{"id":"6b2998e1-a45b-489a-9263-4a0f8d78a850","arxiv_id":"2504.16709","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For single-qubit quantum secret sharing protocols, encoding each qubit three times and taking a majority vote corrects bit-flip, phase-flip, and amplitude-damping noise with only three physical qubits per logical qubit.","lead":"The paper analyzes how three types of quantum noise corrupt a simple multiparty quantum secret sharing protocol and proposes guarding each qubit with a 3-qubit repetition code instead of Shor's 9-qubit code. The resource reduction is real, but the headline claim that this outperforms existing quantum error correction codes rests on an unfair comparison and a numerical mistake.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed advantage over existing QEC codes is not established: Figs. 5/6 compare a single-cycle 3-qubit repetition code with 5- and 4-qubit codes run in a different multi-cycle regime and with an unreproducible recovery operation, so the central comparison is uncontrolled.","rationale":"The reader's verdict of REJECT is appropriate; my concern does not move it. I focus on the benchmark comparison because the paper's headline claim is explicitly a comparative one. The independence assumption identified by the reader is less decisive: in the paper's own bit-flip/phase-flip noise model, independent physical errors are the standard assumption, and even granting independence the comparative conclusion fails. The paper's own Section V.C admits that multi-cycle QEC is expected to outperform single-cycle QEC, yet it applies multi-cycle to the 5- and 4-qubit benchmarks and single-cycle to the repetition code; moreover, the listed recovery operators in Section III.B.b are not a verifiable implementation of the code in Eq. (6). A correct [[5,1,3]] code should reduce logical error below the single-qubit no-encoding line for sufficiently small p, so the paper's claim that the 5-qubit code 'performs worse than no encoding' is a red flag that the benchmark is not the standard code. Eq. (22) compounds this by giving a wrong small-gamma coefficient. The proposed test distinguishes an artifact of cycle count from a real property of the codes; if the 5-qubit code beats no encoding at small p in either mode, Figs. 5/6 cannot support the abstract's 'lower average error rates than existing QEC methods.' Until such a controlled comparison is shown, rejection is warranted.","tokens_in":17247,"tokens_out":18548,"duration_ms":179218,"concrete_test":"Implement the [[5,1,3]] perfect code with a standard syndrome-based recovery for the code states in Eq. (6), and run it in both (i) the single-cycle protocol of Section V.B and (ii) the multi-cycle protocol of Section V.C, under bit-flip/phase-flip noise at p = 0.01, 0.05, and 0.1. Compare the reconstructed-secret error to the 3-qubit repetition code (Eq. (18)) and to no encoding (Eq. (11)). If the 5-qubit code is below no encoding at p = 0.01, or if its error changes materially between the two modes, the paper's Figs. 5/6 do not support the claimed advantage over existing QEC methods.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central comparative claim—that the 3-qubit repetition code achieves lower average error rates than existing QEC methods—rests on Figs. 5 and 6, but those figures do not compare like with like. Sections V.A–V.B evaluate the repetition code in a single cycle: encode once at Bob, run the whole multi-channel path, then decode and majority vote at the end. Section V.C evaluates the [[5,1,3]] perfect code and the [[4,1]] approximate code in a multi-cycle mode, with encoding, recovery, and re-encoding applied to every channel separately. The paper itself states that multi-cycle QEC performs better than single-cycle QEC, so the comparison changes both the cycle count and the number of physical qubits exposed per channel. The claimed result that the 5-qubit code is worse than no encoding is inconsistent with the standard behavior of a [[5,1,3]] code, whose logical error should scale as O(p^2) for small physical error p; this signals that the recovery operators in Section III.B.b (which are not accompanied by a stabilizer table or circuit description) are not implementing the code as intended. As an independent check, substituting Eq. (13) into Eq. (16) gives a leading coefficient 243/64 gamma^2 for Eq. (22), not 3/32 gamma^2, so the amplitude-damping comparison is also not quantitatively reliable.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes the effect of bit-flip, phase-flip, and amplitude-damping noise on the multiparty QSSCM and SSQI protocols of Zhang et al. (Phys. Rev. A 71, 044301). It proposes a 3-qubit repetition code derived from Shor's code by exploiting the fact that each transmitted state is in a known basis, so only bit-flip or only phase-flip correction is needed. The paper derives analytic error expressions, provides simulations, and claims that the 3-qubit repetition code achieves lower average error rates than the five-qubit perfect code, the four-qubit approximate code, and no encoding.","tokens_in":17541,"tokens_out":12635,"duration_ms":121569,"significance":"The resource-reduction idea is potentially useful: for these single-qubit, known-basis protocols, replacing the 9-qubit Shor code with a 3-qubit repetition code is a natural and practical simplification. The bit-flip and phase-flip analysis is largely sound, including the threshold condition e < 1/2 derived in Eq. (17), and the formulas are derived from first-principles counting rather than fitted parameters. The n-party generalization in Eq. (20) is also a useful closed form. However, the central comparative claim against existing QEC codes is not currently supported: the amplitude-damping expansion has a large numerical error, and the comparison in Figs. 5 and 6 mixes different QEC cycle regimes and an unreproducible recovery operation.","major_comments":[{"comment":"The amplitude-damping error after encoding is numerically wrong. Expanding Eq. (13) gives e_a1 = (9/8)γ − (77/96)γ² + O(γ³), and substituting into e_af = 3(e_a1)²(1−e_a1) + (e_a1)³ gives a leading term 243/64 γ², not 3/32 γ². The displayed coefficient is too small by a factor of about 40, so the quantitative comparison for amplitude damping in Fig. 4 and the abstract's claim of lower average error rates are not supported as written.","section":"V.B.b, Eq. (22)"},{"comment":"The central comparison is uncontrolled. The repetition code is evaluated in a single cycle, encoding once at Bob and decoding after the entire multi-channel path, whereas the five- and four-qubit codes are evaluated in Section V.C in a multi-cycle mode with encoding, recovery, and re-encoding applied to each channel separately. Because the paper states that multi-cycle QEC performs better than single-cycle QEC, Figs. 5 and 6 change both the number of cycles and the number of physical qubits exposed per channel, so they do not establish that the 3-qubit code outperforms existing QEC methods.","section":"V.A–V.C, Figs. 5 and 6"},{"comment":"The recovery operation for the [[5,1,3]] code is not reproducible and the reported simulation result is implausible. The operators R_0,...,R_15 are listed without a stabilizer table, circuit, or syndrome-to-recovery mapping, and they do not match the standard recovery of the five-qubit code. The claim in Fig. 5 that the five-qubit code is worse than no encoding contradicts the expected logical-error scaling O(p²) of a distance-3 code under independent bit/phase flips; this strongly suggests the recovery as implemented is not the intended code. A corrected implementation and a fair comparison must be provided before the comparative claim can be assessed.","section":"III.B.b and Fig. 5"},{"comment":"Equation (16) assumes the three physical copies of each logical qubit suffer statistically independent errors, but this assumption is never stated. If the three transmissions are correlated, for example through shared channel noise or common-mode fluctuations, the majority-vote error rate will be higher than computed and the claimed improvement over no encoding may disappear. The independence assumption should be stated explicitly or derived from the channel model.","section":"V.B, Eq. (16)"}],"minor_comments":[{"comment":"The phase-flip channel is written with C_b and p_b; it should be C_p and p_p.","section":"III.A.b, Eq. (3)"},{"comment":"The simulation curves do not report the number of Monte Carlo shots, the error model for encoding/decoding gates, or whether density-matrix simulation was used, which makes the numerical results unreproducible.","section":"Figs. 3–6"},{"comment":"The scheme is described as a version of Shor's code; more precisely it is a basis-dependent classical repetition code, and the paper should say so to avoid overstating the quantum error-correction content.","section":"V.A"},{"comment":"The SSQI derivation assumes that the teleportation fidelity factor (1−e_t) is unchanged by the correction and that the QSSCM error is independent; this should be stated as an explicit assumption.","section":"VI, Eq. (24)"}],"recommendation":"major_revision","confidential_remarks":"The numerical error in Eq. (22) and the uncontrolled comparison are serious, but they are fixable in principle if the authors provide a verifiable recovery circuit for the five-qubit code, correct the amplitude-damping expansion, and rerun the comparison in a single consistent QEC regime. I see no circularity: the error formulas are derived from first principles, and the self-citations [20,59] are background. If the corrected simulations cannot be made reproducible, the paper should not be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my read on arXiv:2504.16709. The paper has a genuinely useful observation: for the Zhang et al. QSSCM protocol, where Bob prepares known basis states and all operations are Pauli/Hadamard, you can separate bit-flip and phase-flip correction and replace Shor's 9-qubit code with a 3-qubit repetition code. The bit-flip/phase-flip noise analysis and the n-party error formulas are correct as far as I can tell, and the threshold e<1/2 for repetition decoding is right. That part is worth keeping.\n\nThe problems start when they claim the repetition code outperforms the five- and four-qubit codes. The comparison in Figs. 5 and 6 is apples-to-oranges: the repetition code is run in single-cycle mode (encode once, decode at the end), while the 5- and 4-qubit codes are run in multi-cycle mode (encode/recover/re-encode per channel). The paper itself acknowledges multi-cycle QEC performs better, so this comparison does not support the 'lower average error rates' claim. Also, the five-qubit code is reported as worse than no encoding at small p, which is inconsistent with a [[5,1,3]] code's expected O(p^2) logical error; the recovery operators in Section III.B.b are not shown to be a valid set for that code, so I suspect the simulation is not implementing the code correctly.\n\nThere is also a clear numerical error in eq. (22). Substituting eq. (13) into eq. (16) gives a leading coefficient of 243/64 γ^2, not 3/32 γ^2. That is a factor of ~40, not a typo, and it undermines the amplitude-damping plot in Fig. 4. A minor additional worry: the independence assumption in eq. (16) is not stated; correlated errors across the three copies would reduce the benefit of majority voting.\n\nFinally, the SSQI section is underdeveloped. The abstract promises noise analysis for the SSQI protocol, but Section VI just reduces the condition to e_QEC < e_noise and does not analyze the teleportation step at all.\n\nThe citation pattern is fine, and I don't see fabricated entities. The paper's core idea is sensible, but the central comparative claim is not established. I would send it to peer review because the basis-separation trick and the analytic formulas are useful to the QSS community, but I would require a major revision: fix eq. (22), redo the comparisons with the same cycle structure, and either provide a real SSQI analysis or remove the claim. If the authors can do that, it could be a reasonable paper. As it stands, I would not cite it yet.","headline":"A sensible basis-separation trick for known-basis QSS, but the central comparison against 5- and 4-qubit codes is uncontrolled and eq. (22) is off by a factor of ~40.","tokens_in":18043,"tokens_out":3894,"would_cite":false,"duration_ms":34699,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.67.Dd","03.67.Pp"],"model":"deepseek-v4-flash","headline":"A 3-qubit repetition code, not the usual 9-qubit code, suffices to protect multiparty quantum secret sharing from bit-flip, phase-flip, and amplitude-damping noise.","keywords":["quantum secret sharing","multiparty secret sharing","repetition code","quantum error correction","bit-flip noise","phase-flip noise","amplitude damping","qubit overhead"],"falsifier":"Transmit three copies of the same state through the same fluctuating channel and record the joint error pattern; if two or three copies fail together more often than $3e^2(1-e)+e^3$ predicts for the measured per-copy error rate $e$, the repetition code's advantage over no encoding disappears under correlated noise.","tokens_in":17029,"feed_emoji":"🔐","tokens_out":16094,"duration_ms":138268,"temperature":0.7,"pith_summary":"The paper tries to establish that the two multiparty quantum secret-sharing protocols studied here can be made more noise-resistant with only three physical qubits per logical qubit, instead of the nine usually required for full protection. Because every transmitted state is eventually measured in the basis in which it was prepared, the authors argue, bit-flip and phase-flip corrections separate cleanly, so a plain repetition code with majority voting handles the dominant error. They further claim that this three-qubit code yields lower average error rates for the reconstructed secret than the five-qubit perfect code, the four-qubit approximate code, and no encoding at all in their simulations, while cutting qubit overhead by two-thirds. If the claim holds, entanglement-free multiparty secret sharing becomes substantially cheaper to run, and the same basis-aware repetition trick can be ported to other single-qubit-based quantum protocols.","feed_headline":"Secret sharing needs only 3 qubits, not 9, under noise","feed_subtitle":"An abridged 9-qubit code cuts qubit overhead by two-thirds and beats larger error-correcting codes on secret error.","key_machinery":"The machinery is a basis-aware 3-qubit repetition code obtained by truncating the standard 9-qubit code: each logical state is sent as three identical physical copies, and the receivers decode by measuring each copy in the preparation basis and accepting the majority outcome. Its role is to replace the two concatenated layers of the 9-qubit code with just one layer, because in these protocols the basis of every state is known and only one Pauli error type can act per transmission; this is what cuts the overhead from nine qubits to three. The quantitative engine is the majority-vote formula $e_{\\text{QEC}}=3e^2(1-e)+e^3$, which says two or three simultaneous copy errors defeat the vote and which is smaller than the uncorrected error $e$ precisely when $e<1/2$. The same repetition code also suppresses amplitude-damping noise in this protocol, something a generic 3-qubit repetition code cannot do; the paper attributes this to the structure of the QSS protocol.","core_discovery":"The discovery is that the structure of these secret-sharing protocols makes a 3-qubit repetition code act like a full quantum error-correcting code, even though a generic 3-qubit repetition code cannot correct amplitude-damping noise. In the first protocol, each qubit is prepared in a known basis and all parties apply single-qubit operations that either commute or anticommute, so the only relevant error on each transmission is either a bit flip or a phase flip depending on the basis; the sender and receivers therefore never need the combined bit-and-phase code. Encoding each state as three copies and decoding by majority vote gives corrected error $e_{\\text{QEC}}=3e^2(1-e)+e^3$, which is smaller than the raw error $e$ whenever $e<1/2$; for the 3-party bit-flip/phase-flip case the paper derives $27p^2+O(p^3)$, and for amplitude damping $3\\gamma^2/32+O(\\gamma^3)$, with the correction effective for all damping strengths in $(0,1)$. Simulations reported in the paper compare this scheme with the five-qubit perfect code, the smallest code that corrects an arbitrary single-qubit error, and the four-qubit approximate code optimized for amplitude damping, and find the repetition code has the lowest reconstructed-secret error, while the larger codes can even perform worse than no encoding because every physical qubit they use is exposed to noise. The same error reduction carries over to the quantum-information sharing protocol because the teleportation noise multiplies the secret-sharing fidelity and hence drops out of the comparison.","pith_inferences":["A direct extension not explored in the paper is to benchmark the same three-copy repetition code on other basis-aware protocols, such as quantum key distribution or secure direct communication, where the known-basis structure should permit the same overhead reduction.","The independence assumption behind $e_{\\text{QEC}}=3e^2(1-e)+e^3$ is the practical weak point: if three copies share a channel or a time slot, correlated noise can make two-copy failures more common than the binomial prediction, so deployments should interleave copies over independently fluctuating channels or randomize their order.","Because the repetition code is basis-dependent, the paper notes it cannot be used in the multi-cycle form applied to the four- and five-qubit codes; a testable question is whether a four-qubit basis-independent code could close that gap with less overhead than five qubits.","The reported comparison is in terms of average error; a worst-case robustness check would be whether majority voting still beats no encoding when an adversary can correlate flips across copies."],"forward_implications":["If the central claim is correct, the QSSCM and SSQI protocols can be run with one-third the qubit overhead of the full 9-qubit code while still correcting the dominant noise channels.","For bit-flip and phase-flip noise with per-channel error probability $p<1/2$, the corrected error rate is $O(p^2)$ (e.g., $27p^2+O(p^3)$ for three parties), so the protocols remain reliable at noise levels where unencoded transmission fails.","Under amplitude damping, the repetition code is claimed to reduce the reconstructed-secret error for every damping strength in $(0,1)$, which would make these protocols usable on energy-loss-dominated hardware such as superconducting qubits.","Because the scheme only assumes single-qubit transmissions with a known measurement basis, it can be applied to other single-qubit-based quantum protocols, including quantum key distribution (QKD), quantum secure direct communication, and quantum authentication, where the final output is a classical bit string.","The paper's comparison implies that simply using a larger perfect code is not automatically beneficial in this setting: if all physical qubits are exposed to noise, a five- or four-qubit code can raise the error above the no-encoding level, while the repetition code keeps it below."],"supporting_citations":[{"why":"Supplies the two multiparty secret-sharing protocols (QSSCM and SSQI) whose noise resilience is analyzed.","marker":"[11]"},{"why":"Supplies the 9-qubit code that the paper abridges into the 3-qubit repetition code.","marker":"[31]"},{"why":"Supplies the five-qubit perfect code used as the main comparison baseline.","marker":"[54]"},{"why":"Supplies the four-qubit approximate code used as the second comparison baseline.","marker":"[56]"},{"why":"Supplies the quantum singleton bound that motivates why correcting arbitrary errors normally costs at least five qubits.","marker":"[70]"},{"why":"Supplies the recovery Kraus operators used when implementing the five-qubit code in the comparison.","marker":"[71]"},{"why":"Supplies the standard teleportation step that the SSQI protocol builds on.","marker":"[77]"}],"fun_headline_variants":["3-qubit code outperforms 9-qubit for secret sharing","Error-correcting secret sharing with only 3 qubits","Shor's code cut to 3 qubits for quantum secret sharing","Noise-proof secret sharing needs just 3 qubits","Quantum secret sharing: 3-qubit error correction beats 9"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The three repeated copies of each message qubit are assumed to be corrupted independently by the channel, so a majority vote can outvote a single faulty copy; if the noise on the three copies is correlated, the claimed error reduction does not follow.","fun_headline_variants_meta":{"raw":{"variants":["3-qubit code outperforms 9-qubit for secret sharing","Error-correcting secret sharing with only 3 qubits","Shor's code cut to 3 qubits for quantum secret sharing","Noise-proof secret sharing needs just 3 qubits","Quantum secret sharing: 3-qubit error correction beats 9"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000415,"raw_usage":{"total_tokens":2197,"prompt_tokens":1056,"completion_tokens":1141,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":672,"completion_tokens_details":{"reasoning_tokens":1052}},"tokens_in":672,"tokens_out":1141,"duration_ms":8028,"temperature":1.0,"reasoning_tokens":1052,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:58:27.124143+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Transmit three copies of the same state through the same fluctuating channel and record the joint error pattern; if two or three copies fail together more often than $3e^2(1-e)+e^3$ predicts for the measured per-copy error rate $e$, the repetition code's advantage over no encoding disappears under correlated noise.","supporting_citations":[{"cited_title":"How to fairly share multiple secrets stage by stage,","cited_arxiv_id":null,"evidence_quote":"Supplies the two multiparty secret-sharing protocols (QSSCM and SSQI) whose noise resilience is analyzed."},{"cited_title":"Implementation of quantum secret sharing and quantum binary voting protocol in the ibm quantum computer,","cited_arxiv_id":null,"evidence_quote":"Supplies the 9-qubit code that the paper abridges into the 3-qubit repetition code."},{"cited_title":"A practical protocol for three-party authenticated quantum key distribution,","cited_arxiv_id":null,"evidence_quote":"Supplies the five-qubit perfect code used as the main comparison baseline."},{"cited_title":"Authenticated multi-user quantum key distribution with single particles,","cited_arxiv_id":null,"evidence_quote":"Supplies the four-qubit approximate code used as the second comparison baseline."},{"cited_title":"Ultrahigh error threshold for surface codes with biased noise,","cited_arxiv_id":null,"evidence_quote":"Supplies the quantum singleton bound that motivates why correcting arbitrary errors normally costs at least five qubits."},{"cited_title":"Fault-tolerant thresholds for the surface code in excess of 5% under biased noise,","cited_arxiv_id":null,"evidence_quote":"Supplies the recovery Kraus operators used when implementing the five-qubit code in the comparison."},{"cited_title":"Experimental repeti- tive quantum error correction,","cited_arxiv_id":null,"evidence_quote":"Supplies the standard teleportation step that the SSQI protocol builds on."}],"review_version":1}