{"id":"8442ac68-0897-49c1-9fbf-6a62e294932f","arxiv_id":"2505.22142","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"An α-parameterized interpolation of quantum polar and Reed-Muller CSS codes gives valid entanglement-free codes with lower simulated logical error rates than polarization-weight quantum polar codes at blocklength 1024.","lead":"This paper designs a family of quantum error-correcting codes that blends quantum polar codes with quantum Reed-Muller codes, tuned by a parameter α. In simulations at blocklength 1024, some of these blended codes outperform a prior quantum polar code construction without needing shared entanglement.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed BSC-to-QRM limiting behavior is unproved and actually ill-defined at the simulated dimensions; the finite-size gains do not require it, so the interpolation claim needs qualification rather than rejection.","rationale":"The reader's weakest assumption identifies the BSC-to-RM limit, and I agree that it is the most load-bearing conceptual weak point. The stress-test sharpens it: for the actual simulated parameters (N=1024, k1=k2=533), the information set size 533 splits the Hamming-weight-5 class, so the α→0 limit is not even a standard RM code unless a specific tie-breaking rule is shown to coincide with the RM construction of [10]. Since the asymptotic ordering result in [7] is stated and proven for BEC(αε) and the paper does not supply a proof for BSC(αq), the interpolation claim is unsupported at the parameter values used in the main comparison. The finite-size numerical claim is not endangered because Table I uses α* values between 0.41 and 0.75, far from the questionable limit; the improvement over PW-QPC stands independently of whether the family converges to QRM at α=0. Thus the correct disposition is to require the authors to prove or carefully qualify the BSC-to-RM limiting claim, exactly matching the reader's CONDITIONAL verdict. Secondary statistical concerns (confidence intervals, the protocol for choosing '10 random valid α values') are worth mentioning but are not the single most load-bearing issue.","tokens_in":9507,"tokens_out":12366,"duration_ms":149158,"concrete_test":"Fix N=1024, k1=k2=533, q=0.06, and reconstruct F_Z(α) and F_X(α) with the Tal-Vardy approximation for BSC(αq) at α=10^-1, 10^-2, ..., 10^-6. Compute f_RM(α)=|F_Z(α)∩F_Z(0)|/|F_Z(0)| using the [10] RM-order frozen set as F_Z(0). If f_RM(α) does not tend to 1, or if the weight-5 rows selected at small α are not the first 147 by the +i/N rule, the claimed α→0 QRM limit is false at these parameters. Separately, derive the leading coefficient c_i of the BSC bit-channel error probability for the split weight-5 rows and check whether sorting by c_i reproduces the i/N tie-break; this identifies the actual limit code.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section V defines the interpolating family from BSC(αq), and the Appendix states 'Recall that when α=0, the interpolating code is an RM code.' The classical ordering limit in [7] is proven for BEC(αε), not for BSC(αq). More sharply, the main simulated code has N=1024, k1=k2=533: the information set cuts through the 252 rows of Hamming weight 5 (cumulative count up to weight 4 is 386). No standard RM code has this dimension, and any BSC-as-α→0 limit among those equal-weight rows is governed by the subchannel constants c_i arising from the Tal-Vardy approximation, not by the +i/N tie-break of [10] used in the paper. Thus 'α=0 is a QRM code' is not a proven consequence of the construction, and at the simulated parameters it is not even the standard QRM code. Table I's α* values lie in [0.41,0.75], so the displayed logical-error-rate improvement over PW-QPC does not depend on the questionable limit; but the title, abstract, and the claimed interpolation mechanism do.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9754,"tokens_out":9713,"duration_ms":116586,"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":[{"comment":"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.","section":"Section V and Appendix"},{"comment":"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.","section":"Section V-A, Table I, and Figure 1"},{"comment":"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.","section":"Section V-D and Figure 2"}],"minor_comments":[{"comment":"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.","section":"Figure 1 caption and Table I"},{"comment":"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.","section":"Section IV"},{"comment":"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.","section":"Appendix, Figures 3 and 4"},{"comment":"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.","section":"Section V-E"},{"comment":"No seeds, code, or confidence intervals are provided for the simulations; reporting these would materially improve reproducibility, especially given the random search over α.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper's core CSS construction and commutation check are sound, and the finite-size gains are plausible, but the numerical comparison is weakened by the post-selection protocol and the interpolation-to-QRM claim rests on an unproved BSC analogue of a BEC result. Both issues are fixable in revision: the authors can soften the framing or supply a proof for the BSC limit, and they can redo the comparison with a pre-specified or deterministic α selection and confidence intervals. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid engineering paper that does something new and useful—it adapts the classical polar-to-RM interpolation to quantum CSS codes, gives a clean commutation check, and demonstrates better SCL decoding than the polarization-weight construction at N=1024. The improvement is real, as far as the numerics go, and does not depend on the shaky RM limit. But the title and abstract promise an interpolation to quantum RM codes, and that part is under-supported: the α→0 limit is proven in [7] only for BEC, and this paper extends it to BSC without proof. Moreover, at the simulated parameters (N=1024, k1=k2=533) the 'RM code' is not a standard QRM code—no RM code has that dimension, and the tie-breaking by +i/N is an arbitrary choice. So the connection to RM is at best heuristic, and the paper should say so.\n\nWhat is genuinely good: the construction is simple, the commutation condition (F_X ∩ F_Z empty) is correctly derived from GG=I, and the positive-rate non-intersection proof for QRM codes is plausible. The authors also do useful diligence by reporting the mixing factor and automorphism-group tradeoffs. The comparison against PW-QPC is meaningful—same blocklength, same decoder, external benchmark. The gains in Table I, from α*=0.41 to 0.75, are substantial, and the list-size behavior in Fig. 2 is coherent with the RM/polar MAP heuristic.\n\nThe soft spots, in order of severity. First, the BSC-to-RM limit is asserted in the Appendix ('Recall that when α=0, the interpolating code is an RM code') but not shown; given that [7] proves it for BEC only, this is a real gap. Second, and sharper, the simulated code parameters do not correspond to any standard RM code, so even if the limit held in some asymptotic sense, it is not the code they simulate. Third, the α* values are the best of 10 random draws, with no confidence intervals or description of how the draws were made; this is an empirical search, not a fitted parameter, and the reported error rate is the same objective used for selection—so the numbers are optimistic by construction. These are not fatal to the contribution, but they need to be stated.\n\nFor a reader working on quantum polar codes or finite-size QEC, this paper is worth the time. It deserves peer review, and a good referee should push for a proof or a qualified statement of the RM limit, a precise description of the α-search, and error bars. I'd recommend engaging with it.","headline":"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.","tokens_in":10287,"tokens_out":2596,"would_cite":true,"duration_ms":25646,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P70","94B05","94B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["quantum error correction","CSS codes","quantum polar codes","quantum Reed-Muller codes","code interpolation","successive cancellation list decoding","Pauli noise","entanglement-free quantum codes"],"falsifier":"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.","tokens_in":9286,"feed_emoji":"⚛️","tokens_out":7333,"duration_ms":67874,"temperature":0.7,"pith_summary":"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.","feed_headline":"One extra parameter shrinks quantum polar code error rates","feed_subtitle":"Tuning alpha lowers logical error rates under list decoding at practical blocklengths.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the channel polarization framework, the polar transform G, and the Hamming-weight formula for rows of G.","marker":"[1]"},{"why":"Introduces the classical polar-to-Reed-Muller interpolation and the α→0 RM limit that the quantum construction adapts.","marker":"[7]"},{"why":"Defines the polarization-weight quantum polar codes and SCL-C decoder baseline that the proposed codes are compared against.","marker":"[10]"},{"why":"Provides the list-decoder implementation used to simulate logical error rates.","marker":"[11]"},{"why":"Provides the approximation algorithm used to estimate virtual channel error probabilities for BSC(αq).","marker":"[15]"},{"why":"Supplies the automorphism-group computation for decreasing monomial codes used to measure symmetry as α varies.","marker":"[13]"}],"fun_headline_variants":["Tuning α reduces quantum polar code errors","Interpolation between polar and Reed-Muller codes","A single parameter sharpens quantum codes","Bridging quantum code families cuts error rates","No entanglement needed: better polar codes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Tuning α reduces quantum polar code errors","Interpolation between polar and Reed-Muller codes","A single parameter sharpens quantum codes","Bridging quantum code families cuts error rates","No entanglement needed: better polar codes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000619,"raw_usage":{"total_tokens":2836,"prompt_tokens":877,"completion_tokens":1959,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":1893}},"tokens_in":493,"tokens_out":1959,"duration_ms":14744,"temperature":1.0,"reasoning_tokens":1893,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:14:37.205842+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Channel polarization: A method for constructing capacity- achieving codes for symmetric binary-input memoryless channels,","cited_arxiv_id":null,"evidence_quote":"Supplies the channel polarization framework, the polar transform G, and the Hamming-weight formula for rows of G."},{"cited_title":"From polar to reed- muller codes: a technique to improve the finite-length performance,","cited_arxiv_id":null,"evidence_quote":"Introduces the classical polar-to-Reed-Muller interpolation and the α→0 RM limit that the quantum construction adapts."},{"cited_title":"Improved logical error rate via list decoding of quantum polar codes,","cited_arxiv_id":null,"evidence_quote":"Defines the polarization-weight quantum polar codes and SCL-C decoder baseline that the proposed codes are compared against."},{"cited_title":"Pw-qpc-list-decoder: List decoder for the polarization weight family of quantum polar code, github,","cited_arxiv_id":null,"evidence_quote":"Provides the list-decoder implementation used to simulate logical error rates."},{"cited_title":"On the Automorphism Group of Polar Codes","cited_arxiv_id":"2101.09679","evidence_quote":"Supplies the automorphism-group computation for decreasing monomial codes used to measure symmetry as α varies."}],"review_version":1}