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REVIEW 3 major objections 6 minor 26 references

Optimized Clifford Noise Reduction: Theory, Simulations and Experiments

T0 review · 3 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2504.13356 v2 pith:MJ64ESQJ submitted 2025-04-17 quant-ph

Edwin Tham , Nicolas Delfosse This is my paper · ORCID
classification quant-ph MSC 81P68
keywords CliNRpartialerrorcorrectionverificationsequenceGrassmanngraphtabusearchlogicalratetrapped-ionquantumcomputerCliffordcircuits
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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 / 6 minor

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.

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 (3)
  1. [Section 6, Fig. 2] 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.
  2. [Section 5.1, Eq. (6) and Section 5.4, Algorithm 3] 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.
  3. [Section 7, Figs. 3 and 4] 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.
minor comments (6)
  1. [Section 6] 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.
  2. [Section 5.3, Algorithm 2] 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.
  3. [Section 7] 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.
  4. [Abstract and Section 6] The '25% improvement' is relative to CliNR with random verification, not to direct implementation; the abstract should specify this baseline to avoid overstatement.
  5. [Section 5.5] 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.
  6. [General] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proxy optimization is scored by independent Monte Carlo logical error rates, not by the proxy itself.

full rationale

The paper's load-bearing comparison is between optimizers that minimize plog directly (Algorithm 1) and a two-step procedure (Algorithm 3) that first minimizes the proxy Proxy(V) from Eq. (6) and then refines the resulting subgroup using the actual logical error rate. The final numerical results in Section 6 report plog estimated with 50,000 Monte-Carlo shots per evaluation, so the claimed 25% improvement and 64% reduction in plog evaluations are scored by an independent stochastic simulator, not by the proxy. Proxy(V) is defined directly from the circuit-level depolarizing noise parameters and Pauli commutation relations; it contains no parameter fitted to the plog data it is meant to approximate. The GL_r(Z2) invariance in Proposition 1 is not smuggled in: the proof shows Proxy(V) depends only on V^perp, so the quotient to the Grassmann graph is exact and the 168x/20,160x search-space reductions are a mathematical consequence rather than an imported assumption. The self-citations to [9] and [25] provide the CliNR baseline and the ion-chain noise model respectively, but neither is used to force the paper's improvement claims; the optimization gains are measured against an independently simulated logical error rate. The skeptic concern that Eq. (6) omits faults during verification and injection is a legitimate correctness/robustness question about whether the proxy ranks sequences faithfully, but it is not circularity because the proxy is not the metric being reported as the final result. No derivation step reduces by construction to its own inputs or to a fitted value passed off as a prediction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central results depend on a small number of explicit modeling choices: the ion chain noise model (including p=1e-4 and tau_m=30), the proxy cost function that neglects verification and injection faults, and the representativeness of the two sampled Clifford circuits. None of these is fitted to the target result, but all are hand-chosen and only weakly validated, which limits how far the quantitative claims generalize.

free parameters (4)
  • two-qubit depolarizing rate p = 1e-4
    Input circuit-level noise rate used in all simulations; only one value is tested, and the reported improvement percentages could depend on it.
  • single-qubit noise rate ratio = p/10 = 1e-5
    Ratio of single-qubit to two-qubit noise in the ion chain model; chosen by hand.
  • idle noise rate ratio = p/100 = 1e-6
    Ratio of idle to two-qubit noise in the ion chain model; chosen by hand.
  • measurement idle noise factor tau_m = 30
    Multiplier for idle noise during measurement in the ion chain model; set to 30 by hand.
assumptions (5)
  • standard math The stabilizer formalism and Gottesman-Knill propagation correctly compute output error distributions for Clifford circuits.
    Used throughout Section 2 and to define the proxy via propagated error sets Omega_i.
  • domain assumption The circuit-level depolarizing noise model is an adequate description of noise for the ion chain model simulations.
    Section 2 and Section 6; all numerical results use this model with p=1e-4.
  • domain assumption The ion chain noise model [25] is an appropriate model for long chains of trapped ions and transfers to the optimization results.
    Section 6 states simulations use the ion chain model from ref [25], co-authored by Delfosse; self-cited but an explicit model.
  • ad hoc to paper The proxy cost Proxy(V) in Eq. (6), which neglects faults during verification and injection, ranks verification sequences similarly to the true logical error rate for the purpose of optimization.
    Section 5.1 defines the proxy as a first-order approximation only when no fault occurs during verification/injection; Algorithm 3 relies on it to guide the search.
  • domain assumption Two randomly chosen Clifford circuits (one per size s=n^1.8 and s=n^2) are representative of general Clifford circuits for assessing optimization gains.
    Section 6 reports results for two circuits only; the abstract generalizes the improvement to CliNR performance.

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Pith. "Pith review of Optimized Clifford Noise Reduction: Theory, Simulations and Experiments." pith.science (2026). https://pith.science/paper/MJ64ESQJ

@misc{pith2026250413356,
  author       = {Pith},
  title        = {Pith review of: Optimized Clifford Noise Reduction: Theory, Simulations and Experiments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MJ64ESQJ}},
  note         = {Machine review of arXiv:2504.13356}
}
abstract

We propose several optimizations of the CliNR partial error correction scheme which implements Clifford circuits by consuming a resource state. Errors are corrected by measuring a sequence of Pauli operators that we refer to as the verification sequence. We first propose a global optimization algorithm searching for a verification sequence resulting in a low logical error rate using tabu search. Then, we introduce a proxy for the logical error rate which is easier to evaluate and we design a two-step optimization algorithm. First, a verification sequence minimizing the proxy is computed, then this sequence is refined by reintroducing the logical error rate. Finally, we identify a large group of automorphisms of the search space which preserve the proxy and we use this symmetry to reduce the size of the search space. This results in a 168 $\times$ (respectively 20,160 $\times$) reduction of the size of the search space for the optimization of verification sequences with three (respectively four) Pauli operators. Our numerical simulations for 20-qubit Clifford circuits with size 400 under the ion chain model show that our optimization algorithms improve the performance of CliNR by 25% and that the two-step optimization achieves the same results as the global optimization with 64% fewer evaluations of the logical error rate. Finally, we perform experiments on a 36-qubit trapped ion quantum computer, without mid-circuit measurements, showing that the CZNR variant of CliNR is at breakeven.

Figures

Figures reproduced from arXiv: 2504.13356 by the authors.

Figure 1
Figure 1. The CliNR implementation of the circuit C = H1CX1,2CZ1,3. The operation Q is a Pauli operator that depends on the measurement outcomes. The circuit C appears in (a). The resource state is prepared in (b) by applying the circuit CR = I ⊗ C to n Bell states. This CliNR implementation uses the verification sequence V = (XIIZII, IZIZZI). The measurement outcome o1 is the outcome of V1 = XIIZII and o2 is the outcome of V… view at source ↗
Figure 2
Figure 2. Comparison of the direct implementation, the CliNR implementation with a uniform random verification sequence [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 4
Figure 4. Comparison of the logical error rate of the CliNR [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: Inferred restart rates (see Appendix B.3) for CZNR circuits. R (1) res and R (2) res is non-trivial. A comparable implemen￾tation of CliNR in the presence of mid-circuit measure￾ment and restart will yield much higher circuit through￾put, with an effective restart rate…

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