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REVIEW 4 major objections 4 minor 88 references

Resource Reduction in Multiparty Quantum Secret Sharing of both Classical and Quantum Information under Noisy Scenario

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2504.16709 v1 pith:HIWRWDZX submitted 2025-04-23 quant-ph cs.CR

Nirupam Basak , Goutam Paul This is my paper · ORCID
classification quant-phcs.CR PACS 03.67.Dd03.67.Pp
keywords quantumsecretsharingmultipartyrepetitioncodeerrorcorrectionbit-flipnoisephase-flipamplitudedampingqubitoverhead
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

4 major / 4 minor

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.

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 (4)
  1. [V.B.b, Eq. (22)] 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.
  2. [V.A–V.C, Figs. 5 and 6] 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.
  3. [III.B.b and Fig. 5] 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.
  4. [V.B, Eq. (16)] 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.
minor comments (4)
  1. [III.A.b, Eq. (3)] The phase-flip channel is written with C_b and p_b; it should be C_p and p_p.
  2. [Figs. 3–6] 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.
  3. [V.A] 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.
  4. [VI, Eq. (24)] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the error-rate comparison derives from first-principles binomial counting and externally defined benchmark codes, not from the paper's own fitted values or self-citations.

full rationale

The central derivation is self-contained. The no-encoding error formulas (Eqs. (11), (13), and (15)) are obtained by direct binomial or Kraus counting over the protocol channels, and the repetition-code formulas (Eqs. (16)-(22)) are the standard majority-vote binomial error expression applied to those per-qubit errors; no parameter is fitted to the quantities being predicted. The five-qubit and four-qubit benchmark codes are taken from published external sources (Laflamme et al. [54] and Leung et al. [56]) and are compared through simulation, not derived from the present authors' own results. The only self-citations, [20] and [59], occur in background lists on QSS security and QEC families and do not support any load-bearing step. Thus there is no self-definitional, fitted-prediction, or self-citation-driven circularity. Possible arithmetic discrepancies, such as the coefficient in Eq. (22), and the single-cycle-versus-multi-cycle comparison in Figs. 5-6 would be correctness concerns rather than circularity.

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

No free parameters are fitted; channel strengths p and gamma are inputs. The analysis relies on error independence across copies, known preparation basis, and a mathematically questionable averaging step in the amplitude-damping majority-vote calculation. No new entities are introduced.

assumptions (4)
  • domain assumption The three copies of each logical qubit undergo statistically independent noise with the same channel parameters.
    Eq (16) applies the binomial majority-vote formula, which requires independence across copies; the paper never states or justifies this.
  • standard math The unitary operations I, sigma_y, and H commute or anticommute so their order can be rearranged up to a global phase.
    Used in eq (1) and eq (9) to place all noise before the unitaries.
  • domain assumption The preparation basis of each transmitted qubit is known or recoverable at the decoding stage, so bit-flip and phase-flip correction can be separated.
    Section V A splits Shor's code into two 3-qubit repetition codes depending on whether the state is in the computational or Hadamard basis; this basis knowledge is assumed.
  • ad hoc to paper Inserting the average single-copy error into the majority-vote formula gives the correct average majority-vote error.
    Eq (22) computes 3(ea1)^2(1-ea1)+(ea1)^3 using the average ea1 from eq (13); this averaging of a convex function is not generally valid and is the source of the numerical error.

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Cite this review

Pith. "Pith review of Resource Reduction in Multiparty Quantum Secret Sharing of both Classical and Quantum Information under Noisy Scenario." pith.science (2026). https://pith.science/paper/HIWRWDZX

@misc{pith2026250416709,
  author       = {Pith},
  title        = {Pith review of: Resource Reduction in Multiparty Quantum Secret Sharing of both Classical and Quantum Information under Noisy Scenario},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HIWRWDZX}},
  note         = {Machine review of arXiv:2504.16709}
}
read the original abstract

Quantum secret sharing (QSS) enables secure distribution of information among multiple parties but remains vulnerable to noise. We analyze the effects of bit-flip, phase-flip, and amplitude damping noise on the multiparty QSS for classical message (QSSCM) and secret sharing of quantum information (SSQI) protocols proposed by Zhang et al. (Phys. Rev. A, 71:044301, 2005). To scale down these effects, we introduce an efficient quantum error correction (QEC) scheme based on a simplified version of Shor's code. Leveraging the specific structure of the QSS protocols, we reduce the qubit overhead from the standard 9 of Shor's code to as few as 3 while still achieving lower average error rates than existing QEC methods. Thus, our approach can also be adopted for other single-qubit-based quantum protocols. Simulations demonstrate that our approach significantly enhances the protocols' resilience, improving their practicality for real-world deployment.

Figures

Figures reproduced from arXiv: 2504.16709 by the authors.

Figure 1
Figure 1. FIG. 1. Error on secret [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Error on reconstructed secret is plotted against channel error probability [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Error on reconstructed secret is plotted against damping probability [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figures from the paper (2 more)
Figure 6
Figure 6. Figure 6: FIG. 6. Plots show the simulated error of the secret as a [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Plots show the simulated errors on reconstructed se [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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