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REVIEW 3 major objections 4 minor 47 references

A new family of rate-1/5 quantum LDPC codes built from a non-abelian group symmetry approaches the teraquop memory regime with fewer than 1000 physical qubits.

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2026-08-01 04:11 UTC pith:2YHS75W3

load-bearing objection A solid construction with honest benchmarks; the teraquop claim is a conditional extrapolation that deserves referee scrutiny, not dismissal. the 3 major comments →

arxiv 2607.27644 v1 pith:2YHS75W3 submitted 2026-07-30 quant-ph

Quantum LDPC codes with design rate 1/5 and good performance below 1000 physical qubits

Yifan Hong This is my paper
classification quant-ph MSC 81P7094B0594B35 PACS 03.67.Pp
keywords quantum LDPC codesbalanced productnon-abelian group codesmetacyclic groupconstant-rate quantum error correctioncircuit-level noisebelief-propagation decodingreconfigurable atom arrays
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.

This paper introduces a family of quantum LDPC codes with design rate 1/5 and check weight 9, built from the balanced product of classical rate-1/2 LDPC codes that share a non-abelian ZSZ group symmetry. With under 1000 physical qubits per block, the best instances are reported to approach the gigaquop and teraquop memory regimes under idling-free circuit-level noise at error strength 0.1%, with GPU-accelerated belief-propagation decoding in about 1-2 ms. If correct, this would mean practical, high-rate quantum memories do not require thousands of physical qubits or a 2D layout. The paper also provides equivariant logical Pauli bases, fold-transversal Clifford gates, and syndrome-extraction schedules tailored to reconfigurable atom arrays. The central design principle is that the non-abelian twist avoids an abelian low-distance obstruction that would otherwise cap the code's minimum distance.

Core claim

The central claim is that taking the balanced product of two classical (3,6)-regular LDPC codes over a non-abelian metacyclic group Z_{l1} rt Z_{l2} yields five-block CSS quantum LDPC codes whose finite-length instances - such as [[160,32,10]], [[320,64,14]], and [[550,110,<=18]] - achieve the highest code-capacity memory efficiency kd^2/n among codes below 1000 physical qubits. Under idling-free circuit-level depolarizing noise with two-qubit gate and SPAM errors, the simulated memory performance shows a pseudothreshold around 0.5%, and extrapolation places the per-logical-qubit per-round error rate at roughly 2x10^-11, 2x10^-12, and 7x10^-14 at p=0.1% for the 320-, 390-, and 550-qubit code

What carries the argument

The central object is the ZSZ group Z_{l1} rt_q Z_{l2}, a semidirect product of cyclic groups with a non-abelian twist q, together with its left- and right-regular representations. Two classical two-block group-algebra codes with trinomial parity checks are built over this group, and their balanced product - quotienting out a diagonal subgroup action - produces the 5-block quantum CSS code. The non-abelian twist is what prevents the low-weight syzygies that force constant distance in abelian constructions, and the same group algebra supplies the equivariant logical Pauli bases (one-generator minimum-weight and one-generator symplectic) and the automorphism-fold structure used for Hadamard- a

Load-bearing premise

The teraquop claim rests on extrapolated logical error rates that assume the distance upper bounds for the two largest codes are exact and that the decoder's subthreshold slope continues down to 10^-13 without an error floor.

What would settle it

Compute the true minimum distance of ZSZ-LP-550 (and ZSZ-LP-390) with an exact distance-finding method rather than a heuristic estimator; if the distance is smaller than 18 (or 16), or if a circuit-level simulation at p about 5e-5 shows the logical error rate flattening above 10^-13, the extrapolated teraquop regime is not reached.

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

If this is right

  • If the teraquop extrapolation holds, a rate-1/5 quantum memory below 1000 physical qubits can reach per-logical-qubit per-round error rates near 10^-13 at 0.1% noise, a regime previously associated with codes thousands of qubits larger.
  • The construction gives a concrete template for constant-rate qLDPC codes with small check weight (9) and modest block length, relevant for trapped-ion and neutral-atom processors.
  • The classical ZSZ-2BGA seed codes are competitive (3,6)-regular LDPC codes at finite lengths, with higher estimated distance than standard random and progressive-edge-growth ensembles and low error floors, so they may be independently useful in classical error correction.
  • Equivariant logical Pauli bases and fold-transversal gates offer low-overhead paths to logical readout, surgery, and Clifford operations without requiring the full quantum code to retain the group automorphism.
  • Measured decoding latencies of 1-2 ms (with p99 around 2-5 ms) suggest real-time decoding is feasible for architectures with syndrome-extraction rounds in the tens of milliseconds.

Where Pith is reading between the lines

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

  • One consequence the author leaves implicit is that the same filtered-random-search pipeline could be applied to other non-abelian metacyclic groups or to co-designed trinomials with lower rearrangement costs, likely yielding even better code instances than the specific candidates reported.
  • The distance preservation between classical seeds and quantum product is described as special to the selected codes; a natural extension is to characterize when the balanced product inherits the classical minimum distance, which would turn the 'surprising' observation into a design rule.
  • Because the noise model omits idling and single-qubit errors, a testable extension is to re-benchmark these codes with realistic idling and measurement models; the gap between the reported optimistic curves and full hardware noise will determine how much of the teraquop margin survives.
  • The fold-transversal gate construction rests on a nontrivial automorphism of the ZSZ group; searching for codes that simultaneously admit good distance, low short-cycle density, and a useful fold could be a productive direction for practical constant-rate computation.

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

3 major / 4 minor

Summary. The manuscript introduces a new family of constant-rate quantum LDPC codes, ZSZ-LP, with design rate 1/5 and check weight 9, obtained as balanced products of classical two-block ZSZ group-algebra LDPC codes. It reports several concrete instances below 1000 physical (data + ancilla) qubits, and benchmarks their idling-free circuit-level memory performance under a depolarizing noise model using a GPU-accelerated Relay-BP decoder. The central numerical claim is that ZSZ-LP-550 may reach a per-logical-qubit per-round LER of roughly 7e-14 at p=0.1%, thereby approaching the teraquop memory regime; ZSZ-LP-320 and -390 are claimed to approach the gigaquop regime. The paper additionally provides syndrome-extraction schedules for reconfigurable atom arrays, equivariant logical Pauli bases, surgery gadgets, and fold-transversal ZX-duality structures, together with classical ZSZ-2BGA codes benchmarked on the AWGN channel.

Significance. If the central performance claim holds, this is a genuinely useful step for practical constant-rate qLDPC memories: it shows that a non-abelian group-algebra construction can bypass the abelian low-distance obstruction while retaining enough algebraic structure for logical-operator design, scheduling, and surgery. The manuscript is unusually transparent: it ships explicit trinomials and group parameters for every code, reports error bars (Clopper-Pearson), gives full decoder settings, and includes an honest list of caveats. The classical ZSZ-2BGA codes and the surgery-gadget analyses are also potentially valuable independently. However, the headline gigaquop/teraquop claims are not directly measured; they rest on extrapolations whose validity depends on distance tightness and decoder floor assumptions that are not yet demonstrated.

major comments (3)
  1. [§6.1, Table 1] The extrapolated LERs for ZSZ-LP-390 and ZSZ-LP-550 use distance values that are explicitly reported only as upper bounds (d≤16 and d≤18), and Section 6.1 sets d_circ = d−2 based on circuit-distance estimates for the small codes. The teraquop claim for ZSZ-LP-550 depends on the leading subthreshold slope p^{d_circ/2}=p^8. If the true distance is 16 rather than 18, the slope becomes p^7 and the extrapolated LER at p=0.1% shifts by roughly an order of magnitude, which can push the value above the 10^-13 teraquop threshold. Please certify the distance of ZSZ-LP-550 exactly (e.g., with pySATDist), or report the extrapolation under both d=16 and d=18 and state explicitly which claim survives.
  2. [§6.1, §1.1.1 caveat (i)] The extrapolation assumes that Relay-BP's subthreshold behavior is controlled by the code distance down to LER ~10^-13. The manuscript's own caveat (i) states that 'the Relay-BP decoder can behave suboptimally at low noise due to trapping sets and may not be able to correct all errors of weight below half the circuit distance.' No data are provided below p≈2e-4, and no analysis of decoder nonconvergence or trapping-set floors is given. This is a load-bearing assumption for the teraquop claim. Please add low-p nonconvergence statistics, importance-sampling checks, or a comparison with a stronger decoder (e.g., OSD post-processing) at selected points, or weaken the claim to explicitly depend on the absence of a decoder floor.
  3. [§1.1.1 caveat (ii)] The reported per-logical-qubit LER is computed as the block LER divided by k. If logical failures are strongly correlated across the block, this quantity can underestimate the failure probability of a randomly selected logical operation, which is what the gigaquop/teraquop definitions require. The paper acknowledges this in caveat (ii), but the central claim is still stated in these units. Please quantify the possible correlation effect (e.g., by reporting logical failure statistics for individual logical qubits) or explicitly limit the claim to 'average per-logical-qubit' LER as a figure of merit rather than a guaranteed per-operation error rate.
minor comments (4)
  1. [Section 7] The text contains an unresolved citation placeholder: 'the GB codes in??' in the Outlook paragraph on OGM+OGS bases. Please fill in the reference.
  2. [Section 5, footnote 5] Footnote 5 ends with 'minimal circuit depth 12.5' which appears to be an incomplete sentence or a missing bound. Please clarify.
  3. [Table 1] The final column caption says 'actual or extrapolated LER'; it would be clearer to mark each entry explicitly as measured or extrapolated, since the teraquop discussion depends on which entries are extrapolated.
  4. [Section 6.1] The fit parameters c0, c1, c2 and the chosen d_circ values for each code are not tabulated. Since the extrapolation is central to the main claim, please include the fit values and the p-range used for each fit.

Circularity Check

0 steps flagged

No significant circularity; LER data are genuine benchmarks and extrapolation is explicitly conditional.

full rationale

The central quantitative claims rest on numerical simulations of the candidate codes under circuit-level noise and on a phenomenological extrapolation of those measured LER curves. The extrapolation uses the code distance d as an exponent (d_circ=d-2), but d is computed independently of the LER data via pySATDist/QDistEvol, so the fit does not reuse the LER data as input; only the three parameters c0,c1,c2 are fit to the measured curves. The paper explicitly marks the distances of ZSZ-LP-390 and ZSZ-LP-550 as upper bounds and lists caveats (Relay-BP trapping sets, correlated logical failures, memory-only benchmarking) that make the teraquop claim conditional rather than definitionally forced. The ZSZ group construction is inherited from prior work [GHKL26], including a co-author, but the paper supplies its own bounds (Appendices C, D), code instances, and benchmarks; the citation is motivational and algorithmic, not a load-bearing proof of the claimed performance. The code search optimizes for distance/girth, and the reported kd^2/n metric reuses that distance, but this is a selection-effect feature of any code search, not a fitted input masquerading as a prediction. No equation reduces to its own input, and no self-citation chain is used to forbid alternatives. The measured LER curves against surface and BB codes are independent benchmarks, so the honest finding is no significant circularity.

Axiom & Free-Parameter Ledger

5 free parameters · 7 axioms · 0 invented entities

The central claims rest on search-fitted construction parameters (trinomials, group parameters), external decoder and distance-estimation tools, and an explicitly optimistic noise model; the extrapolation coefficients are the most direct fitted inputs to the headline performance numbers. No new physical entities are postulated: ZSZ-LP codes, OGM/OGS bases, and surgery ancillas are explicit constructions with specified stabilizer checks, not entities with independent falsifiable handles.

free parameters (5)
  • Trinomials a,b,c,d per candidate code = Table 3, e.g., ZSZ-LP-390: a=1+x^17 y+x^14 y^2, b=1+x^16+x^3 y, c=1+x^24+x^8 y, d=1+x^21+y
    Construction inputs selected by filtered random search over girth and distance (Section 2.5); all reported instances and their performance are conditioned on these specific choices, so they are search-fitted parameters rather than universal design rules.
  • ZSZ group parameters (ℓ1, ℓ2, q) = (16,2,9), (12,4,7), (16,4,3), (26,3,3), (22,5,3), (31,5,2)
    Chosen by hand per the Section 2.5 heuristics (ℓ1≫ℓ2, avoid symmetric a=c,b=d, low-order element scarcity); these choices set the distance bounds and girths in Appendices C-D.
  • Extrapolation coefficients c0, c1, c2 per LER curve = Fitted to subthreshold data at p ≤ 0.4% (not fully tabulated in the main text)
    The gigaquop/teraquop numbers in Table 1 and Figure 1 are evaluations of these fits at p=0.1%, several to many decades below the measured data; they are genuine fit parameters.
  • Decoder gamma distribution (gamma_dist) = Individually tuned per code via grid search (Table 15)
    Section 6.1: 'The gamma distribution ... is individually tuned for each code's memory circuit through a grid search.' The measured LERs depend on this tuning.
  • Circuit-distance estimate d_circ = d − 2 = d−2 per code, e.g., 8 for ZSZ-LP-160
    Section 6.1 heuristic relating code distance to circuit distance, used in the fit exponent p^{(d−2)/2}; a different mapping would change every extrapolated LER.
axioms (7)
  • standard math Left- and right-regular representations of the group algebra F2[ZSZ] commute (L[g]R[h]=R[h]L[g]), giving the CSS orthogonality condition HX HZ^T = 0.
    Invoked in Section 2.4, Eqs. (12)-(13); follows from associativity of group multiplication. Verified by inspection of the block structure.
  • standard math When the classical seeds H_left, H_right have full rank, the balanced product gives a CSS code with k=ℓ logical qubits and rate exactly 1/5.
    Section 2.4, Eq. (14); the reported k/n=1/5 depends on this rank argument, and all listed candidates are asserted to have full-rank seeds.
  • domain assumption The circuit-level noise model is two-qubit depolarizing with probability p, SPAM errors with probability p, and NO single-qubit or idling noise.
    Section 6.1; the paper itself calls this 'an optimistic best-case scenario.' The teraquop claim is conditioned on this model, which omits the idling errors that 30-60 ms atomic rearrangements would plausibly induce.
  • domain assumption The GPU Relay-BP decoder [NVI26] with the stated hyperparameters (gamma0=0.125, pre_iter=80, num_sets=1000, max_iterations=60, stop_nconv=5) and code-specific gamma_dist yields the reported LER and latency.
    Section 6.1 and Table 7; the central numerical results depend on this external decoder implementation, which is not shipped with the paper.
  • domain assumption QDistEvol (10^6 iterations) and pySATDist estimates give the true minimum distances; the upper bounds d≤16 and d≤18 for ZSZ-LP-390/550 are treated as exact in the LER extrapolations.
    Table 1, Table 2, and footnote 3 in Section 3.1 ('we are nonetheless quite confident of its tightness'); the fit exponent p^{(d−2)/2} inherits this assumption, which is load-bearing for the teraquop claim.
  • domain assumption AOD movement cost model: 5 μm storage-zone spacing, 5500 m/s² acceleration, with a uniform cost rule counting cyclic shifts and one-step riffle shuffles as comparable primitives.
    Section 5 and Table 6; the 30-60 ms rearrangement times and the conclusion that 1-2 ms decoding suffices for real-time decoding rest on this hardware model from [ZDZ+25].
  • ad hoc to paper Code-selection filters (girth thresholds 6/8, distance thresholds, fallback rules) are applied and the reported instances are the survivors; no complete theory predicts the observed performance.
    Section 2.5 and Figure 4; the reported codes are the output of a heuristic search pipeline, so claims about achievable performance are conditioned on the specific filter choices.

pith-pipeline@v1.3.0-daily-deepseek · 58452 in / 28357 out tokens · 282308 ms · 2026-08-01T04:11:43.330140+00:00 · methodology

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read the original abstract

Constant-rate quantum low-density parity-check (LDPC) codes promise fault-tolerant quantum computation with constant spatial overhead in the asymptotic limit. Nonetheless, discovering finite-length code instances with good practical performance remains challenging. We introduce a new family of quantum LDPC codes with design rate $1/5$ and check weight $9$ that approaches the teraquop memory regime per qubit-round with several hundred physical qubits, under idling-free circuit-level noise of strength $0.1\%$ and GPU-accelerated Relay-belief-propagation (Relay-BP) decoding with average latencies around 1-2 ms, a regime relevant to trapped-ion and neutral-atom processors. The construction involves the balanced product of classical LDPC codes with design rate $1/2$ that share non-abelian $\mathbb{Z}_\ell \rtimes \mathbb{Z}_m$ group symmetries, which may be of independent interest for classical error correction. We build syndrome extraction circuits tailored to reconfigurable atom arrays using a simple greedy scheduler, with single-round rearrangement times around 30-60 ms using present hardware specifications, and substantial room for future improvements. We also construct logical Pauli bases that are equivariant with respect to the group symmetry, which can significantly compress the design space for code surgery. Together, these results further advance the practicality of constant-rate quantum LDPC codes for near-term, fault-tolerant quantum computers.

Figures

Figures reproduced from arXiv: 2607.27644 by Yifan Hong.

Figure 1
Figure 1. Figure 1: Left: The memory efficiency improvement kd2/n over the surface code is plotted as a function of the code length n for several high-rate qLDPC codes proposed for near-term implementation, including bivariate bicycle (BB) [BCG+24], generalized bicycle (GB) [WBC+26], quasi-cyclic lifted product (QC-LP) [XAP+24, CXK+26] and Kasai’s construction [Kas26] with affine permutation matrices (APM) [ZDG+26]. Note that… view at source ↗
Figure 2
Figure 2. Figure 2: Left: The 95th percentile distance as a function of the code length n is plotted for three random (3, 6)-regular LDPC ensembles: classical two-block ZSZ (ZSZ-2BGA) codes, the bipartite configuration model [RU08], and the progressive-edge-growth (PEG) algorithm [HEA01]. 1000 codes are sampled from each ensemble, and minimum distances are estimated with 10,000 iterations of QDistEvol. A linear fit is applied… view at source ↗
Figure 3
Figure 3. Figure 3: The left and right Cayley graphs of the ZSZ group [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Filter pipeline for candidate ZSZ-LP code discovery. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Pareto frontiers are plotted for two ZSZ-LP code ensembles, sampled using parts of the code [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Schematic of the two X-surgery gadgets constructed for the ZSZ-LP-390 data code Q (centre), drawn carrying three copies X¯ 1, X¯ 2, X¯ 3 of the logical X¯ operator. For a graph ancilla (left), the vertices, edges, and faces (cycles) are respectively the ancilla X-checks, qubits, and Z-checks; here Q′ graph measures a single logical (attached to X¯ 2). For a hypergraph-product ancilla (right), the lattice o… view at source ↗
Figure 7
Figure 7. Figure 7: Schematic of the automorphism-fold (green) on the data and check subblocks. The data subblocks [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Physical layout used for syndrome extraction on a reconfigurable atom array. The five data-qubit [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Left: The block logical error rate (LER) per syndrome-extraction round, averaged over X-memory and Z-memory simulations, for the candidate ZSZ-LP codes in [PITH_FULL_IMAGE:figures/full_fig_p033_9.png] view at source ↗

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