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Random quantum codes saturate the hashing bound

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2026-08-04 10:56 UTC pith:YZQEPQZR

load-bearing objection Main threshold result is solid and independently derived; the postselected-recovery claim is the genuinely weak spot and should be either proven or labeled as a conjecture. the 2 major comments →

arxiv 2510.07396 v3 pith:YZQEPQZR submitted 2025-10-08 quant-ph cond-mat.stat-mech

Spectral properties and coding transitions of Haar-random quantum codes

Grace M. Sommers , J. Alexander Jacoby , Zack Weinstein , David A. Huse , Sarang Gopalakrishnan This is my paper
classification quant-ph cond-mat.stat-mech MSC 81P7381P45 PACS 03.67.Pp03.67.-a
keywords Haar-random quantum codeshashing bounderror-correction thresholdmixed-state phase transitionspectral bandspostselected error correctionRényi entropyquantum MacWilliams identity
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.

Random quantum codes—codes that put logical information in a random subspace of many noisy qubits—lose the ability to correct errors at exactly the hashing bound, the same threshold as random stabilizer codes. For a single qubit encoded in many qubits, that threshold is p_c≈0.189. The paper establishes this by tracking the eigenvalue spectrum of the corrupted density matrix, which splits into bands labeled by how many qubits an error touched. Below threshold these bands stay separated; at threshold the typical error band fills all available space. Above threshold, typical errors are uncorrectable, but if one postselects on low-weight error bands, information remains safe until a much higher detection threshold p_d=1/2.

Core claim

The paper's central claim is that a Haar-random encoding—a code whose logical space is a random subspace of the physical Hilbert space—has an error-correction transition at the hashing bound H(p_c)=1, matching random stabilizer codes. The argument runs through the spectrum of the decohered density matrix: each Pauli error of weight w creates an approximately orthogonal copy of the logical space, so the spectrum is a sum of bands of eigenvalues at scale p^w; the threshold is where the typical band's dimension exhausts the Hilbert space. The claim is backed two independent ways: a perturbative 'mean-shift' ansatz for the band positions, validated numerically for up to N=13 qubits, and an expli

What carries the argument

The central object is the band spectrum of the decohered density matrix. The depolarizing channel is decomposed as a convex sum of fixed-weight error channels; each weight-w channel maps the logical subspace to a nearly orthogonal subspace, producing a 'band' of eigenvalues of size ~p^w with multiplicity (q^2−1)^w C(N,w). Each band's internal level density is taken to be Marchenko-Pastur (random-matrix), and inter-band overlaps are treated as small perturbations. The threshold calculation is independently confirmed by Weingarten calculus, which evaluates Haar-averaged purities; the detection and Rényi-2 thresholds are fixed by a quantum MacWilliams identity, a duality between the error pictu

Load-bearing premise

The load-bearing premise is that images of the logical subspace under distinct Pauli errors stay nearly orthogonal, and inter-band mixing stays negligible, for all weights below the hashing weight; the paper supports this by perturbation theory and numerics for up to 13 qubits, but does not prove it rigorously.

What would settle it

Evaluate the exact or well-converged coherent information of a Haar-random code at N=20–30 under depolarizing noise for p between 0.19 and 0.5. If the low-weight bands fail to stay orthogonal—say the projected density matrix Π_w ρ Π_w loses logical information below p=1/2—the postselected-recovery claim is false; if the threshold drifts away from p_c≈0.189 as N grows, the hashing-bound claim for Haar-random codes is false.

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

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If this is right

  • For zero-rate qubit codes, the error threshold is p_c≈0.189, and the transition broadens as 1/√N at finite size.
  • For error rates between p_c and p_d=1/2, logical information is unrecoverable without postselection but is protected if one projects onto low-weight error bands; for finite-rate codes the detection threshold becomes p_d=1−q^{r−1}.
  • Rényi-α entropies of the decohered state are nonanalytic at H_α(p)=1, and these singularities are exactly the success/failure transition of a 'soft' postselection POVM.
  • The result generalizes to qudits: correction threshold at H(p)=1, detection at p=1−1/q.
  • The two postselection protocols—weight-projecting and Rényi-reweighting—are inequivalent, mirroring ensemble inequivalence in long-range statistical mechanics.

Where Pith is reading between the lines

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

  • A concrete test beyond the paper: at N≈20–30, simulate Haar-random codes and check whether the postselected coherent information on the w=0 and w=1 bands stays maximal up to p=1/2; no exact result at that size is currently at hand.
  • The derivation via weight enumerators suggests that the hashing-bound saturation is generic for nondegenerate subspace codes, not a special property of stabilizer structure; if true, other non-stabilizer code families should show the same threshold.
  • The paper's band picture indicates that the low-weight bands in local codes like the surface code will be modified by error degeneracy; whether a similar postselection window survives there is testable with existing statistical-mechanics mappings.
  • The finite-rate detection formula p_d=1−q^{r−1} implies a shrinking postselection window as the code rate grows, which could inform practical rate-versus-overhead tradeoffs in fault-tolerant designs.

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 / 3 minor

Summary. The paper studies the spectral and coding properties of Haar-random quantum codes subjected to independent per-qudit depolarizing noise of strength p. It claims that the spectrum of the decohered physical density matrix decomposes into Pauli-weight bands, that the coding threshold is the hashing bound H(p)=1-k/N (p_c≈0.189 for zero-rate qubit codes), and that the same bound holds for Haar-random codes as for random stabilizer codes. It further claims that for p above the hashing bound typical errors are uncorrectable but postselected error correction survives until a detection threshold p_d=(q-1)/q (1/2 for qubits), and that soft postselection by a Rényi-type reweighting produces transitions at H_alpha(p)=1-k/N. The primary threshold is supported by an explicit Weingarten replica calculation and by exact quantum MacWilliams identities for alpha=2,infinity, while the postselected coherent-information transition is supported only by numerics and a heuristic band-alignment assumption.

Significance. If the results hold, the paper establishes that the hashing bound is not special to stabilizer codes and gives an operational, information-theoretic meaning to Rényi-entropy singularities that have been observed in decohered quantum codes. The manuscript is unusually concrete: the Weingarten calculation is explicit, the MacWilliams-identity discussion is self-contained and potentially useful, no parameters are fitted, and the numerical data include scaling collapses. These strengths make the primary hashing-bound claim credible. The main weakness is that the advertised postselected-recovery transition is not derived analytically and rests on a band-alignment assumption that the authors themselves state is unproved.

major comments (2)
  1. [Sec. V (main) and Sec. III (Supplement), Eqs. (12)-(16), (S33)] The claim that soft-postselected error correction has a transition at p_c^(alpha), and that postselected recovery persists beyond the hashing bound, is not analytically established. The coherent information of the postselected state requires both S_vN(sigma_Q,alpha) and S_vN(sigma_RQ,alpha). The paper explicitly states in Sec. V of the main text 'We have not been able to derive an analytic ansatz for the coherent information of the post-selected state' and repeats in the Supplement 'We do not have an ansatz for this quantity.' The heuristic substitute, namely that each eigenvector in band w* of rho_RQ corresponds to q^k degenerate eigenvectors in band w* of rho_Q, is exactly the band-alignment/isolation assumption that needs testing; it does not follow from the Marchenko-Pastur or mean-shift analysis of the reduced spectra. The only evidence is Fig. 4(d) for N<=13. Since the abstract and
  2. [Supplement Sec. IV, Eqs. (S34), (S45)-(S46)] The Weingarten derivation computes annealed averages log E_U tr rho^n and then invokes the replica limit; the claim that these equal the quenched entropies E_U S_vN(rho) is an unproved concentration assertion. Rare encodings can dominate E tr rho^n even when typical encodings behave differently, and the coding threshold is a property of typical codes. The footnote [33] states that the quantities are self-averaging 'in practice' and the N<=13 numerics support this, but the analytic statement that Haar-random codes saturate the hashing bound should be accompanied by a variance bound or by an explicit remark that the annealed average is the standard random-coding object. Please either prove or clearly state this annealed/quenched equivalence and its limitations.
minor comments (3)
  1. [Fig. S3 caption] The caption is a leftover placeholder: 'FIG. 1. Your caption here.' This should be replaced with a proper caption.
  2. [Abstract and Sec. V heading] Typographical issues: 'randomstabilizercodes' in the abstract and 'P AST THRESHOLD' in the Sec. V heading should be corrected.
  3. [Supplement Sec. VII, Eq. (S73)] The quantity I_c^(infinity)(R>Q) is used but not defined in the main text; a one-line definition would improve readability.

Circularity Check

0 steps flagged

No significant circularity: the hashing-bound threshold is independently derived via Weingarten calculus, and no fitted parameter is renamed as a prediction.

full rationale

The central claim that Haar-random codes saturate the hashing bound is derived in two independent ways: the band/orthogonality picture (Eq. 6) and the explicit Haar-average of purities in Supplemental Sec. IV, where the dominant permutations sigma=tau=I,C yield E_U S_vN(rho_Q)=k+N H(p) below the hashing bound and N above it. This calculation uses standard Weingarten functions and involves no fitted parameters and no appeal to the band ansatz, so the threshold H(p_c)=1 is not an input to that calculation. The Rényi thresholds are likewise obtained from the same Weingarten saddle point and from the quantum MacWilliams identity, both external to the paper. The paper explicitly labels as heuristic the postselected coherent-information claim (main text Sec. V: 'We have not been able to derive an analytic ansatz for the coherent information of the post-selected state...'; Supplemental Sec. III: 'We do not have an ansatz for this quantity.'). That is a stated limitation and an unverified band-alignment assumption, but it is not a reduction of the prediction to its inputs: the threshold location itself is fixed by the independently computed Rényi transition, and the numerical collapse in Fig. 4(d) is presented as evidence rather than as a fitted parameter renamed as a prediction. The only self-citations (e.g., Ref. [52] in the operator-size-distribution list, Ref. [7] in the introduction) are contextual/definitional and carry no load-bearing argument. Therefore no construction-level circularity is present.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters are fitted: p is the physical noise rate, and p_c, p_d, p_c^(α) are derived. The central claims rest on statistical-orthogonality assumptions about random states and on the annealed/quenched replica step, which are the main non-rigorous inputs.

axioms (5)
  • domain assumption Typical error images are nearly orthogonal: overlaps between E_μ|ψ> and E_{μ'}|ψ'> are O(q^{-N/2}) for distinct Pauli errors.
    Used throughout Secs. III–V to treat bands as independent; quantified perturbatively in SM Sec. II. Verified numerically only for N≤13.
  • domain assumption Annealed replica average E_U log trρ^n equals the quenched entropy, and only identity/cyclic permutations dominate the Weingarten sum.
    SM Sec. IV uses this to obtain exact leading-N entropies; footnote [33] notes self-averaging but no rigorous proof is given.
  • domain assumption Marchenko-Pastur law applies to the error-corrupted density matrices despite correlations in the matrix elements.
    SM Sec. I states the elements are neither Gaussian nor independent but correlations are negligible 'in practice'; used to assign band shapes.
  • domain assumption The error-weight distribution under local depolarizing noise concentrates in an O(√N) window, so fixed-weight channels capture the local-noise physics.
    Used in Secs. III and IV to connect fixed-weight ensembles to the full depolarizing channel; central-limit behavior is standard but the replacement is heuristic.
  • standard math Weingarten formula for Haar moments and the quantum MacWilliams identity are valid and applicable.
    Standard background results used in SM Secs. IV, VI, and VII; the MacWilliams identity is derived for projector codes.

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Cite this review

Pith. "Pith review of Spectral properties and coding transitions of Haar-random quantum codes." pith.science (2026). https://pith.science/paper/YZQEPQZR

@misc{pith2026251007396,
  author       = {Pith},
  title        = {Pith review of: Spectral properties and coding transitions of Haar-random quantum codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YZQEPQZR}},
  note         = {Machine review of arXiv:2510.07396}
}
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read the original abstract

A quantum error-correcting code with a nonzero error threshold undergoes a mixed-state phase transition when the error rate reaches that threshold. We explore this phase transition for Haar-random quantum codes, in which the logical information is encoded in a random subspace of the physical Hilbert space. We focus on the spectrum of the encoded system density matrix as a function of the rate of uncorrelated, single-qudit errors. For low error rates, this spectrum consists of well-separated bands, representing errors of different weights. As the error rate increases, the bands for high-weight errors merge. The evolution of these bands with increasing error rate is well described by a simple analytic ansatz. Using this ansatz, as well as an explicit calculation, we show that the threshold for Haar-random quantum codes saturates the hashing bound, and thus coincides with that for random $\textit{stabilizer}$ codes. For error rates that exceed the hashing bound, typical errors are uncorrectable, but postselected error correction remains possible until a much higher $\textit{detection}$ threshold. Postselection can in principle be implemented by projecting onto subspaces corresponding to low-weight errors, which remain correctable past the hashing bound.

Figures

Figures reproduced from arXiv: 2510.07396 by David A. Huse, Grace M. Sommers, J. Alexander Jacoby, Sarang Gopalakrishnan, Zack Weinstein.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic circuit diagram of the Haar-random codes [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Fixed-weight error ensemble, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Depolarizing error channel, [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Theoretical phase diagram for postselection in a [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Your caption here. The initial encoded state |Ψ N k ditA iitili [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗

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