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REVIEW 3 major objections 5 minor 20 references

Interpolation of Quantum Polar Codes and Quantum Reed-Muller Codes

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Pith's one-line read Interpolating between quantum polar and Reed-Muller codes with one parameter α yields entanglement-free quantum codes that beat the polarization-weight construction's logical error rate at blocklength 1024 under SCL decoding.

desk verdict Useful, practical construction of entanglement-free quantum CSS codes with real finite-size gains, but the claimed interpolation to QRM is not proven and is ill-defined at the simulated parameters. read the letter →

arxiv 2505.22142 v2 pith:CWJL5IP7 submitted 2025-05-28 quant-ph cs.ITmath.IT

Keita Hidaka , Dina Abdelhadi , Ruediger Urbanke This is my paper
classification quant-phcs.ITmath.IT MSC 81P7094B0594B35
keywords quantumerrorcorrectionCSScodespolarReed-MullercodeinterpolationsuccessivecancellationlistdecodingPaulinoiseentanglement-free
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 aims to solve two problems in quantum error correction: quantum polar codes either need pre-shared entanglement or have poor finite-size performance, while quantum Reed-Muller codes have no known efficient decoder. It proposes a one-parameter family of quantum CSS codes indexed by α, built by ordering the classical bit-flip and phase-flip channels synthesized from the Pauli channel according to BSC(αq) rather than the physical noise level q. Scanning α identifies valid codes with commuting stabilizers and no entanglement assistance, and selected α values yield logical error rates below those of polarization-weight quantum polar codes under successive cancellation list decoding at blocklength N=1024. The practical point is that the family offers a tunable tradeoff among finite-size error rate, list size, and code symmetry, which matters for small quantum devices.

What carries the argument

The load-bearing object is the interpolating classical channel family W_X^α = BSC(α(p_X+p_Y)) and W_Z^α, a flagged mixture of BSCs, which for the equal XZ noise model reduces to two independent copies of BSC(αq). The construction maps each α to two frozen sets F_Z(α) and F_X(α); the validity condition F_Z(α) ∩ F_X(α) = ∅ follows from GG = I over GF(2) and is what guarantees a commuting set of stabilizers without pre-shared entanglement. The parameter α interpolates between the quantum polar code (α=1) and the quantum Reed-Muller code (α=0), and the machinery's work is to turn the search over α into a practical code-design procedure: estimate virtual channel error probabilities via an approximation algorithm, sort, freeze, check commutation, and then select α* by SCL-C decoding.

What would settle it

Compute the frozen sets F_Z(α) and F_X(α) for BSC(αq) as α approaches zero at several noise levels q and blocklengths; if they do not converge to the Hamming-weight-threshold frozen sets of the corresponding quantum Reed-Muller code, the claimed interpolation to QRM is refuted, even though the finite-size comparisons could still hold.

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Extended reading notes

Core claim

At the center of the paper is the observation that the classical interpolation idea of [7] carries over to quantum CSS constructions. For independent equal XZ noise, the two induced classical channels both become BSC(αq). For each α, the paper computes virtual channel parameters, freezes the worst N−k1 channels in the Z-basis and the worst N−k2 channels in the X-basis (with order reversed to account for the transpose action of the quantum polar transform), and keeps only those α for which no index is frozen in both bases, which guarantees commuting stabilizers. The discovered result is that the best valid α is usually strictly between 0 and 1: at N=1024, k1=k2=533, and list size 16, the tuned codes reduce the logical X error rate relative to the polarization-weight quantum polar code construction for all tested noise levels q=0.04 to 0.10, and even when α=1 is a valid polar code, a smaller α improves performance. The paper also reports that smaller α enlarges the automorphism group and lowers the mixing factor.

Load-bearing premise

The construction assumes that the classical interpolation result proven for the binary erasure channel also holds for the binary symmetric channel BSC(αq), so that α→0 really delivers the quantum Reed-Muller code; the paper states this limit for BEC but gives no proof for BSC.

Editorial extensions

If this is right

  • At blocklength 1024 with k1=k2=533 and list size 16, the α-tuned codes achieve lower logical X error rates than the polarization-weight quantum polar codes for every tested noise level q between 0.04 and 0.10.
  • Even when α=1 produces a valid quantum polar code, choosing a smaller α improves the finite-size logical error rate, so the interpolation parameter acts as a design knob independent of validity.
  • The mixing factor of the best α codes is 406 or 414, below the value 470 for the PW-QPC reference, meaning smaller list sizes should suffice to approach ML decoding.
  • Decreasing α from 1 to 0.1 grows the automorphism group from roughly 3.6×10^16 to 1.08×10^17, which supports fault-tolerant gate implementations.
  • At higher rate (k=94), the optimal α decreases as the list size L grows, consistent with RM codes outperforming polar codes under MAP decoding.

Reading between the lines

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

  • The same construction should apply to biased Pauli noise by treating W_X^α and W_Z^α as two different interpolating channels instead of two copies of BSC(αq); the paper does not test this, but the framework does not require the two channels to be identical.
  • The observed decrease of optimal α with list size suggests that in the infinite-list (MAP) limit the best code in the family approaches the quantum Reed-Muller code; if true, the family could serve as an efficiently decodable approximation to QRM codes.
  • The validity check F_Z ∩ F_X = ∅ is purely combinatorial, so the α-selection procedure could be automated for arbitrary rates and channel models, turning the interpolation into a general code-search method.
  • The automorphism-group growth with decreasing α, measured on the classical induced code, may translate into a larger set of transversal gates for the quantum CSS code, though the paper only reports the group size and not the gates.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a family of quantum CSS codes obtained by ordering the rows of the polar transform according to virtual-channel parameters of BSC(αq), where q is the Pauli noise parameter for independent equal-XZ noise. By varying the interpolation parameter α, the frozen sets F_Z(α) and F_X(α) change; the authors search over α for values for which F_Z∩F_X=∅, which yields a valid commuting CSS code without pre-shared entanglement, and then select the α that minimizes the SCL-C logical error rate. At blocklength N=1024 with k1=k2=533, they report lower logical error rates than the polarization-weight quantum polar codes (PW-QPC) of [10] for q between 0.04 and 0.10, and they study list-size dependence, mixing factors, and automorphism-group growth. The appendix quantifies how the frozen set at intermediate α overlaps with the frozen sets at α=1 and α=0.

Significance. If the numerical comparisons are accepted, the paper offers a simple entanglement-free quantum CSS code construction with finite-size SCL performance better than the PW-QPC benchmark at the listed noise rates, together with a tunable trade-off between code structure and performance. The commutation condition is clearly stated, and the positive-rate non-intersection argument in Section IV is a useful structural observation. The claimed interpolation to quantum Reed-Muller codes is, however, not yet supported: the α→0 limit is asserted for BSC(αq) without proof, and at the simulated parameters the limiting code is not a standard threshold QRM code. In addition, the reported improvements are obtained by selecting the best of ten random α draws per noise parameter, so the comparison is subject to selection bias. The paper does not ship code, seeds, or confidence intervals, which limits independent verification of the numerical claims.

major comments (3)
  1. [Section V and Appendix] The claim that setting α=0 yields a quantum Reed-Muller code is asserted rather than proved for the BSC ordering. Section III.B cites [7] for the statement that polar codes designed for BEC(αε) tend to RM codes as α→0, but the present construction in Section V uses BSC(αq), and the Appendix repeats 'Recall that when α=0, the interpolating code is an RM code' without a proof or reference for the BSC case. The extension from BEC to BMS channels is not automatic. Moreover, at the simulated parameters N=1024 and k1=k2=533 the information set cuts through the Hamming-weight-5 rows (the cumulative count up to weight 4 is 386), so the α→0 limit is governed by tie-breaking among equal-weight rows, and it is not a standard RM(m,r) code of dimension 533. The Appendix itself acknowledges that tie-breaking matters when the limit is not a full threshold set. I therefore ask the authors to either prove or cite a proof of the BSC ordering limit, or to qualify the title, abstract, and Appendix statements by saying that the family interpolates between polar and RM-like orderings. I note that the reported α* values in Table I lie in [0.41,0.75], away from 0, so the finite-size gains do not logically depend on this questionable limit; the issue affects the interpolation mechanism and framing rather than the numerical comparison itself.
  2. [Section V-A, Table I, and Figure 1] The reported α* error rates are minima over ten randomly drawn α values per noise parameter q, and these same post-selected values are then presented as the performance of the proposed construction. Selecting the best of ten decoder evaluations overstates the expected improvement relative to a fixed benchmark and makes the comparison sensitive to sampling noise; for example, Table I reports P_e,SCL-C(α*) ≈ 0 at q=0.04 with no confidence interval. To support the claim of outperforming PW-QPC, the authors should report the full set of tested α values, the distribution of error rates over those draws, confidence intervals, and a precise description of the random-sampling protocol (including seeds). Alternatively, they could fix α(q) by a deterministic rule based on channel parameters or on a separate validation noise value, and then evaluate the selected codes on the reported q values without further selection.
  3. [Section V-D and Figure 2] The interpretation that 'α→0 corresponds to RM codes' is used to explain why the best α decreases as the list size L increases. This again relies on the unproved BSC-to-RM limit, and the statement that 'L→∞ corresponds to MAP decoding' describes a fixed code while the comparison here varies α with L. The sentence in Section V-D should be reworded as a heuristic motivation rather than a consequence, and the figure should be described as showing the L-dependent behavior of the constructed family rather than as evidence for an RM limit.
minor comments (5)
  1. [Figure 1 caption and Table I] The α* values listed in the Figure 1 caption do not match Table I (for example, the caption lists five values including 0.49 and 0.41, while Table I has seven q-values with α* = 0.61 for q=0.04 and 0.60 for q=0.10). Please align the figure labels with the table.
  2. [Section IV] The positive-rate non-intersection proof compresses the step from the dimension sums to the inequality m−w > w′−1; please expand this derivation so that the contradiction is fully transparent.
  3. [Appendix, Figures 3 and 4] The definitions of F_Z(0) and F_Z(1) are implicit; please state explicitly how the α=0 and α=1 frozen sets are obtained when the limiting code is not a full threshold RM code.
  4. [Section V-E] The automorphism-group computation assumes the constructed codes are decreasing monomial codes, citing [13]; this assumption should be stated as an assumption in the text, since the BSC-designed polar codes are not shown to satisfy it for all α.
  5. [General] No seeds, code, or confidence intervals are provided for the simulations; reporting these would materially improve reproducibility, especially given the random search over α.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the finite-size gain is an explicitly selected search result benchmarked against an external code; the unproved BSC(αq)-to-RM limit is a framing/correctness gap, not a circular reduction.

full rationale

The numerical claim is self-contained against external benchmarks. Section V explicitly says: 'For each noise parameter q, we pick 10 random values of α that produce a valid quantum code, and evaluate the performance of the SCL-C decoder to decide the best value of α∗.' The reported α∗ and P_e(α∗) are therefore selected minima, not predictions obtained from the same fit, and the comparison code PW-QPC [10] and decoder [11] are external. The construction itself is explicit: define W^α_X = BSC(α(p_X+p_Y)), W^α_Z as the corresponding mixture, compute Tal–Vardy estimates, form F_Z(α) and F_X(α), and retain only α values whose frozen sets do not overlap. No equation in the paper reduces the claimed gain to its own input. The main weakness is an unsupported extrapolation: [7] proves the α→0 RM limit for BEC(αε), but Section V applies the same interpolation to BSC(αq), and the Appendix asserts 'Recall that when α=0, the interpolating code is an RM code.' That limit is not proven and is ill-defined at Table I's parameters (N=1024, k1=k2=533 is not an RM dimension). This is a missing-proof/framing issue, not circularity: the displayed gains use α∗ ∈ [0.41,0.75] and do not depend on the α=0 limit. Minor self-citations ([7], [17]) are not load-bearing for the finite-size comparison. Score 1.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The construction rests on standard CSS and polar-code machinery plus one unproved interpolation assumption. The only fitted parameter introduced by this paper is α, chosen per channel by search; k1,k2 are fixed for comparison. No new physical entities, particles, or forces are introduced.

free parameters (2)
  • α (interpolation/noise-scaling parameter) = Table I: 0.61, 0.49, 0.41, 0.75, 0.65, 0.60, 0.60 for q=0.04..0.10
    Chosen per noise level q as the best of 10 random valid values under SCL-C logical error rate; this is fitted to the reported performance metric.
  • k1=k2 (classical code dimensions) = 533 for N=1024 (k=42); 559 for N=1024 (k=94)
    Fixed by hand to match the comparison code from [10] and to achieve target quantum dimension; not optimized in this paper.
assumptions (4)
  • domain assumption Pauli channel decomposes into two classical BMS channels W_X and W_Z as in [14].
    Section II-A uses this standard QEC reduction to map quantum channel design to classical polar code design.
  • standard math CSS construction with C1^⊥⊆C2 yields a valid quantum code, and commutation is equivalent to F_Z∩F_X=∅.
    Section II-B and Section IV rely on GG=I over GF(2) and the dot-product commutation criterion.
  • domain assumption The Tal-Vardy/Pedarsani approximation of P(E_i) gives the correct virtual-channel ordering.
    Section V uses the approximation algorithm of [15,16] to sort channels; the validity and performance of the construction depend on this ordering being accurate.
  • ad hoc to paper Polar codes designed for BSC(α q) converge to RM codes as α→0.
    Section V and the Appendix assert this, but [7] proves the interpolation limit for BEC(αε), not for BSC. The QRM interpolation label depends on this unproved extension.

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Cite this review

Pith. "Pith review of Interpolation of Quantum Polar Codes and Quantum Reed-Muller Codes." pith.science (2026). https://pith.science/paper/CWJL5IP7

@misc{pith2026250522142,
  author       = {Pith},
  title        = {Pith review of: Interpolation of Quantum Polar Codes and Quantum Reed-Muller Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CWJL5IP7}},
  note         = {Machine review of arXiv:2505.22142}
}
read the original abstract

Good quantum error-correcting codes that fulfill practical considerations, such as simple encoding circuits and efficient decoders, are essential for functional quantum information processing systems. Quantum polar codes satisfy some of these requirements but lack certain critical features, thereby hindering their widespread use. Existing constructions either require entanglement assistance to produce valid quantum codes, suffer from poor finite-size performance, or fail to tailor polar codes to the underlying channel properties. Meanwhile, quantum Reed-Muller (RM) codes demonstrate strong performance, though no known efficient decoding algorithm exists for them. In this work, we propose strategies to interpolate between quantum polar codes and quantum RM codes, thus addressing the challenges of designing valid quantum polar codes without entanglement assistance and improving finite-size code performance.

Figures

Figures reproduced from arXiv: 2505.22142 by the authors.

Figure 1
Figure 1. The logical X error rate of J1024, 42K codes with different constructions. Each data point is evaluated by aver￾aging over 106 samples with an SCL-C decoder of list size 16. For each q, we search over multiple α values and show the best performing codes on the plot. The α values of the corresponding codes are in Table I. implementation, P(Ei) is estimated using the approximation algorithm discussed in [15], [16]. Th… view at source ↗
Figure 2
Figure 2. Logical X error rate is simulated at q = 0.0425, for J1024, 94K d) Effect of list-size: Here we focus on the performance of α-QPC-QRM codes of blocklength N = 1024, k1 = k2 = 559 yielding a quantum code of higher dimension k = 94. At this high rate, many values of α lead to valid quantum code constructions. For q = 0.0425, we vary α and the list size L, and observe the value of the resulting SCL-C decoder logical er… view at source ↗
Figure 3
Figure 3. The proportion of frozen indices of α codes shared with polar code. tion of frozen indices of an α code that overlaps with those of polar code and RM code, respectively, defined as follows: fpolar(α) = |FZ(α) ∩ FZ(1)| |FZ(1)| fRM(α) = |FZ(α) ∩ FZ(0)| |FZ(0)| Recall that when α = 0, the interpolating code is an RM code, and when α = 1, it is a polar code. These two metrics indicate how much ”polar-like” and ”RM-like”… view at source ↗
Figures from the paper (2 more)
Figure 2
Figure 2. Figure 2: When we construct an α-QPC-QRM interpolated code with α = 1, its channel parameters are polar code parameters, which gives us the upper bound of the SC decoding error. In other words, the upper bound of the SC decoding error (or its approximation) is minimized under α …
Figure 4
Figure 4. Figure 4: The proportion of frozen indices of α codes shared with RM code. I, while the corresponding high-quality virtual channels are excluded, thereby degrading the SC decoding performance. This observation suggests that improved performance cannot be expected from the propos…

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