Scalable decoding protocols for fast transversal logic in the surface code

Earl T. Campbell; Joan Camps; Mark L. Turner; Neil I. Gillespie; Ophelia Crawford

REVIEW 4 major objections 6 minor 69 references

Scalable decoding protocols for fast transversal logic in the surface code

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Ghost decoding and patience make $O(1)$-round transversal logic in the surface code decodable at scale, restoring modularity and unlocking an order-of-magnitude speed-up over lattice surgery.

desk verdict Genuinely new decoding ideas for fast transversal logic, but the scalability claim rests on an empirically tuned pass count that grows with distance. read the letter →

arxiv 2505.23567 v1 pith:RFRQJ6QM submitted 2025-05-29 quant-ph

Mark L. Turner , Earl T. Campbell , Ophelia Crawford , Neil I. Gillespie , Joan Camps This is my paper

classification quant-ph MSC 81P7081P68 PACS 03.67.Pp

keywords transversallogicsurfacecodequantumerrorcorrectionreal-timedecodingwindowingghostprotocolpatienceAMOcomputing

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 tries to establish that fast transversal logic in the surface code can be decoded scalably in real time, even though transversal entangling gates destroy the locality that makes lattice-surgery decoding efficient. Transversal gates can run with only $O(1)$ rounds of syndrome extraction between them, but each entangling gate fuses the detector error models of separate logical qubits into cross-qubit hyperedges, inflating the decoding volume. The ghost protocol decomposes those hyperedges so that each logical qubit decodes independently and exchanges sparse messages, turning the decoding problem back into a modular, windowed one. A second protocol, patience, removes the accuracy penalty that aggressive temporal windowing imposes on teleportation-based $T$ gates by rarely postponing a classically controlled gate when the decoder heralds a failure. If the claims hold, slow-clock AMO hardware runs Clifford+T circuits at an order-of-magnitude higher logical clock rate than lattice surgery, with a small space overhead.

What carries the argument

The central object is the ghost-edge decomposition of the detector error model — the representation of which physical errors flip which syndrome detectors. Transversal entangling gates create order-three hyperedges that span logical qubits; the protocol splits each into a time-like ghost edge confined to one logical qubit and a ghost singleton (an edge to the boundary) confined to the other, so that the spatially delocalised decoding problem becomes a sparse message-passing problem among single-qubit decoder cores. The companion patience mechanism is a heralding procedure: two inferences — growth in correction weight between decoding iterations, and a complementary matching on a closed temporal boundary — flag the rare distance-reducing failures identified by Eq. (1), and only then does the decoder postpone a conditional gate to collect further syndrome rounds. The paper also identifies the topological source of resilience that makes this work: error strings that pierce a transversal entangling gate fork in space, creating logically trivial loops that substitute for the buffer regions a windowed memory decoder would otherwise need.

What would settle it

Run the deep-Clifford benchmark circuit behind Fig. 3 at distances $d=15$ and $d=17$ with eight logical qubits and count how many ghost-decoding passes are needed to keep the logical error per layer falling exponentially with distance; if that count grows with distance or qubit count instead of staying constant, the per-window reconfiguration overhead grows with system size and the scalability claim fails.

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Extended reading notes

Core claim

The central claim is that $O(1)$-round transversal Clifford+T logic in the rotated surface code can be decoded with fully windowed, scalable decoders — something prior work saw as blocked either by exponential decoding volume or by the need for global, non-windowed correlated decoding. The paper's key move is to decompose the order-three hyperedges created by transversal entangling gates into ghost edges: a time-like edge confined to one logical qubit plus a ghost singleton confined to the other, so that decoding factorises into sparse message passing between independent single-qubit decoders. Iterating this ghost protocol purges the inter-qubit hyperedges from the syndrome, and exposing ghost singletons only during carefully chosen passes keeps low-weight logical loops out of the final decoding problem. For teleportation-based $T$ gates the paper shows the time overhead is a buffering problem, that aggressive temporal windowing with $n_{\mathrm{buf}}=1$ still suppresses errors in the distance, and that the residual failure modes, bounded by $w \ge (n_{\mathrm{buf}} + 1 + \lceil d/2\rceil)/2$, halve the effective distance. The patience protocol heralds those rare failure modes and postpones only the affected conditional gate, recovering accuracy close to global decoding at negligible average cost and delivering the claimed order-of-magnitude speed-up with small space overhead.

Load-bearing premise

Everything rests on the assumption that a small, hand-chosen number of decoding passes will keep working for larger circuits and bigger code distances; the paper admits it currently picks these schedules by trial and error rather than by rule.

Editorial extensions

If this is right

Fully windowed decoding of $O(1)$-round transversal Clifford+T logic becomes possible: each decoder core owns one logical qubit, and the per-core decoding problem looks like a single-qubit memory experiment regardless of the surrounding circuit.
On slow-clock AMO hardware, Clifford+T circuits run at more than an order of magnitude higher logical clock rate than under lattice surgery, at a small space overhead instead of the large overhead of time-optimal lattice surgery.
Fast $T$ gates survive aggressive temporal windowing: with $n_{\mathrm{buf}}=1$ errors are still suppressed in the distance (Eq. (1) predicts a halved effective distance of $\lceil d/4\rceil$ at larger $d$), and patience restores accuracy close to global decoding at an average cost below $3\times10^{-3}$ extra syndrome rounds per $T$ gate at $d=9$, $p=0.1\%$.
Because the protocols do not depend on graph-based decoders, the ghost decomposition can be paired with fast hypergraph decoders, which the paper suggests as the route toward applying the approach beyond the surface code.

Reading between the lines

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

The scalability claim currently hangs on pass-count tuning: Appendix B says parameters are found empirically per distance and round count, and a rule for setting them automatically would turn the scheme from a demonstrated benchmark into a generally testable one.
The mechanism behind the resilience — error strings forking at transversal gates create extra syndrome information that substitutes for buffer regions — suggests a design principle for other $O(1)$-depth gadgets, potentially extensible to transversal $S$ gates, magic-state teleportation, and codes beyond the rotated surface code.
Patience is effectively speculative execution — proceed at minimal buffer size, stall only on a herald — and the paper reports only the mean delay, so the natural follow-up is the delay tail that a real-time controller would actually schedule against.
As the paper's own note-added records, closely related decoding schemes appeared while this work was being finalised; the decisive comparison will be whether windowed ghost decoding keeps its accuracy and bounded pass counts at larger distances and qubit counts where tuning is currently manual.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing to know: this paper is worth a serious referee. The ghost protocol and patience protocol are real new ideas, and the numerics are unusually careful for this area. The central claim about O(1)-round transversal logic being decodable in a windowed, modular way is supported by the simulations. But the word “scalable” is doing more work than the evidence supports: the number of decoding passes is chosen empirically per (distance, number of rounds), and in the deep Clifford benchmark it rises from 4 to 8 when d goes from 9 to 11. That is a soft spot you need to keep in mind.

New and good: the decomposition of interpatch order-three hyperedges into ghost edges plus singletons, with iterative message passing between per-qubit decoders, is a fresh way to break the non-locality that made fast transversal logic look undecodable in a windowed setting. The patience protocol — heralded, rare buffer extension — is clever and the measured average delay is tiny. The topological analysis that leads to Eq. (1) is plausible, and the claim that they have verified by search that the smallest failing patterns saturate that inequality is good practice. The T-gate proxy data showing strong suppression with nbuf=1 is surprising and the figure of merit is reasonable.

Soft spots, in order: (1) App. B admits passes are empirical and grow with d. The paper says they have never seen passes exceed d, which is not a bound. If passes scale linearly with d, then the per-window classical work grows, and the order-of-magnitude speed-up claim needs a throughput analysis to survive. This is the main gap. (2) The non-Clifford results are for a Clifford proxy circuit, not the actual teleportation circuit. The authors explain why, and it is a reasonable proxy, but the headline claim about Clifford+T speed is one step indirect. (3) The promised Stim circuits on Zenodo are not linked in the text we have; that should be fixed for reproducibility. (4) There is no full space/time comparison to lattice surgery, so the speed-up is argued from round counts rather than end-to-end resource estimates.

None of this kills the paper. The ideas are concrete, the simulations are honest, and the authors flag their own limitations. For peer review: yes, send it out. The main thing I would ask a referee to demand is a clear statement about the scaling of pass count — or better, an automatic way to set passes — and some throughput numbers. I’d cite it for the ghost protocol if I were working on decoder windowing, and I’d bring it to reading group.

Referee Report

4 major / 6 minor

Summary. The paper proposes two windowed decoding protocols, the ghost protocol and patience, to make real-time decoding of fast transversal logic in the surface code scalable. The ghost protocol decomposes order-three hyperedges that arise at transversal entangling gates into ghost edges and ghost singletons, allowing each logical qubit to be decoded independently with iterative message passing. Patience augments this by heralding distance-reducing failure modes and postponing conditional gates until more syndrome data are available. The authors report Stim simulations of deep Clifford circuits and a Clifford proxy for teleportation-based T gates, showing subthreshold error suppression and claiming that, with a very small space overhead, transversal Clifford+T logic can be executed in O(1) time per gate, giving an order-of-magnitude speed-up over lattice surgery.

Significance. If the central claims hold, the paper addresses an important bottleneck for AMO-based fault-tolerant computing: transversal logic can in principle provide a fast logical clock, but decoding was previously thought to require either global correlated decoding with untenable decoding volumes or O(d) syndrome-extraction rounds between gates. The ghost-protocol decomposition is a genuinely new idea, and the topological arguments in Sec. V for inherent resilience are plausible and thought-provoking. The paper ships reproducible Stim circuits, uses a realistic circuit-level noise model, and provides numerical evidence of subthreshold scaling for both Clifford and T-gate-proxy circuits. However, the significance depends on resolving the scalability of the empirically tuned decoding passes and on validating the T-gate proxy, so the contribution is promising but not yet fully established.

major comments (4)

[Appendix B (Fig. 3)] Appendix B reports that the number of decoding passes is an empirically tuned parameter, with the default four passes used only for d ≤ 9; at d = 11 the deep Clifford circuits require six passes for nr = 2,3 and eight passes for nr = 1. Since each pass is a full independent decoding round over every logical qubit, the per-window decoding cost is not demonstrated to be constant or bounded as d grows; the statement that the pass count has not been seen to exceed the code distance actually allows growth with d. No deep-Clifford data are shown for d > 11. This is load-bearing for the abstract claim of very small space overhead and an order-of-magnitude speed-up, because the speed-up over lattice surgery assumes O(1) decoding overhead per logical gate. The authors should either prove or numerically demonstrate that the required pass count remains small with d, or substantially weaken the scalability claim.
[Sec. IV (Fig. 4b)] The T-gate time-complexity analysis is based entirely on a proxy circuit that strips out the classically controlled S gates, the H gates between consecutive T gates, and the injected |T> magic states, replacing them with a Clifford circuit in which all qubits are prepared in |0>. The paper asserts that this proxy captures the structural properties of teleportation-based T gates, including severances, but provides no evidence that the decoder's behaviour on the proxy matches the real circuit. In particular, the classically controlled S gate changes the Pauli frame and can affect which corrections are logically trivial, so it is not obvious that the proxy's severance structure is representative. Since the central non-Clifford claim (O(1)-time T gates with small buffers) rests on this proxy, the authors should validate it against a full teleportation circuit at least at small distances, or provide a precise equivalence argument.
[Sec. VI (Fig. 9, Table I)] Patience is introduced as the mechanism that restores near-global accuracy by heralding distance-reducing failures, but the two herald conditions are heuristic and no analysis is given of their false-negative rate. A false negative leads directly to a logical error, as the text concedes, so the claim that patience recovers the code distance requires either a proof that all failure modes of the form in Fig. 8b are heralded, or a numerical estimate of the false-negative rate and its contribution to the logical error rate. Table I also omits entries for p = 0.001 at d = 13 and d = 15, which are exactly the distances where the distance-halving effect of Eq. (1) should be most visible; these data should be reported.
[Sec. II and Sec. III] The scalability argument for general fast transversal logic is benchmarked only on four-qubit random Clifford circuits and a single T-gate proxy. The worst-case binary-tree circuits considered by Sahay et al. [17] are dismissed as unrealistic, but the manuscript does not show that the ghost protocol's message-passing schedule remains efficient for arbitrary circuits with O(1) spacing; indeed App. B shows the schedule must be re-tuned with circuit structure, nr, and d. To support the claim of scalable decoding for transversal logic in general, the authors should either extend the numerical evidence to more adversarial connectivity patterns or state the class of circuits for which the protocol is intended.

minor comments (6)

[Abstract and Sec. I] The abstract says 'O(1) time' while the introduction says 'O(1) rounds of syndrome extraction'; this should be made consistent and explicit about which resource is being held constant.
[Fig. 5b] The 'TW error' is defined as the probability that the windowed decoder disagrees with the global decoder, which is not itself a logical error rate; the caption should make this distinction explicit to avoid over-interpretation.
[Eq. (1)] Eq. (1) introduces w, c, and ch without a formal definition in the main text; define these quantities clearly before the inequality is used.
[Table I] The dash entries in Table I are not explained; state whether the data are missing, not simulated, or statistically indistinguishable from zero, and report the number of samples used for the averages.
[Appendix A] The statement that all error bars represent likelihoods within a factor of 1000 of the max likelihood hypothesis should be clarified, and the number of shots per data point should be stated.
[Sec. V] The claim that avoiding a buffer of size nbuf ≈ d/2 leads to a fourfold space overhead in the number of qubits is not derived; provide the counting argument behind this factor.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ghost protocol and patience are constructive protocols whose error rates are measured, not derived from the target claim; App. B pass-count tuning is a disclosed scalability caveat, not a fitted prediction.

full rationale

The derivation chain is self-contained. The ghost protocol is defined constructively: order-three hyperedges from transversal CNOTs are decomposed into ghost edges and ghost singletons, and decoding proceeds by iterative message passing and syndrome refinement (Sec. III). The reported logical-error rates in Figs. 3, 5 and 9 are simulated outcomes under a circuit-level depolarizing noise model; no plotted logical-error quantity is an input to the protocol or to Eq. (1). Eq. (1) is derived from the geometry of correction strings relative to the open temporal boundary and is then checked by exhaustive search, and the patience delay schedule in Table I follows from Eq. (1) rather than from fitting the logical-error data. The closest circularity-adjacent issue is Appendix B, where pass counts and ghost-singleton exposure are set empirically: 'We currently determine these parameters empirically and further study is needed to work out how to set these parameters automatically in regimes that are hard to simulate.' This is a genuine limitation on the strength of the scalability claim beyond the simulated distances, but it is a hyperparameter-dependence and a missing automatic-tuning result, not an equation-level reduction of the claimed prediction to the fit. The self-citations present in the paper, e.g. [16], [19], [33], and the Riverlane noise model [63], are background or engineering-comparison references and do not carry the central argument; the principal benchmarks are external ([12], global correlated decoding) and the results are measured against those baselines. No uniqueness theorem from the authors' prior work is invoked to force the protocol choice. Therefore no circular step can be exhibited from the paper's own equations or citations.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central claim rests on the noise model, the proxy circuit, the postponability of conditional gates, the linear-volume assumption for realistic circuits, and the distributed decoding architecture. The only free parameters are the empirically tuned decoding pass schedules and the chosen small buffer constants. The ghost edges and singletons are algorithmic artifacts, not physical entities.

free parameters (3)

Decoding pass count and ghost-singleton exposure schedule = Default 4 passes; for Fig. 3 at d=11, nr=2 or 3: 6 passes with singletons on passes 1 and 4; for d=11, nr=1: 8 passes…
Chosen empirically per (d, nr) to maintain exponential error suppression (App. B). The paper states it cannot yet set these parameters automatically, so they are hand-tuned to the data.
T-gate proxy operating points nbuf and nsep = nbuf=1, nsep=3
These constants define the O(1)-round operating point for the T gate proxy. They are chosen by the authors to demonstrate fast logic, not fitted to error rates, but they are the target regime for the speed-up claim.
Patience delay (nbuf increase when heralded) = Delay = 0,1,2,3,4,5,6 for d = 3,5,7,9,11,13,15 (Table I)
Computed from Eq. (1) so that w = ceil(d/2) in the rare cases patience is triggered. These are deterministic delays derived from the distance, not fitted to error data, but they set the worst-case latency of the protocol.

assumptions (5)

domain assumption Circuit-level noise model with two-qubit depolarising error rate p, single-qubit and idle error rate p/10, and measurement error p (App. A)
Used for all Stim simulations; a standard circuit-level noise model but it is an assumption about hardware behaviour that affects all reported logical error rates.
ad hoc to paper The T-gate proxy circuit (Fig. 4b) captures the structural decoding properties of full teleportation-based T gates, including severances, despite omitting classically controlled gates and non-stabiliser states
The paper strips out the non-Clifford feedback to make the circuit Clifford-simulable. The fidelity of this proxy to real T-gate teleportation is not proven, and the authors acknowledge non-Clifford simulations are future work.
domain assumption The decoder can postpone classically controlled S gates while additional syndrome rounds are collected (property 1 of patience)
Patience relies on the freedom to delay a conditional gate without changing the logical operation. This is architecturally plausible but depends on having bufferable idle qubits and flexible scheduling.
domain assumption Realistic algorithms such as Toffoli ladders keep the decoding volume growing linearly with window depth, not exponentially as in the binary-tree worst case of Sahay et al.
The scalability argument assumes worst-case exponential error-spreading circuits are not representative of practical algorithms. This is a stated belief, not a proven bound.
domain assumption Each logical qubit is decoded on its own core with sparse messages, and distributed decoding can be reconfigured with minimal overhead between windows
The modularity claim assumes one decoder core per logical qubit and low message-passing overhead. This is an architectural assumption about the classical control system.

invented entities (1)

Ghost edges and ghost singletons
purpose: Artificial decomposition of order-three hyperedges that span logical qubits into graphlike edges, enabling independent per-qubit decoding with message passing.
These are constructs of the decoder model, not physical entities, and have no falsifiable observable outside the decoding algorithm itself.

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Cite this review

Pith. "Pith review of Scalable decoding protocols for fast transversal logic in the surface code." pith.science (2026). https://pith.science/paper/RFRQJ6QM

@misc{pith2026250523567,
  author       = {Pith},
  title        = {Pith review of: Scalable decoding protocols for fast transversal logic in the surface code},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RFRQJ6QM}},
  note         = {Machine review of arXiv:2505.23567}
}

read the original abstract

Atomic, molecular and optical (AMO) approaches to quantum computing are promising due to their increased connectivity, long coherence times and apparent scalability. However, they have a significantly reduced cadence of syndrome extraction compared to superconducting devices, a potentially crippling slow-down given the substantial logical gate counts required for quantum advantage. Transversal logic, which exploits higher connectivity, has the potential to significantly speed up the logical clock rate by reducing the number of syndrome extraction rounds required, but current decoders for fast transversal logic are not scalable. This is not just because existing decoders are too slow to handle the large decoding volumes resulting from fast logic; transversal logic breaks the key structural properties that make real-time decoding of lattice surgery efficient. We introduce two new, windowed decoding protocols for transversal logic in the surface code that restore modularity and locality to the decoding problem. Using our protocols, we show that, with a very small space overhead, our scalable decoders unlock an order of magnitude speed-up for transversal logic compared to lattice surgery. Taken together, our results provide key evidence for the viability of large-scale algorithms on AMO qubits.

Figures

Figures reproduced from arXiv: 2505.23567 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** shows the subthreshold scaling of ghost decoding of deep Clifford logic between four rotated surface codes under circuit level noise (p = 0.1%)—a benchmark first proposed in [12]. In our numerics, the best accuracy is seen with nr = 3 but the spacetime volume of the computation is minimised at nr = 1 in the regimes we study, results that are consistent with what was previously observed with global correlated deco… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: , two quantities allow us to quantify the time overhead of a T gate realised with transversal gates: 1. nbuf captures the number of rounds of syndrome extraction between a transversal CNOT that creates the entanglement necessary for teleportation and the logical meas…

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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nbuf capturesthenumberofroundsofsyndromeex- traction between a transversal CNOT that creates the entanglement necessary for teleportation and the logical measurement that informs a classically controlled S gate

work page

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severance

nsep defines the number of rounds that separate consecutive T gates. Notably, nbuf functions as a buffer region with respect to the decoding problem centred on the logical measure- ments in the circuit, presenting new structural challenges for decoding. This is because the logical measurement can be flipped by errors before the transversal CNOT on both qu...

work page

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That is, we are free to accumulate more syndrome informa- tion than originally planned for via further syn- drome extraction rounds before applying a condi- tional gate

The freedom to postpone the application of con- ditional logic during teleportation. That is, we are free to accumulate more syndrome informa- tion than originally planned for via further syn- drome extraction rounds before applying a condi- tional gate

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The inherent rarity of the distance reducing failure modes described in Sec. V. This rarity is evident in the low TW error probability observed in Fig. 5b. Property (1) means nbuf can be a runtime parameter without changing the underlying logical operation (in the absence of error, aT gate remains aT gate regardless of how many rounds separate a CNOT from...

work page

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Patient GP

We run the ghost protocol as usual, but track whether the weight of the correction string in- creases between the first and final decoding iter- ations performed by the ghost protocol. This func- tions to detect most cases where aggressive tem- poral windowing spuriously triggers a syndrome refinement operation (of the form described in Sec. III) despite ...

work page

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With this complementary match- ing, we can herald cases that yield a different cor- rection bit to the previous decoding with the open temporal boundary

Inspired by soft-output decoding constructions [48, 49], we perform a further complementary match- ing on a modified DEM, where the open temporal boundary is turned into a closed boundary, mean- ing error strings can no longer terminate at a tem- poral boundary. With this complementary match- ing, we can herald cases that yield a different cor- rection bi...

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Bluvstein, S

D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kali- nowski, 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, Logical quantum processor based on recon- figurable atom arrays, Nature626, 5...

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