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REVIEW 3 major objections 4 minor 89 references

This paper shows that jointly measuring commuting logical operators in quantum LDPC codes with a classical scheduler code cuts cat-state cost per measurement by up to 3x and Clifford-circuit cost by up to 74x in simulation.

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2026-08-01 21:10 UTC pith:JCILLMWP

load-bearing objection Genuinely new scheduler-code framework for joint cat-based logical measurements, but the headline ~3x measurement speed-up is computed with unconstrained classical codes and may not survive the walking cat's weight-<=30 accessibility limit; the CliNR speed-ups are on firmer ground. the 3 major comments →

arxiv 2607.16166 v2 pith:JCILLMWP submitted 2026-07-17 quant-ph

Fast logical operations in quantum LDPC codes using simple resource states

Mark Webster , Nicolas Delfosse This is my paper
classification quant-ph MSC 81P7094B05 PACS 03.67.Pp03.67.Lx
keywords quantum LDPC codeslogical measurementcat statesscheduler codejoint measurementCliNRToffoli gatewalking cat architecture
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish that the main bottleneck of quantum LDPC codes—accessing many logical qubits in the same block—can be removed by measuring many commuting logical operators at once using only the simplest resource state, the cat state. The key innovation is a classical 'scheduler code' whose columns specify which products of logical Paulis to measure, allowing all outcomes to be decoded together. In simulations on the walking-cat codes Q70 and Q102, the protocol measures 20 commuting logical operators in 1.71 cat states per operator versus 5.06 for the Viterbi protocol, a 2.96x reduction. Combined with a logical-level CliNR scheme, random Clifford circuits on Q102 need 81.4 cat states instead of 6056.3, a 74.4x speed-up, and Toffoli gates gain 4–5x. The paper argues these gains follow directly from the distance properties of the scheduler code and from executing CliNR resource-state preparation at the logical level while keeping injection physical.

Core claim

The central discovery is that joint measurement of ℓ commuting logical operators in an LDPC code can be reduced to decoding a classical binary linear code. A measurement schedule is an ℓ×m generator matrix G: column j specifies which product of target Paulis to measure with a cat state, and the rows relate logical outcomes u to measured outcomes v by v = uG. The protocol selects G so that the undetectable error rate (MEDM, an error-detected restart scheme) or the logical error rate (MECM, an error-corrected scheme) falls below a target ε with the fewest average cat measurements; truncated versions stop as soon as the posterior probability of the most likely outcome exceeds 1−ε. Using best-kn

What carries the argument

The scheduler code: a binary linear code given by an ℓ×m generator matrix G, where each column is the binary vector of the product of target logical Paulis to measure in one cat-based step and the first ℓ columns form an information set so that u = vG⁻¹ in the noiseless case. The code's weight enumerators and minimum distance determine how many extra measurements are needed to reach a target logical error rate; in the truncated protocols, the prefix codes G|i set the early-termination threshold via their undetectable error rates, while the full code sets the final error floor. Concrete speed-ups come from choosing G0 from best-known-distance tables to meet the undetectable-error condition an

Load-bearing premise

The scheduler matrices are taken from the best-known-distance classical code tables, but the paper does not verify that every product of the target logical Paulis appearing in those schedules has a representative of weight at most 30 in Q70/Q102, the maximum cat-state size in the walking cat architecture; if the accessible subcodes have worse parameters, the 2.8–3.0x measurement speed-ups shrink.

What would settle it

Enumerate all column products of the ℓ=20 scheduler matrix as logical Paulis in Q102 and compute their minimum-weight representatives: if any column product has weight >30, the schedule is not executable under the walking cat architecture's accessibility limit, directly shrinking the claimed speed-up. For the CliNR claim, re-run the random Clifford benchmark with a full physical error model that includes cat-missing errors and correlated measurement flips; if the 74x reduction drops below, say, 10x, the speed-up is not robust to realistic noise.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Measuring ℓ=20 commuting logical operators costs 1.71–1.81 cat states per operator instead of 5.06, at flip rate 6.3e-3 and target error 1e-10.
  • Random Clifford circuits on Q102 run with 81.4 cat states using MEDM-based logical CliNR versus 6056.3 for gate-based Viterbi synthesis, a 74.4x reduction.
  • Logical Toffoli gates (via Clifford+T decomposition) gain 4.2–5.0x on Q70 and Q102.
  • The same scheduler-code framework applies to stabilizer measurements, yielding a fast protocol for preparing stabilizer states by measuring their generators.
  • Because only cat states and transversal physical CNOTs are required, the protocols suit moving-qubit platforms such as trapped ions, neutral atoms, and spin qubits.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The scheduler-code reduction suggests a general design rule: any classical code with sufficiently good distance properties can be turned into a logical measurement schedule, so exploring code families beyond best-known-distance tables (e.g., LDPC or product codes) could yield even smaller schedules for large ℓ.
  • A direct experimental test would be to implement the paper's ℓ=4 example schedule on Q70 and compare the measured logical error rate and average attempt count with the predicted 11.5 cat measurements.
  • If the independence-of-flips assumption (rate p_F) holds only approximately for codes without single-shot properties, the protocol still works but with a modified effective flip rate; measuring actual flip correlations in walking-cat hardware would calibrate the speed-ups.
  • The CliNR gain hints at a broader principle: moving resource-state preparation to the logical level while keeping injection physical can convert partial error correction from overhead into speed-up, potentially applying to other resource states beyond stabilizer states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper introduces protocols for joint cat-state-based measurement of l commuting logical Pauli operators in quantum LDPC codes. The key idea is a classical binary linear 'scheduler code' G whose columns specify which products of the target Paulis are measured. Two protocols are analyzed: MEDM (error-detected measurement, with restart) and MECM (error-corrected measurement, with decoding). The authors derive exact expressions for average cat-state counts for MEDM, use weight enumerators and lookup decoding for MECM for small l, and select scheduler codes from Grassl's best-known-distance tables. They report a ~2.8x speed-up (MEDM) and ~2.96x speed-up (MECM) over single-Pauli Viterbi measurements for l=20 at p_F=6.3e-3 and epsilon=1e-10. They then apply these fast measurements to a variant of the CliNR scheme, reporting up to 74.4x speed-up for random Clifford circuits on Q102 and 4-5x for Toffoli circuits. The appendices contain proofs, simulation details, and an algorithm for accommodating logical-Pauli accessibility constraints.

Significance. If the quantitative claims hold, this is a practically useful step toward reducing the overhead of logical operations in high-rate LDPC architectures. The paper's framework is transparent and largely parameter-free: the MEDM counts come from closed-form expressions with exact weight enumerators, the l=1 limit reproduces the results of [17], and the scheduler-code construction is a natural and reusable idea. The CliNR application is also creative. However, the headline measurement speed-ups are computed for abstract best-known-distance classical codes, not for the accessible logical subcodes of Q70/Q102; the MECM numbers for l>=14 are heuristic; and the CliNR injection step leaves physical measurement errors unmodeled. These points are load-bearing for the stated claims and need to be addressed before the numbers can be accepted as demonstrated. The core protocol concept is sound and the issues are fixable, so the paper merits a major revision rather than rejection.

major comments (3)
  1. [Sec. III D, App. C 4, Tables II-III] The scheduler matrices used in Tables II and III are selected as best-known-distance binary linear codes from Grassl's tables, with no check that each column of G can be realized as an accessible logical Pauli in Q70/Q102. Section A 2 restricts cat-based measurements to logical operators with minimum-weight representative <=30, and states that Q102 has non-CSS logicals of minimum weight 30; products of l=20 commuting logical Paulis will in general exceed this bound. Appendix C 4 describes an evolutionary search over accessible columns but reports no resulting matrices or speed-ups. Consequently, the abstract's claim of a 'speed-up of nearly 3x' from 'numerical simulations with Q70 and Q102' is not supported by the simulations as presented: those simulations use abstract binary codes, not the architecture's accessible subcodes. Please either supply accessible scheduler matrices for Q70/Q1
  2. [App. C 1, Table III] For l>=14, the paper states that mAvg is estimated by sampling at most 4096 errors of each weight and that the LER is not verified explicitly. The l=20 row of Table III (mAvg=34.2, Viterbi/MECM=2.96x) therefore rests on a heuristic estimate, not an exact simulation. Equation (4) provides a rigorous upper bound for the LER only if d is the exact distance and all errors up to (d-1)/2 are correctable, but the reported mAvg is an approximation with no quantified error bar or independent confirmation. Since this row is the headline measurement speed-up, please provide a certified or exact calculation of mAvg for the [82,20,19] scheduler code, or at minimum a sensitivity analysis showing that the ratio 2.96x is robust to the sampling heuristic.
  3. [Sec. IV B, App. D 1] The injection step (RSI) in logical CliNR uses destructive physical X-basis and Z-basis measurements on the data and auxiliary blocks to infer the logical outcomes x_i and z_i. The paper's noise model only includes cat-based measurement flips at rate p_F; no allowance is made for errors in these destructive physical measurements. A single wrong outcome in x_i or z_i would lead to an incorrect Pauli correction Q and hence a logical error on the output. At target epsilon=1e-10, physical measurement errors at the ~1e-3 level are non-negligible. The reported CliNR speed-ups (18.5x and 74.4x) count only cat-based measurements, so the comparison may be optimistic. Please either include physical measurement errors in the model or explicitly state and justify an assumption that these measurements are noiseless.
minor comments (4)
  1. [Eq. (4)] The text says 'assuming all errors of weight (d-1)/2 are correctable' and writes LER = 1 - sum_{wt(e)<=(d-1)/2} P(e). For a code of distance d, all errors of weight at most (d-1)/2 are correctable, so the right-hand side is an upper bound on LER, not the exact LER. Please clarify this to avoid implying equality.
  2. [Table III caption] The final column header reads 'Viterbi/MEDM' but the table reports MECM results; it should be 'Viterbi/MECM'.
  3. [Abstract] Typo: 'Quantum LPDC codes' should be 'quantum LDPC codes'.
  4. [Sec. III D / App. C 4] If accessible scheduler matrices are found, it would be helpful to include them in an ancillary file or appendix, since the current Tables II and III only give aggregate length/distance/count data, which limits reproducibility.

Circularity Check

0 steps flagged

No significant circularity: scheduler codes are external Grassl-table inputs and the flip-rate model is inherited from [17]; the speed-up ratios are computed from the stated formulas, not fitted.

full rationale

After walking the derivation chain, I find no circular step. The MEDM/MECM protocols take as inputs the flip rate p_F = 6.3e-3 (from [17]'s numerical model), the target logical error rate ε = 1e-10, and classical scheduler codes selected from Grassl's tables [47]. The reported m_Avg values and speed-up ratios are then computed from the paper's own formulas (Eqs. 1-4, B1-B5), not fitted to produce the claimed numbers. The same p_F and ε are used for the Viterbi baseline and the new protocols, so the ratios are not forced by a parameter adjusted after the fact. The ℓ=1 repetition-code example in Appendix C2 exactly reproduces the EDM/ECM/Viterbi results of [17], providing an external cross-check. The paper does rely heavily on same-group prior work ([17], [38,39], [52,57]), but that reliance is load-bearing only as external numerical/algorithmic input, not as an unverified self-citation chain: the flip-rate independence assumption is explicitly supported by numerical verification in [17], and the Q70/Q102 codes are fixed architecture inputs. The skeptic's accessibility concern — that Tables II–III use unconstrained Grassl scheduler codes without verifying that every column product has a weight ≤30 representative in Q70/Q102 — is a real realizability/correctness risk, and the paper itself flags it and gives only a partial adaptation in Appendix C4. But that is a missing verification, not a definitional or fitted-input circularity. No step in the derivation reduces to its own input by construction.

Axiom & Free-Parameter Ledger

4 free parameters · 7 axioms · 1 invented entities

The protocol's quantitative claims rest on five inputs: the flip-rate model pF=C1*w_bar*p (C1 fitted in [17]), the target error rate, the maximum cat size, the external classical code tables, and the sampling budget for large-l MECM. The scheduler code itself is a new construct but is not a free parameter — it is selected to satisfy fixed UER/LER targets, not fitted to the speed-up numbers.

free parameters (4)
  • cat-based measurement flip rate p_F = 6.3e-3 (= C1*w_bar*p with C1=2.1, w_bar=30, p=1e-4)
    Input noise model inherited from [17]; C1=2.1 is a constant fit to walking-cat numerical data in [17]. Every count in Tables II-IV is computed at this rate; ratios are less sensitive to it than absolute counts.
  • target logical error rate epsilon = 1e-10
    Specified target, not fitted; determines scheduler-code block sizes and the Viterbi baseline of 5.06 cat states per logical measurement.
  • maximum cat-state size w_bar = 30
    Walking-cat hardware parameter from [17]; decides which products of target Paulis are measurable in the architecture.
  • MECM error-sampling budget (l>=14) = 4096 errors per weight
    Sampling budget truncates the mAvg estimate for l>=14; the paper states the LER was not explicitly verified for these cases (App C1).
axioms (7)
  • domain assumption Independent bit-flip noise on cat-based measurement outcomes
    Section II: 'we assume that... measurement outcomes suffer from independent bit-flips with flip rate pF'; verified numerically in [17] for Q70/Q102; the paper says 'We expect this assumption to hold for quantum LDPC codes with single-shot properties [40].' If flips are correlated, all UER/LER/mAvg formulas fail.
  • domain assumption One cat-based measurement per SEC; SEC interleaving does not change relative cost
    Section II and App C1 defer SEC rounds explicitly ('we do not explicitly consider these here'); the stated metric is cat-state count, a proxy for time.
  • ad hoc to paper Scheduler columns correspond to accessible logical Paulis (weight <= w_bar) in Q70/Q102
    Tables II-III use abstract best-known-distance codes with no check that every column product has a weight-<=30 representative; App C4 gives a constraint-aware search procedure, but the headlined results do not use it.
  • ad hoc to paper All errors of weight <= (d-1)/2 are correctable
    Eq. (4) estimates LER as 1 - P(wt(e) <= (d-1)/2). This is a conservative upper bound on LER for minimum-distance decoding, used to size G1; conservative in direction but an approximation.
  • standard math Grassl's code tables [47] provide valid best-known-distance codes
    External, maintained resource; the paper states the access date (2026-06-22). Results depend on these tables' correctness.
  • standard math CSS structure supports transversal CNOT and inference of logical X/Z from destructive physical X/Z measurement
    App D1's stabilizer-tracking proof assumes logical operators are CSS; also used in the accessibility argument for CliNR stabilizers (App D3).
  • standard math Concatenated generator [G0|G1] has distance at least d0+d1
    Used for MECM code sizing (Section III D): for any nonzero u, wt(uG0)>=d0 and wt(uG1)>=d1, so the bound holds; conservative if the true distance is larger.
invented entities (1)
  • Scheduler code (scheduler matrix G) independent evidence
    purpose: Classical binary linear code whose generator matrix defines the sequence of product measurements and whose UER/LER determine the quantum protocol's error performance and measurement count.
    A mathematical construct, not a physical entity. It has falsifiable handles: the predicted per-Pauli measurement counts are checkable by simulation (the l=1 case reproduces [17]'s EDM/ECM numbers), and the error-rate conditions are computable from the code tables.

pith-pipeline@v1.3.0-alltime-deepseek · 20547 in / 25255 out tokens · 209846 ms · 2026-08-01T21:10:01.651060+00:00 · methodology

0 comments
read the original abstract

Quantum LPDC codes provide a substantial reduction in qubit overhead required for fault-tolerant quantum computation compared to surface code, thanks to their high encoding rate. However, operating simultaneously on multiple logical qubits encoded in the same block is more challenging and may slow down logical operations. Prior work addresses this problem by designing complex resource states to perform logical measurements in LDPC codes. Here, we propose an approach that only consumes cat states. Whereas previous work on cat-based measurements focuses on a single logical measurement, we design a protocol for the joint measurement of $\ell$ commuting logical operators. The key ingredient is the design of a scheduler code determining the measurement sequence and allowing for the decoding of all logical measurement outcomes. Numerical simulations with the LDPC codes Q70 and Q102 of the walking cat architecture show a speed-up of nearly $3\times$ over Viterbi measurements for the measurement of $\ell=20$ commuting logical operators. Combining our fast logical measurements with a new variant of the CliNR partial error correction scheme, we achieve a speed-up of up to $74\times$ for random Clifford circuits. Our approach also applies to non-Clifford gates, producing a speed-up of up to $5\times$ for Toffoli gates.

Figures

Figures reproduced from arXiv: 2607.16166 by Mark Webster, Nicolas Delfosse.

Figure 2
Figure 2. Figure 2: FIG. 2: Viterbi Gate-Based Protocol for implementing [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Physical and Logical CZNR Circuits: see main [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Toffoli Circuit Used for Simulation of Logical [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

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Reference graph

Works this paper leans on

89 extracted references · 12 linked inside Pith

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    This method does not require any physical opera- tions but instead involves maintaining a Pauli frame

    Clifford Operations via F rame T racking In the walking cat architecture [17], logical Clifford op- erators are implemented where possible viaframe track- ing. This method does not require any physical opera- tions but instead involves maintaining a Pauli frame. For a code block withklogical qubits, the Pauli frame is a tableau comprising a binary 2k×2ksy...

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    The input to this protocol is a logical Clifford operator, which we decompose into a circuit of one and two-qubit logical Clifford gates (see [48, 50–52])

    Viterbi Gate-Based Protocol for Logical Clifford Operations Where a Clifford operation cannot be implemented via frame tracking, theViterbi gate-based protocolis used (see Figure 2). The input to this protocol is a logical Clifford operator, which we decompose into a circuit of one and two-qubit logical Clifford gates (see [48, 50–52]). Single-qubit logic...

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    These are used to calculate the undetectable error rate and the posterior probability used in the MECM protocol of Section III C

    W eight Enumerators, UER of Classical Codes and Posterior Probability Calculation In this section we explain how to calculate weight enu- merators for cosets of binary linear codes. These are used to calculate the undetectable error rate and the posterior probability used in the MECM protocol of Section III C. Given a binary vectorvof lengthmand full-rank...

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    This produces anideal maximum likelihood decoder which is used for simulations of the MECM protocol and to calculate the logical error rate of binary linear codes exactly

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    Heuristic Enumeration of Cosets Using Random Information Sets The exact methods for calculating coset probabilities of Section B 1 have exponential complexity inℓ, m. For large codes, we instead use heuristic methods. The ran- dom information set algorithm for distance-finding [54– 56] can be modified to sample from low-weight elements of the cosetv+⟨G⟩. ...

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    Modeling the MEDM and MECM Protocols In this section we describe the methodology used to model the MEDM and MECM protocols of Sections III B and III C. The first step of modeling MECM is to choose a scheduler code (see Section III A). Forℓ∈[2,4, ..20], we pre-calculated the weight enumerators of the best- known-distance binary linear codes from [47] with ...

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    MEDM/MECM Example - Repetition Code for Single Pauli Measurements In this example, we show how MEDM and MECM ap- ply for single Pauli measurements (ℓ= 1) and recover the results of [17] for the EDM, ECM and Viterbi mea- surement protocols. To apply the MEDM protocol, we require⟨G⟩to meetU ⟨G⟩ = UER⟨G⟩/(P(0) + UER⟨G⟩)< ε= 10 −10. The codewords of the lengt...

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    The resulting codeG= [G 0|G1] has distanced=d 0 +d 1 = 15 and block sizem=m 0 +m 1 = 31

    Accordingly, we select forG 1 the smallestℓ= 4 code with distanced 1 = 10 which has block sizem 1 = 20. The resulting codeG= [G 0|G1] has distanced=d 0 +d 1 = 15 and block sizem=m 0 +m 1 = 31. To verify the truncated MECM protocol, we simulate all errorsewith weight wt(e)≤7 and check that the logical error rate of the protocol is at the target logical err...

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    Accommodating Pauli Measurement Accessibility in MEDM/MECM We now show how to adapt the MEDM and MECM protocols of Sections III B and III C to take into account accessibility constraints set out in Section A for logical Pauli measurements. We assume that a logical Pauli ba- sisBhas been selected and that any product ofwbasis elements is accessible. LetP i...

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    X A: 0#$ *%! +&'(#B: 0#$+, XXZZZ Data: (

    Logical CliNR Proof Here we prove that the logical CliNR protocol outlined in Section IV B has the desired logical action and show how to adapt the protocol to accommodate frame track- ing. To verify the protocol, we check that each element of the stabilizer groupSof the QECC stabilizes the output block B throughout the protocol. We then verify the logica...

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    Logical CZNR Protocol The logical CZNR protocol can be used for Clifford circuits composed of CZ and S gates and is set out in Figure 3. Logical CZNR is simpler to execute than the logical CliNR protocol of Section IV B - only one auxil- iary code block is required and so only we execute only one transversal CNOT and one destructive Z measure- ment. The s...

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    We assume logical widthwso that any product of welements of the logical Pauli basis is accessible and that any CSS logical operator is accessible (see Section A)

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    Each of the logical stabilizers was then measured using the Viterbi measurement protocol giving a total number of cat-based measurements (CM)

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    Full results of the simulations are set out in Table IV

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    We used Q70 and Q102 code blocks from [17]

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