Diamond Circuits for Surface Codes

Dripto M. Debroy

REVIEW 3 major objections 5 minor 19 references

Diamond Circuits for Surface Codes

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Diamond circuits run a surface code on a sparse qubit grid with about 25% fewer qubits, preserving spacelike distance at a timelike cost.

desk verdict New surface-code circuit family cuts qubit/coupler counts by dropping half the measure qubits, but the distance claim needs a proof rather than being inherited from the LUCI framework. read the letter →

arxiv 2502.10355 v1 pith:3FSR4SDV submitted 2025-02-14 quant-ph

Dripto M. Debroy This is my paper · ORCID

classification quant-ph

keywords surfacecodeLieblatticeHeavy-Squarediamondcircuitssubsystemlogicalerrorratequantumcorrectioncontrollineefficiency

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper presents 'diamond circuits': a way to run the standard surface code on the sparse Heavy-Square (Lieb) lattice, where two-thirds of the qubits couple to only two neighbors. The construction removes half of the measurement qubits from a rotated surface-code grid, reducing the resource count for a distance-$d$ code from roughly $2d^2-1$ qubits and $4d^2$ couplers to $1.5d^2$ qubits and $2d^2$ couplers. The author argues through the LUCI framework that these circuits preserve the code's spacelike distance, while paying a roughly fourfold penalty in timelike distance, and benchmarks them under a superconducting-inspired noise model. The payoff is architectural: on a fixed budget of control lines or qubits, a diamond circuit can support a distance about 1.31 times larger than a standard surface code, making it competitive when wiring, frequency collisions, or fabrication constraints dominate.

What carries the argument

The load-bearing mechanism is the LUCI framework applied to the mid-cycle state of a subsystem surface code. In this construction the mid-cycle state is a set of weight-3 gauge operators (the 'facets' of the cut diamond) on a Lieb lattice; dotted-line pairings fuse adjacent gauges into weight-6 superstabilizers in the bulk, while unpaired boundary gauges become weight-3 stabilizers. Each measure qubit is shared by four stabilizers, so the circuit interleaves two layers of CNOTs, a measurement and reset, and the reverse CNOTs over four rounds to close one detecting region while opening the next. The framework is what certifies that dropping half the measure qubits still leaves a valid surface-code circuit with spacelike distance $d$.

What would settle it

For a distance-5 diamond circuit, build the full detector-error model from the compiled circuit and compute the minimum weight of an undetectable X-type and Z-type error chain; if any such chain has weight less than 5, the spacelike-distance claim is false. A second check is to run the memory experiment at several distances and look for the expected timelike factor: diamond logical error rates should match a standard surface code only after roughly 4x more rounds, so a measured timelike penalty much smaller or larger would contradict the detecting-region picture.

Watch

Extended reading notes

Core claim

The central claim is that the Heavy-Square lattice is not a bad home for the surface code. By starting from a subsystem surface code mid-cycle state in which weight-3 gauge operators tile the lattice, pairing gauge operators across each square to form weight-6 superstabilizers, and measuring and resetting only half of the would-be measure qubits, the author constructs a circuit whose end-cycle state is exactly the usual surface code state. The resulting diamond circuits extract both X- and Z-type stabilizers with each measurement qubit serving four stabilizers, at the cost of making detecting regions about four times longer in time. The proof of principle is the LUCI diagram and detector slices for distance 5, plus numerical logical-error-rate curves: the diamond circuit threshold is roughly three times lower than the standard surface code, but when line count rather than distance is the fixed resource, the reduced qubit and coupler counts allow a larger distance and, below the crossover error rate, a lower logical error rate per code block.

Load-bearing premise

The construction assumes the LUCI framework's guarantee: dropping exactly half the measure qubits and pairing the gauge operators as drawn still yields a valid distance-$d$ surface code with the claimed spacelike distance; this paper applies that framework rather than proving it from scratch.

Editorial extensions

If this is right

A distance-$d$ diamond circuit uses $1.5d^2$ qubits and $2d^2$ couplers, a more than 40% reduction in control lines compared with the standard circuit, so a line-limited machine can implement a larger code distance.
The timelike distance is degraded by roughly a factor of four, so diamond circuits need a decoder and error rates that tolerate longer detecting regions; the measured threshold is about three times lower.
If physical error rates lie below both thresholds and hardware resources are held fixed, the distance ratio asymptotes to $\sqrt{6/3.5} \approx 1.31$, which is enough for diamond circuits to win on logical error rate in the line-limited regime.
Because every data qubit has only two couplers, the architecture is expected to reduce crosstalk and frequency collisions in superconducting implementations, although the numerics in this paper deliberately do not assume that advantage.

Reading between the lines

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

If line count is the true scaling bottleneck, the right metric is not threshold but the crossover curve: diamond circuits only pay off below a physical error rate that shrinks as the system grows, so a hardware team should measure its per-qubit and per-coupler wiring cost before choosing.
The same LUCI 'dropout' recipe could be applied to other topological subsystem codes and to defect-tolerant surface codes, not just to the standard surface code; the author hints at this, but the generality is not demonstrated here.
A natural testable extension is a time-varying gauge pairing that swaps which gauges are paired each round; symmetry suggests this might halve the four-round timelike penalty without adding qubits, though the detecting-region overlap would need rechecking.

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.

Desk Editor's Note

read the letter

Quick take: this is a real new circuit family, not a trivial restatement. The diamond circuit on the Heavy-Square lattice drops half the measure qubits and still ends each round in a surface code state, which is clever and worth knowing. The resource counts are concrete: about 1.5d² qubits and 2d² couplers versus 2d² and 4d² for the standard surface code, so for line-limited hardware the trade-off is meaningful.

The paper does two things well. First, it spells out the LUCI construction for this extremal case with clear diagrams and detector slices; the figures are readable and the logic of pairing weight-3 gauges into weight-6 superstabilizers is transparent. Second, the simulations are honest: they compare d×d×d standard blocks to d×d×4d diamond blocks, which fairly accounts for the timelike penalty, and they use a standard noise model (SI1000) and decoder. The threshold being roughly 3× lower is consistent with the larger detecting regions.

The soft spots are real but not disqualifying. The main one is that the central distance-preservation claim is inherited from the LUCI framework, not demonstrated here. The paper asserts that spacelike distance is preserved, and the resource-count crossover in Fig. 4 depends on that label being genuine. There is no formal proof that every logical Pauli below weight d triggers a detector, and the worry about paired gauge components cancelling an error across rounds is worth checking. The simulations are evidence, but without raw data or error bars they can't fully verify the distance claim. A serious referee should ask for either a proof or a more direct verification, e.g., a distance-checking script against the circuit's detector error model.

A second, minor issue is that the paper leans heavily on the same group's earlier LUCI paper. That's fine if the framework is sound, but it would help to have an independent check or at least a self-contained statement of the relevant theorem.

Bottom line: this deserves a serious referee. It's an architecturally interesting result with a plausible performance trade-off, and the missing distance proof is fixable. I'd bring it to a reading group.

Referee Report

3 major / 5 minor

Summary. The manuscript introduces 'diamond circuits', a family of surface-code memory circuits implemented on the Lieb (Heavy-Square) lattice using the LUCI framework. The mid-cycle state is a subsystem surface code whose weight-3 gauge checks are paired into weight-6 superstabilizers; by dropping half of the measure qubits, the construction uses roughly 1.5d^2 qubits and 2d^2 couplers for a distance-d code, compared with roughly 2d^2 qubits and 4d^2 couplers for the standard surface code circuit. The paper claims that the spacelike distance is preserved at the cost of a timelike-distance penalty, benchmarks the circuits under SI1000 noise using stim and sparse blossom, reports a threshold roughly three times lower than the standard surface code, and argues that a line-count-limited architecture can nevertheless favor diamond circuits at sufficiently low physical error rates.

Significance. If the distance-preservation claim holds, this is a useful architecture result: it gives a concrete way to realize a surface-code memory on a lower-connectivity lattice and quantifies the trade-off between hardware components and logical performance. The paper is commendably explicit about the fairness of the comparison (d x d x d versus d x d x 4d), about the timelike penalty, and about the fact that the numerical results ignore any noise improvement from reduced connectivity. The LUCI diagrams and detector slices make the construction reproducible in principle. The central weakness is that the key distance property is inherited from Ref. [9] rather than demonstrated here, and the numerical evidence lacks error bars, raw data, and a quantified threshold estimate.

major comments (3)

[Abstract and §1, Fig. 2] The abstract and §1 state that diamond circuits 'preserve the spacelike distance of the code', and the resource counts (1.5d^2 qubits, 2d^2 couplers) and the crossover in Fig. 4 all depend on that distance being genuine. The only evidence offered is the LUCI framework of Ref. [9] and the detector slices of Fig. 2; no proof or independent check is supplied that every logical Pauli operator of weight < d triggers at least one detector. In particular, the paper does not rule out the possibility that an error on one weight-3 gauge component is later cancelled by an error on the paired component of the same weight-6 superstabilizer, producing an undetected sub-d logical operator. Please provide a proof or a computational certificate, for example an exhaustive circuit-level distance check using stim that enumerates logical operators of weight < d and verifies detector coverage, or a precise detector-graph argument showing that the paired gauge structure cannot cancel in that way.
[Section 'Logical error rate comparison', Fig. 3] The threshold comparison in Fig. 3 is central to the paper's numerical claims, but the plot has no error bars, no number of shots, no fitting procedure, and no raw data. Without these, the reader cannot assess whether the apparent crossing is statistically significant or whether the crossover curve in Fig. 4 is reliable. Please report shot counts, error bars (for example binomial or bootstrap confidence intervals), a quantified threshold estimate with confidence interval, and make the simulation data and code available.
[Section 'Logical error rate comparison'] The factor-of-four timelike penalty is asserted but not derived. The paper compares d x d x d for the standard surface code with d x d x 4d for diamond circuits, claiming that the detecting regions last four times as long. Because this volume comparison is part of the fairness of the benchmark and because the resource trade-off in Fig. 4 depends on the effective timelike distance, the paper should derive the factor of four from the detector graph or demonstrate it directly, for example by showing logical failure rates as a function of the number of rounds at fixed physical error rate, or by computing the shortest timelike logical operator in the circuit-level detector graph.

minor comments (5)

[Conclusions] The final paragraph contains an incomplete sentence: 'This result shows that the LUCI framework contains architecturally interesting beyond considering dropout.' Please rephrase to state the intended claim.
[Fig. 4 caption] The caption contains a duplicated word: 'The bottom plot plot shows ...' should be 'The bottom plot shows ...'.
[Main text, 'Crumble link'] The text mentions a 'Crumble link for a distance-5 circuit diamond circuit memory experiment' but no URL is given in the manuscript; please provide the link or remove the reference.
[§1 and Fig. 2] The exact compilation of the LUCI diagram shapes into gates is deferred to Ref. [9], Figs. 3 and 4; for self-containedness, at least one representative example showing the compilation of a single weight-3 gauge measurement would help the reader verify the construction without consulting a separate paper.
[Appendix C] Appendix C describes two even-distance variants but does not state explicitly which variant is used in the simulations of Fig. 3; please clarify that the left variant with weight-3 corner stabilizers is used throughout the numerical results.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central construction cites the prior LUCI framework, but the paper's resource counts and simulated logical-error-rate comparisons are independently derived.

full rationale

The paper's central claim is that a surface code can be implemented on a Lieb lattice using diamond circuits that preserve spacelike distance while requiring fewer qubits and couplers. The construction is described as an instance of the LUCI framework (Ref. [9]), a prior preprint by the same author group. This citation is load-bearing for the validity of the circuit and its claimed distance properties, but it is not circular: Ref. [9] is presented as a general framework for intentionally removing measure qubits, with its own stated assumptions, and the present paper does not define 'diamond circuit' in terms of the target claim, nor does it fit parameters to data it then re-presents as predictions. The resource counts (1.5d² qubits, 2d² couplers) are derived from the lattice geometry and circuit structure, not from the simulated error rates. The logical error rates and threshold are obtained from stim simulations under the SI1000 noise model, decoded with a matching decoder, and compared against the standard surface code; these are external numerical benchmarks rather than self-referential outputs. The paper does not supply raw data or a self-contained proof that every logical Pauli of weight below d triggers a detector, but that is a completeness or correctness concern, not a circularity concern. No equation or construction reduces to its own output by definition, and no fitted parameter is renamed as a prediction. Therefore no circular step is identified.

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

No new physical entities are introduced; 'diamond circuits' is a name for a circuit family, not a new postulated object. The central claims rest on the cited LUCI framework and on the chosen simulation and decoding methodology.

assumptions (3)

domain assumption The LUCI framework correctly describes valid surface-code circuits when measurement qubits are dropped, including the Lieb lattice case.
Used throughout to derive diamond circuits and their distances; supported only by citation to Ref. [9].
domain assumption The SI1000 noise model captures relevant superconducting circuit noise with a single parameter.
Adopted from Ref. [15]; the paper's quantitative claims about threshold depend on this noise model.
standard math The two-pass correlated sparse blossom decoder is a reliable proxy for maximum-likelihood decoding of surface-code detectors.
Used to obtain logical error rates; the paper cites Refs. [17,18] and does not validate decoder behavior on this specific graph beyond the resulting scaling.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Diamond Circuits for Surface Codes." pith.science (2026). https://pith.science/paper/3FSR4SDV

@misc{pith2026250210355,
  author       = {Pith},
  title        = {Pith review of: Diamond Circuits for Surface Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3FSR4SDV}},
  note         = {Machine review of arXiv:2502.10355}
}

read the original abstract

We present and benchmark an interesting circuit family which we call diamond circuits, that use a mid-cycle construction built around the subsystem surface code to implement a surface code on a Lieb or "Heavy-Square" lattice. This makes them more qubit- and measurement-efficient than previous constructions. These circuits are described via the LUCI framework, and are effectively circuits with half the measure qubits dropped out of the grid. These circuits preserve the spacelike distance of the code, but suffer a penalty in timelike distance, and could be useful in regimes where quantum computers are limited by frequency collisions or number of control lines.

Figures

Figures reproduced from arXiv: 2502.10355 by the authors.

**Figure 2.** FIG. 2: LUCI Diagram and detector slices for diamond circuits. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3: Logical error rate comparison [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5: Ratio of maximum distances vs. linecount. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: FIG. 6: Qubit grid and mid-cycle states for distance-6 diamond circuits. [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 10 canonical work pages

[9]

Luci in the surface code with dropouts,

D. M. Debroy, M. McEwen, C. Gidney, N. Shutty, and A. Zalcman, “Luci in the surface code with dropouts,” (2024), arXiv:2410.14891 [quant-ph]

arXiv 2024
[1]

Building logical qubits in a superconducting quantum computing system,

J. M. Gambetta, J. M. Chow, and M. Steffen, “Building logical qubits in a superconducting quantum computing system,” npj quantum information 3, 2 (2017)

work page 2017
[2]

Topological and subsystem codes on low-degree graphs with flag qubits,

C. Chamberland, G. Zhu, T. J. Yoder, J. B. Hertzberg, and A. W. Cross, “Topological and subsystem codes on low-degree graphs with flag qubits,” Phys. Rev. X 10, 011022 (2020)

2020
[3]

Comparative study of quantum error correction strate- gies for the heavy-hexagonal lattice,

C. Benito, E. L´ opez, B. Peropadre, and A. Bermudez, “Comparative study of quantum error correction strate- gies for the heavy-hexagonal lattice,” Quantum 9, 1623 (2025)

work page 2025
[4]

Quantum codes on a lattice with boundary,

S. B. Bravyi and A. Y. Kitaev, “Quantum codes on a lattice with boundary,” (1998), arXiv:quant-ph/9811052 [quant-ph]

arXiv 1998
[5]

Topological quantum memory,

E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,” Journal of Mathemati- cal Physics 43, 4452 (2002)

work page 2002
[6]

Surface codes: Towards practical large-scale quantum computation,

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012)

work page 2012
[7]

Subsystem surface codes with three-qubit check opera- tors,

S. Bravyi, G. Duclos-Cianci, D. Poulin, and M. Suchara, “Subsystem surface codes with three-qubit check opera- tors,” Quantum Info. Comput. 13, 963–985 (2013)

work page 2013

Show all 19 references

[8]

Quantum error correc- tion below the surface code threshold,

R. Acharya, L. Aghababaie-Beni, I. Aleiner, T. I. Ander- sen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, N. As- trakhantsev, J. Atalaya, et al., “Quantum error correc- tion below the surface code threshold,” arXiv preprint arXiv:2408.13687 (2024)

2024 arXiv
[10]

Relaxing hard- ware requirements for surface code circuits using time- dynamics,

M. McEwen, D. Bacon, and C. Gidney, “Relaxing hard- ware requirements for surface code circuits using time- dynamics,” Quantum 7, 1172 (2023)

2023
[11]

New circuits and an open source decoder for the color code,

C. Gidney and C. Jones, “New circuits and an open source decoder for the color code,” arXiv preprint arXiv:2312.08813 (2023)

2023 arXiv
[12]

Lowering connectivity requirements for bivariate bicycle codes using morphing circuits,

M. H. Shaw and B. M. Terhal, “Lowering connectivity requirements for bivariate bicycle codes using morphing circuits,” (2024), arXiv:2407.16336 [quant-ph]

2024 arXiv
[13]

Low-distance surface codes under realistic quantum noise,

Y. Tomita and K. M. Svore, “Low-distance surface codes under realistic quantum noise,” Phys. Rev. A 90, 062320 (2014)

2014
[14]

Snakes and lad- ders: Adapting the surface code to defects,

C. Leroux, S. F. Lin, P. Bienias, K. R. Sankar, A. Ben- hemou, A. Kubica, and J. K. Iverson, “Snakes and lad- ders: Adapting the surface code to defects,” (2024), arXiv:2412.11504 [quant-ph]

2024
[15]

Benchmark- ing the planar honeycomb code,

C. Gidney, M. Newman, and M. McEwen, “Benchmark- ing the planar honeycomb code,” Quantum6, 813 (2022)

2022
[16]

Stim: a fast stabilizer circuit simulator,

C. Gidney, “Stim: a fast stabilizer circuit simulator,” Quantum 5, 497 (2021)

2021
[17]

Optimal complexity correction of corre- lated errors in the surface code,

A. G. Fowler, “Optimal complexity correction of corre- lated errors in the surface code,” (2013), arXiv:1310.0863 [quant-ph]

2013 arXiv
[18]

Sparse blossom: correcting a million errors per core second with minimum-weight matching,

O. Higgott and C. Gidney, “Sparse blossom: correcting a million errors per core second with minimum-weight matching,” Quantum 9, 1600 (2025)

2025
[19]

Constructions and noise threshold of topological subsystem codes,

M. Suchara, S. Bravyi, and B. Terhal, “Constructions and noise threshold of topological subsystem codes,” Journal of Physics A: Mathematical and Theoretical 44, 155301 (2011). 5 Appendix A: SI1000 Noise Model The noise model used in this paper is SI1000, a superconducting insp...

2011

Pith tools

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

[1] [9]

Luci in the surface code with dropouts,

D. M. Debroy, M. McEwen, C. Gidney, N. Shutty, and A. Zalcman, “Luci in the surface code with dropouts,” (2024), arXiv:2410.14891 [quant-ph]

arXiv 2024

[2] [1]

Building logical qubits in a superconducting quantum computing system,

J. M. Gambetta, J. M. Chow, and M. Steffen, “Building logical qubits in a superconducting quantum computing system,” npj quantum information 3, 2 (2017)

work page 2017

[3] [2]

Topological and subsystem codes on low-degree graphs with flag qubits,

C. Chamberland, G. Zhu, T. J. Yoder, J. B. Hertzberg, and A. W. Cross, “Topological and subsystem codes on low-degree graphs with flag qubits,” Phys. Rev. X 10, 011022 (2020)

2020

[4] [3]

Comparative study of quantum error correction strate- gies for the heavy-hexagonal lattice,

C. Benito, E. L´ opez, B. Peropadre, and A. Bermudez, “Comparative study of quantum error correction strate- gies for the heavy-hexagonal lattice,” Quantum 9, 1623 (2025)

work page 2025

[5] [4]

Quantum codes on a lattice with boundary,

S. B. Bravyi and A. Y. Kitaev, “Quantum codes on a lattice with boundary,” (1998), arXiv:quant-ph/9811052 [quant-ph]

arXiv 1998

[6] [5]

Topological quantum memory,

E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,” Journal of Mathemati- cal Physics 43, 4452 (2002)

work page 2002

[7] [6]

Surface codes: Towards practical large-scale quantum computation,

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012)

work page 2012

[8] [7]

Subsystem surface codes with three-qubit check opera- tors,

S. Bravyi, G. Duclos-Cianci, D. Poulin, and M. Suchara, “Subsystem surface codes with three-qubit check opera- tors,” Quantum Info. Comput. 13, 963–985 (2013)

work page 2013