    2015

  74. [83]

    Higgott, T

    O. Higgott, T. C. Bohdanowicz, A. Kubica, S. T. Flammia, and E. T. Campbell, Improved decoding of circuit noise and fragile boundaries of tailored surface codes, Phys. Rev. X13, 031007 (2023)

    2023

  75. [84]

    Higgott, BeliefMatching on GitHub (2024)

    O. Higgott, BeliefMatching on GitHub (2024)

    2024

  76. [85]

    S.-H. Lee, A. Li, and S. D. Bartlett, Color code decoder with improved scaling for correcting circuit-level noise, Quantum 9, 1609 (2025)

    2025

  77. [86]

    Gillett, Parallel and low-latency streaming decoding of quantum error correction codes (2025), APS March Meeting 2025

    N. Gillett, Parallel and low-latency streaming decoding of quantum error correction codes (2025), APS March Meeting 2025

    2025

  78. [87]

    Serra-Peralta, data-lom-decoders on GitHub (2025)

    M. Serra-Peralta, data-lom-decoders on GitHub (2025)

    2025

  79. [88]

    Delft High Performance Computing Centre (DHPC), DelftBlue documentation,https://doc.dhpc.tudelft.nl/ delftblue/(2023), accessed: 2023-05-15

    2023

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.