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REVIEW 2 major objections 4 minor 67 references

The paper claims that an adaptive simulated annealing search can find CSS and SWEL quantum error-correcting codes whose distances frequently meet or exceed the quantum Gilbert-Varshamov bound for block lengths up to 50 qubits.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · deepseek-v4-flash

2026-08-01 12:22 UTC pith:JAQMOJ6E

load-bearing objection Useful code catalog and a promising annealing energy function—but the SWEL-space completeness proof has a concrete bug in Appendix E that needs fixing or softening. the 2 major comments →

arxiv 2607.19574 v1 pith:JAQMOJ6E submitted 2026-07-21 quant-ph

Quantum codes from classical annealing

Michael A. Perlin , Matthew Steinberg , Ben Criger This is my paper
classification quant-ph
keywords quantum error correctionstabilizer codesCSS codesSWEL codessimulated annealingcode distanceGilbert-Varshamov boundtransversal gates
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proposes a method to search for moderately sized quantum error-correcting codes by adaptive simulated annealing, targeting CSS codes and a subclass called SWEL codes. It claims that for block lengths up to 50 qubits, the search discovers codes whose distances frequently meet or exceed the quantum Gilbert-Varshamov bound. These codes matter because they provide concrete, high-rate seed codes for concatenated fault-tolerant architectures and for near-term quantum hardware demonstrations. The search is backed by constructive theorems showing that its elementary moves connect every CSS or SWEL code, so the results are not just heuristic finds but represent a complete exploration of these code families.

Core claim

The central discovery is an annealing-based optimization that, despite the exponential size of code space, finds CSS and SWEL codes with distances meeting or exceeding the stabilizer or CSS Gilbert-Varshamov bounds for many n≤50, k≥4 parameter sets. The energy function combines distance with the count of minimum-weight logical operators, eliminating plateaus and steering the search toward codes with few low-weight errors. The discovered codes include several with only a handful of minimum-weight logical operators, making them natural candidates for further distance improvement, and some whose distances surpass previously indexed codes on public databases.

What carries the argument

The argument rests on three pillars. First, a parametrization of CSS codes by CSS-preserving Clifford gates (GL(n,F2)) and of SWEL codes by SWEL-preserving gates (O(n,F2)), with the search moves being CNOT gates and an elementary four-qubit SWEL-preserving gate; completeness theorems (Theorems 1,2,4,5) establish that these moves connect all codes in the respective classes. Second, an energy function E = log m/log(1/p) − d, where m is the number of minimum-weight nontrivial logical operators and p is a reference physical error rate, which acts as a surrogate logical error rate and resolves the distance plateaus that plague naïve distance maximization. Third, an adaptive annealing schedule tha

Load-bearing premise

The claim that every SWEL code can be reached by the elementary SWEL gates depends on a decomposition theorem whose proof, in one base case, leaves a 4x4 permutation matrix with V[3,3]=V[2,2]=1 unreduced; if that gap is not filled, the annealer may not explore the entire SWEL space.

What would settle it

Test the base case of Algorithm 3 for the 4x4 permutation matrix with V[3,3]=V[2,2]=1: if decompose_orthogonal_matrix returns a non-identity matrix or an infinite loop, Theorem 5's decomposition is incomplete, and the SWEL search may miss some codes.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The listed CSS and SWEL codes provide concrete benchmark parameters that can be used to evaluate or seed future error-correcting code constructions.
  • SWEL codes admit transversal logical Hadamard and phase gates, giving a path to fault-tolerant gate sets from numerically found codes.
  • The codes can be concatenated: a J38,12,6K top-level code with a J8,6,2K inner code yields parameters J304,72,12K, bringing hundreds of logical qubits within reach of thousands of physical qubits.
  • The annealing energy function predicts relative logical error rates at d>4, supporting its use as a design and selection criterion for code search.

Where Pith is reading between the lines

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

  • The same annealing machinery could be extended to search for codes with imposed symmetries (e.g., cyclic or Reed-Muller families), potentially discovering larger codes where distance evaluation can be sped up by the symmetry group.
  • Because the energy function counts minimum-weight logicals, codes that come close to a higher distance may be systematically nudged by deleting or modifying a few logical operators, suggesting a local refine-and-improve step beyond the paper.
  • The greedy stabilizer weight reduction post-processing implies a two-phase strategy for hardware-oriented code design: first optimize distance and logical count, then compress stabilizer weights, a trade-off the paper quantifies only empirically.
  • If the completeness proof for SWEL is repaired, the algorithm could be used as a constructive existence proof for SWEL codes with given parameters, not just a heuristic.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper introduces an adaptive simulated-annealing search for moderate-sized CSS and SWEL quantum error-correcting codes. For each fixed (n,k), the search uses random CSS/SWEL-preserving Clifford gates to sample initial codes, applies CNOT or elementary four-qubit SWEL-preserving gates as local moves, and minimizes an energy function combining code distance with the number of minimum-weight logical operators. The authors report codes for n≤50 whose distances frequently meet or exceed the quantum Gilbert–Varshamov or CSS Gilbert–Varshamov bounds, and validate the energy surrogate against independent Monte Carlo simulations. The paper also provides a full listing of the found codes, a greedy stabilizer-weight-reduction post-processing step, and a discussion of fault-tolerant gate sets enabled by SWEL structure.

Significance. If the numerical results are taken at face value, the paper makes a useful contribution: a catalog of concrete high-rate CSS and SWEL codes with distances that are competitive with (and sometimes better than) previously indexed codes, together with an explicit construction of SWEL codes that admit transversal H and S gates. The manuscript is transparent in listing all stabilizer generators, and the energy surrogate is validated externally via Monte Carlo rather than fitted to the reported codes. The SWEL-preserving-gate framework (Theorem 3 and Theorem 4) is a good conceptual contribution, and the identification of a 2304-element automorphism group for one code is a nice bonus. The main weakness is the proof of the search-completeness claim for SWEL codes, which is not correct as written, and the lack of detail about the actual distance-finding subroutine used for n up to 50.

major comments (2)
  1. [Appendix E, Theorem 5, Algorithms 2 and 3] The proof of Lemma 5 is invalid as written. In Algorithm 3's base case, a 4×4 orthogonal matrix V that permutes qubits 1 and 4 has V[3,3]=V[2,2]=1, so the algorithm returns an empty sequence even though V is not the identity (V=t({1,4})). Algorithm 2 also appends the invalid set {n,n} when the last column is already e_n, producing a weight-1 "transvection" that is not in O(n,F2). Moreover, the claim that all elements of O(4,F2) are t(Z4) or the identity modulo permutations is false: a 3-cycle permutation is a counterexample. Consequently, Theorem 5's assertion that elementary SWEL-preserving gates connect all Jn,kK SWEL codes is unproven. The listed codes are unaffected, but the completeness guarantee for the SWEL annealer must be fixed or softened.
  2. [§3.3, §5] The energy function is said to require brute-force enumeration of all nontrivial logical operators, which scales as 2^{O(n)}. For n=50 (Table 1), naive enumeration would be computationally prohibitive given the reported runtimes of "minutes to a couple of days". The manuscript does not describe the distance-finding or minimum-weight-logical enumeration algorithm actually used. Please provide the algorithm and its complexity; otherwise the numerical search is not reproducible. (The listed codes themselves can still be independently checked.)
minor comments (4)
  1. [Abstract] Typo: "the latter of which which admit" — duplicate "which".
  2. [§2.7.2] Typo: "guarantees the existence of an stabilizer code" should be "a stabilizer code".
  3. [Eq. (23)] The right-hand side is typographically ambiguous; please parenthesize the denominator clearly.
  4. [§G] The Python code example should pin versions of stim and sympy, not only qldpc, for full reproducibility.

Circularity Check

0 steps flagged

No circular derivation; energy heuristic is validated against independent simulation and benchmarks are external, with only a minor non-load-bearing self-citation.

full rationale

The paper's central claim—that annealing finds CSS/SWEL codes with distances at or above QGV bounds—rests on an energy function E(C)=log m/log(1/p)-d (Eq. 36) and move sets whose completeness is argued via Theorems 1-5. The energy function is derived as a heuristic surrogate from Eq. (34) and is checked against Monte Carlo logical-error-rate simulations in Fig. 2; it is not fitted to the codes in Table 1, so this is a consistency check rather than a fitted-parameter prediction. The QGV/CSS bounds and qecdb comparisons are external benchmarks, not outputs of the search. Theorems 1-5 are proved in Appendices A-E from elementary linear algebra; the relevant cited results are non-self (e.g., Ref. [42] for the CSS GV bound, Ref. [35] for prior SWEL properties). There is a minor self-citation (Ref. [36], 'In preparation', by co-author Perlin) supporting the utility of SWEL gate sets, but it is not load-bearing since Ref. [35] independently establishes the transversal-gate properties. A genuine caveat is that Appendix E's Algorithm 3 appears not to reduce some O(n,F2) inputs fully (e.g., a 4x4 swap can be returned with V[3,3]=V[2,2]=1), so the SWEL-search completeness guarantee is not established by the written proof; this is a correctness/completeness gap, not a circular reduction of the predicted codes to the search inputs.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

No new physical entities are introduced; SWEL is a subclass of CSS codes defined by properties, and the elementary gates are Clifford operations, not new particles or forces. The ledger lists the hand-set search parameters and the key existence/decomposition assumptions the completeness claims rely on.

free parameters (3)
  • p in energy function E(C)=log m/log(1/p)-d = 10^-3
    Hand-chosen to reflect realistic physical error rates; sets the relative weight of log m vs d, influencing which codes the annealer favors.
  • Annealing hyperparameters (N=10^4, μ=0.5, log10 r = linspace(-2,-4,K), K∈{10^3,10^4})
    Hand-set schedule parameters; they affect search quality but not the reported code parameters.
  • Equal X/Z stabilizer counts (s_X=s_Z=s)
    Search restricted to CSS codes with equal numbers of X- and Z-type stabilizers; a design choice that excludes unbalanced CSS codes.
axioms (5)
  • standard math Stabilizer code formalism and symplectic binary linear algebra over F2
    Used throughout Sections 2-3 and appendices; standard background.
  • domain assumption Quantum Gilbert-Varshamov bounds (Eq. 22, 23) from Refs [37,41,42] correctly lower-bound achievable distances
    The QGV and CSS-GV bounds are external existence guarantees used as benchmarks in Table 1.
  • domain assumption Code-capacity depolarizing noise with IID Pauli errors and lookup-table decoder is the relevant performance model
    Used for the energy surrogate motivation and the Monte Carlo validation (Eq. 34, Fig. 1).
  • domain assumption The energy surrogate E(C)=log m/log(1/p)-d is a faithful proxy for logical error rate in the explored regime
    The search optimizes this surrogate; the paper validates it on found codes, but it is not proven to guide toward globally optimal codes.
  • ad hoc to paper Lemma 5: every U∈O(n,F2) decomposes into transvections t(Q) with |Q|∈{2,4}
    This is the core of the SWEL completeness proof (Theorem 5); the proof via Algorithm 3 has an incomplete base case, so the decomposition is effectively assumed for the completeness claim.

pith-pipeline@v1.3.0-alltime-deepseek · 51348 in / 41971 out tokens · 321093 ms · 2026-08-01T12:22:38.379663+00:00 · methodology

0 comments
read the original abstract

We introduce an adaptive simulated annealing algorithm to search for moderately-sized quantum error-correcting codes with high encoding rates and large distances. Our search targets two classes of stabilizer codes: (1) CSS codes, and (2) a subclass of CSS codes that we call ``self-dual with equivalent logicals'' (SWEL) codes, the latter of which which admit transversal implementations of logical Hadamard and phase gates that can be leveraged to construct fault-tolerant gate sets. The search is guided by an energy function that acts as a surrogate for the logical error rate in a code-capacity noise model, combining code distance with a count of minimum-weight logical operators to resolve the discrete plateaux that impede na\"ive distance optimization. For block lengths of up to $50$ physical qubits, our search finds state-of-the-art CSS and SWEL codes whose distances frequently meet or exceed the variants of the quantum Gilbert-Varshamov bound. In addition to providing favorable seed codes for fault-tolerant architectures based on code concatenation, the codes found in this work are promising candidates for high-rate code demonstrations on near-term quantum computing hardware.

Figures

Figures reproduced from arXiv: 2607.19574 by Ben Criger, Matthew Steinberg, Michael A. Perlin.

Figure 1
Figure 1. Figure 1: Logical error rates (LER) and discard rates (DR) per logical qubit (LQ) for the codes in Table [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Logical error rates from Fig [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

discussion (0)

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    We will use the following fact about the columns ofU: Lemma 6.Ifvis a column of someU∈ O(n,F 2)withn >1, then∥v∥is odd and v̸=1 n. Here1 n ∈F n 2 is the vector of all ones. In other words,vhas an odd number of ones and at least one zero. Proof.Letu 1 andu 2 be arbitrary but different columns of an arbitraryU∈O(n,F 2). 1.U ⊺U=1 n implies thatu ⊺ 1u1 = 1, s...

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    Ifn >4, we can recursively consider the sub- matrix ˜U=   u1,1 u1,2 ···u 1,n−1 u2,1 u2,2 ···u 2,n−1 ... ... ... ... un−1,1 un−1,2 ···u n−1,n−1  , (119) and again reduce the last column of ˜U. Note that the transvections used to reduce the last col- umn of ˜Udo not, if extended to act onFn×n 2 , af- fect the last (already reduced) column ofU, sin...

  67. [67]

    extendAbyB

    Ifn= 4, then all elements ofO(n,F 2)are equal to eithert(Z 4)or1 4 modulo row and column permutations, so any reduced matrixUfor which un =e n must be equal to the identity matrix modulo permutations. Altogether, we can use Algorithm 2 to recursively re- duce any matrixU∈O(n,F 2)⊂F n×n 2 to a permuta- tion matrix using only transvectionst(Q)with weight |Q...