{"id":"1224f970-af01-4c3b-a7c7-3a93baf8773a","arxiv_id":"2504.13356","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A tabu-search and symmetry-reduced optimization of CliNR verification sequences achieves about 25% lower logical error rates and demonstrable breakeven on a 36-qubit trapped-ion system.","lead":"The authors propose better ways to choose the error-checking measurements in CliNR, a lightweight error-suppression scheme for Clifford quantum circuits. Simulations show about 25% lower logical error rates than random checks, and a 36-qubit trapped-ion experiment shows the scheme performs at breakeven.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Two-step optimization's 64%-evaluation claim depends on an unvalidated proxy: Eq. (6) drops verification/injection faults, and the load-bearing comparison is made on one circuit.","rationale":"The reader's conditional verdict is well founded. The mathematical Proposition 1 and the Grassmann-graph reduction are sound as far as the text shows; those parts are not the problem. The risk is empirical: the proxy's adequacy is asserted but not demonstrated. I checked for independent support: there is no machine-checked proof, no released artifact, and the experiment section explicitly does not use the optimization, so the only support for the central efficiency claim is the Section 6 simulation on two circuits. The paper itself flags the proxy's limitation in Section 5.1 and the need for hardware-adapted cost functions in Section 7, so this is not a manufactured concern. It is also not an 'outside-consensus' issue; it is an internal consistency problem between the approximation used and the claim. I therefore agree with the reader's weakest-assumption identification. No reason to change the verdict: conditional acceptance with a demand for proxy validation and multi-circuit evidence. I would not reject because the construction is plausible and the reported magnitudes may survive such a test.","tokens_in":16239,"tokens_out":7097,"duration_ms":68123,"concrete_test":"Compute both Proxy(V) (Eq. 6) and plog(CliNR(CR,V)) by Monte-Carlo for a fixed n=20, r=4 circuit: draw ~200 verification sequences (e.g. 100 uniform random and 100 low-proxy candidates produced by Algorithm 2), using 5×10^5 shots per plog estimate. Measure rank correlation (Spearman or Kendall tau) and, separately, for the 10 lowest-proxy sequences, record their plog quantiles. If the rank correlation among low-proxy candidates is weak (tau < ~0.3) or the best-plog sequence is not in the low-proxy set, the proxy pre-optimization is unreliable. As a complementary check, add explicit first-order verification/injection fault terms to Proxy and rerun Algorithm 2/3; if the resulting subgroup's best plog worsens or the 180-evaluation comparison disappears, the omitted terms are load-bearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Algorithm 3 matches Algorithm 1's logical-error rate with 64% fewer plog evaluations. The mechanism is the proxy in Eq. (6): Proxy(V) sums first-order preparation errors that are undetected by V and nontrivial on the resource state. Section 5.1 explicitly limits this to the case of no fault during verification or injection. Verification and injection are implemented with real gates (ancilla preparation, controlled-Pauli chains, Bell measurements, feed-forward corrections), so in the ion-chain circuit-level model they carry their own depolarizing noise. Faults there can escape detection, create logical errors, or change which sequences pass; none of these are in Proxy(V). Because Algorithm 2 first selects a subgroup ⟨V⟩ by minimizing Proxy and only then refines inside that subgroup, a proxy ranking that diverges from plog traps the search away from the global optimum no matter how many refinement evaluations are spent. The paper does not report any direct comparison of Proxy(V) with plog for a sample of sequences, nor any sensitivity analysis to noise rate or circuit size. The 64% figure itself is derived from one n=20, r=4 circuit of size s=n^2 (plus one s=n^1.8 circuit), averaged over 250 optimization repeats of that same circuit; it may not generalize. No code or data are provided to test the proxy's fidelity. This is the load-bearing soft spot: if verification/injection faults dominate at realistic parameters, the two-step algorithm's advertised saving does not transfer to equal performance.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses the CliNR partial error correction scheme, which implements Clifford circuits by consuming a resource stabilizer state and measuring a short sequence of Pauli operators (verification sequence). The authors propose a tabu-search global optimizer (Algorithm 1) over verification sequences using Monte Carlo estimates of the logical error rate; a proxy cost function (Eq. (6)) that counts first-order preparation errors which are not detected by the verification sequence and are non-trivial on the resource state; a two-step optimizer (Algorithm 3) that first minimizes this proxy on the Grassmann graph of rank-r subgroups (Algorithm 2) and then refines the sequence inside the returned subgroup using the true logical error rate; and a proof (Proposition 1) that the proxy is invariant under GL_r(Z2), giving a 168-fold (r=3) and 20,160-fold (r=4) reduction of the proxy-search vertex set. Numerical simulations for n=20, r=4 under an ion-chain noise model for two random Clifford circuits report an additional 21-25% reduction in logical error rate over random verification, with the two-step algorithm reaching the global optimizer's level with about 180 instead of 500 logical-error-rate evaluations. Hardware experiments on a 36-qubit trapped-ion system implement CZNR and conclude breakeven performance.","tokens_in":16597,"tokens_out":15119,"duration_ms":134955,"significance":"If the numerical claims hold, the paper provides a practical way to substantially reduce the optimization cost of CliNR while preserving the achieved logical error rate, and the Grassmann-quotient observation is an elegant theoretical contribution. Proposition 1 is proved cleanly, the algorithms are fully specified, and final performance is scored by independent Monte Carlo estimates of the logical error rate rather than by the proxy, so there is no circularity in the evaluation. The hardware experiment, although without mid-circuit measurement, is a useful proof-of-concept with a clear post-processing protocol. The main limitations are empirical: the headline gains are demonstrated on one random circuit per size, the proxy is not validated against the true cost, and the experimental claims lack statistical quantification. These gaps mean the significance is currently qualified rather than fully established.","major_comments":[{"comment":"The headline claims of a 25% improvement and a 64% reduction in plog evaluations are derived from exactly two random Clifford circuits, one per size (s=n^1.8 and s=n^2). The 250 repetitions are all on the same circuit, so they quantify Monte Carlo and optimization-run variability, not variability across circuits. Please repeat the comparison over a sample of random circuits at n=20 and report the mean and spread; the abstract's general statement about '20-qubit Clifford circuits' is not supported by one circuit per size.","section":"Section 6, Fig. 2"},{"comment":"The efficiency gain of the two-step algorithm rests on the proxy's ability to rank verification subgroups similarly to plog. The paper itself notes that Proxy(V) is only a first-order approximation when no fault occurs during verification and injection, and no direct comparison of Proxy(V) with plog (e.g., rank correlation over a sample of sequences), no sensitivity analysis in p, tau_m, or circuit size, and no ablation isolating the proxy contribution is provided. Because Algorithm 2 commits to the proxy-minimizing subgroup before any plog refinement, a proxy that misranks subspaces would trap the search regardless of the refinement step. Please add such a validation or explicitly bound the regime where the proxy is reliable.","section":"Section 5.1, Eq. (6) and Section 5.4, Algorithm 3"},{"comment":"The experimental breakeven conclusion is drawn without error bars or statistical tests, from 2,048 shots per circuit. Visual equality between direct and CZNR and flatness of the logical error rate as r increases need binomial or bootstrap confidence intervals, especially because observation (iii) (removing stabilizers worsens the error rate) is used to rule out noise saturation. Please quantify the uncertainty associated with each plotted rate and the comparisons.","section":"Section 7, Figs. 3 and 4"}],"minor_comments":[{"comment":"The statement that there are 50,625 possible non-trivial verification sequences inside a rank-4 subgroup is arithmetically wrong: an ordered basis of a 4-dimensional subspace over F_2 has 15*14*12*8 = 20,160 elements (or 840 unordered bases), while 50,625 equals 15^4 and counts tuples with repetition and dependence. The conclusion that 500 evaluations is far short of exhaustive search remains true, but the number should be corrected.","section":"Section 6"},{"comment":"The tabu list stores subgroups ⟨V⟩, but line 10 tests whether the tuple V is not in Ltabu; it should test whether ⟨V⟩ is not in Ltabu.","section":"Section 5.3, Algorithm 2"},{"comment":"The statement that the results 'reflect circuit and device performance as is with no additional post-processing, filtering, or error-mitigation applied' is contradicted by Appendix B.3, which describes discarding failed shots, applying Pauli corrections, and inferring restart rates. Please rephrase to state that post-processing is limited to the described verification-conditioning and Pauli-frame update.","section":"Section 7"},{"comment":"The '25% improvement' is relative to CliNR with random verification, not to direct implementation; the abstract should specify this baseline to avoid overstatement.","section":"Abstract and Section 6"},{"comment":"The phrase 'the search graph can then be replaced by the Grassmann graph' should be qualified as applying to the proxy-optimization step only, since the actual cost plog is not invariant under GL_r(Z2) and the refinement step still searches ordered sequences.","section":"Section 5.5"},{"comment":"The paper contains no code or data availability statement; releasing the simulation and experimental data would materially help reproducibility of the numerical and hardware claims.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The theoretical core (Proposition 1 and the Grassmann-quotient argument) is sound, and I see no circularity in the evaluation methodology. My main concern is empirical breadth: the 25%/64% claims rely on one circuit per size and an unvalidated proxy. I recommend major revision rather than rejection because the required changes are additional validation experiments and analyses, not a conceptual overhaul. The paper is within the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things worth knowing about this paper. First, the Grassmann-graph symmetry reduction is a genuine contribution: the proxy in Eq. (6) is exactly invariant under GL_r(Z2), so quotienting the search space is costless and the 168x/20,160x reductions are real. Second, the headline numbers — 25% better than random verification, and the two-step search matching full tabu search with 64% fewer plog evaluations — rest on a thin base: two random 20-qubit circuits, essentially one for the 64% figure, and a proxy whose fidelity is never directly checked against the true logical error rate.\n\nThe paper does a lot right. The algorithms are fully specified and simple to implement. Proposition 1 is correct, and the proof is clear. The proxy is a sensible first-order approximation: it counts preparation errors that are undetected by the verification sequence and non-trivial on the resource. The two-step idea — cheaply narrow to a subspace, then refine with the expensive cost — is reasonable and well motivated. The hardware experiment is honest: they report breakeven, not success, and the noise-saturation control (observation iii) is a good check.\n\nThe soft spots are empirical, not logical. The proxy explicitly ignores faults during verification and injection; Section 5.1 says so. If those faults dominate at realistic parameters, the proxy's ranking could be far from plog, and the two-step algorithm would be trapped in a bad subspace. The paper doesn't report a proxy-vs-plog correlation or a sensitivity analysis, so the 64% saving is an observation on one circuit, not a demonstrated general fact. The hardware data have no error bars; breakeven is supported by point estimates and a saturation test, which is suggestive but not conclusive. There's also a minor inconsistency between the text saying they tested both low- and high-weight stabilizers and Fig. 3 showing a single CZNR curve per r. And no code or data are released, which makes it hard to check the proxy fidelity or reproduce the numbers.\n\nNone of this undermines the core contribution. The symmetry reduction is real, the heuristic is useful, and the writing is careful about what is proven versus what is observed. The paper is a solid engineering contribution, not a deep theoretical one.\n\nWho should read it: anyone implementing CliNR or similar stabilizer-verification schemes, and anyone interested in using group symmetries to shrink combinatorial search in error correction. It deserves peer review; I'd send it out. For the final version, I'd ask for a direct proxy-vs-plog comparison, more circuits, error bars on the hardware points, and a code/data release. But this is revision material, not grounds for rejection.","headline":"A useful engineering advance for CliNR optimization with a solid symmetry reduction, but the headline 25%/64% numbers rest on thin empirical evidence and an unvalidated proxy.","tokens_in":17062,"tokens_out":6587,"would_cite":true,"duration_ms":56196,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68"],"pacs":[],"model":"deepseek-v4-flash","headline":"A cheap proxy cost function, minimized over the Grassmann graph, finds CliNR verification sequences with the same logical error rate as full tabu search while using about 64% fewer evaluations, and hardware experiments put CZNR at…","keywords":["CliNR","partial error correction","verification sequence","Grassmann graph","tabu search","logical error rate","trapped-ion quantum computer","Clifford circuits"],"falsifier":"Increase the noise in verification measurements and injection teleportation relative to preparation noise in the ion-chain simulation, then run both algorithms with equal evaluation budgets; if the two-step sequence's logical error rate is clearly worse than the global search's, the proxy ranking is the weak link.","tokens_in":16046,"feed_emoji":"⚛️","tokens_out":10837,"duration_ms":92493,"temperature":0.7,"pith_summary":"Partial error correction schemes consume a resource state to implement a circuit and rely on a short verification sequence of stabilizer measurements to catch preparation errors before they propagate. This paper's central claim is that the verification sequence itself can be optimized, and that the optimization can be done cheaply: a proxy cost function, minimized over a search space reduced by symmetry to the Grassmann graph, finds verification sequences whose logical error rate matches those found by a global tabu search while using about 64% fewer evaluations of the true logical error rate (roughly 180 instead of 500). In numerical simulations of 20-qubit Clifford circuits under the ion-chain noise model, the optimized sequences give an additional roughly 25% reduction in logical error rate over random verification, and for somewhat shorter circuits they move CliNR from above breakeven to below breakeven. The paper also reports a 36-qubit trapped-ion experiment, without mid-circuit measurement, in which the CZNR variant of CliNR reaches breakeven performance. The practical stake is that near-term users could get the full error suppression of CliNR without paying the full search cost, and that the symmetry-reduction method may generalize to other stabilizer-measurement search problems.","feed_headline":"Two-step search matches CliNR optimization with 64% fewer evaluations","feed_subtitle":"CliNR is a low-overhead error-reduction scheme for Clifford circuits; symmetry shrinks its search space up to 20,160-fold.","key_machinery":"The load-bearing object is the verification sequence $V=(V_1,\\dots,V_r)$, an ordered tuple of independent stabilizers of the resource state, together with the proxy cost function $\\operatorname{Proxy}(V)=\\sum_i\\sum_{P\\in\\Omega_i}\\tilde{p}_i\\,\\delta\\{P\\in V^\\perp\\setminus S^\\perp\\}$, which counts preparation errors that the sequence neither detects nor absorbs. The argument is carried by the group action $\\alpha(M,V)=MV$ of $\\operatorname{GL}_r(\\mathbb{Z}_2)$ on the search space: it preserves the proxy because it preserves the commuting subspace $V^\\perp$, and quotienting by it identifies exactly the rank-$r$ subgroups of the stabilizer group, whose incidence graph is the Grassmann graph (vertices are rank-$r$ subgroups, adjacent when their intersection has rank $r-1$). Working on that quotient shrinks the search space by a factor of 168 for $r=3$ and 20,160 for $r=4$, and the two-step algorithm uses the quotient to find a good subgroup cheaply before re-introducing the true logical error rate inside the subgroup.","core_discovery":"On the authors' own terms, the discovery is that the CliNR optimization problem has exploitable structure rather than requiring brute-force enumeration of verification sequences. The paper defines a proxy $\\operatorname{Proxy}(V)=\\sum_i\\sum_{P\\in\\Omega_i}\\tilde{p}_i\\,\\delta\\{P\\in V^\\perp\\setminus S^\\perp\\}$, which sums, over all error locations in resource-state preparation, the probability of a Pauli error that commutes with every measured stabilizer and is not in the resource stabilizer group; this is a first-order estimate of the logical error rate when verification and injection are noiseless. Proposition 1 shows this proxy is invariant under the natural action of $\\operatorname{GL}_r(\\mathbb{Z}_2)$ on ordered stabilizer tuples, and that identifying sequences in the same orbit yields the vertex set of the Grassmann graph. Algorithm 2 performs tabu search on that reduced graph using the proxy, and Algorithm 3 refines the returned subgroup with the true logical error rate. The quantitative claim is that for 20-qubit Clifford circuits of size $n^2$ under the ion-chain model, the two-step procedure matches the logical error rate of global tabu search using about 180 rather than 500 evaluations of $\\operatorname{p}_{\\log}$, while optimized sequences add roughly 25% error suppression over random verification. The experimental claim is that CZNR on a 36-qubit trapped-ion machine without mid-circuit measurements is at breakeven.","pith_inferences":["If the omitted verification and injection faults dominate the noise budget, the proxy's ranking will drift from the true logical error rate; folding those faults into the proxy is the natural first extension and would preserve the Grassmann-graph quotient only if the added terms respect the same symmetry.","The efficiency gain only pays off when the noise model matches the device; the hardware experiment uses unoptimized sequences, so a hardware-calibrated version of the two-step algorithm is the direct next benchmark rather than a claim of this paper.","The breakeven experiment kept ancillas idle because the machine lacks mid-circuit measurement; the reported restart-rate data suggests that adding mid-circuit restart would reduce both idle time and accepted-shot overhead, likely pushing CZNR below the breakeven threshold before gate fidelities improve."],"forward_implications":["The same logical error rate that global search reaches after 500 evaluations of the true cost is attainable with about 180 evaluations, so optimization on real hardware, where each evaluation is expensive, becomes practical.","For 20-qubit Clifford circuits of size $n^2$ under the ion-chain model, optimized verification sequences add roughly 25% error suppression over random verification; for size $n^{1.8}$, optimization moves CliNR from no improvement to roughly 17\\text{--}20% improvement over direct implementation.","The $\\operatorname{GL}_r(\\mathbb{Z}_2)$ symmetry shrinks the candidate count by factors of 168 for $r=3$ and 20,160 for $r=4$, making longer verification sequences or larger circuits feasible within the same search budget.","Because the quotient construction depends only on the stabilizer structure, the same symmetry reduction can be applied to other stabilizer-measurement search problems, such as choosing syndrome-measurement sequences in quantum error correction."],"supporting_citations":[{"why":"Defines the CliNR scheme and its CZNR variant, supplying the random-verification baseline and the resource-state framework that the optimization improves.","marker":"[9]"},{"why":"Supplies the ion-chain circuit-level noise model used in the numerical simulations.","marker":"[25]"},{"why":"Provides the Grassmann manifold and Grassmann graph background on which the quotient search space and Proposition 1 are built.","marker":"[2]"},{"why":"Introduces tabu search, the metaheuristic used in Algorithms 1 and 2.","marker":"[11, 12]"},{"why":"Provides the efficient Pauli-propagation method used to compute output errors and to precompute the error sets $\\Omega_i$ for the proxy.","marker":"[14]"},{"why":"Provides the stabilizer-circuit simulator used to propagate errors and to estimate logical error rates in simulation.","marker":"[10]"},{"why":"Provides the compilation of Haar-random Clifford circuits into the {H, S, S†, CX} gateset used for the simulation circuits.","marker":"[4]"}],"fun_headline_variants":["Symmetry shrinks CliNR search space 20,160-fold","CliNR two-step optimizer: same results, 64% fewer checks","Proxy-guided tabu search boosts CliNR by 25%","Ion-trap CZNR at breakeven, no mid-circuit reads"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The optimization shortcut works only if the cheap scoring rule, which counts only errors made while preparing the resource state, orders candidate verification sequences almost as well as the true failure rate would.","fun_headline_variants_meta":{"raw":{"variants":["Symmetry shrinks CliNR search space 20,160-fold","CliNR two-step optimizer: same results, 64% fewer checks","Proxy-guided tabu search boosts CliNR by 25%","Ion-trap CZNR at breakeven, no mid-circuit reads"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000791,"raw_usage":{"total_tokens":3570,"prompt_tokens":1116,"completion_tokens":2454,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":732,"completion_tokens_details":{"reasoning_tokens":2374}},"tokens_in":732,"tokens_out":2454,"duration_ms":16853,"temperature":1.0,"reasoning_tokens":2374,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:11:01.446229+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Increase the noise in verification measurements and injection teleportation relative to preparation noise in the ion-chain simulation, then run both algorithms with equal evaluation budgets; if the two-step sequence's logical error rate is clearly worse than the global search's, the proxy ranking is the weak link.","supporting_citations":[],"review_version":1}