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arxiv: 2605.20346 · v1 · pith:MXSUMCGJnew · submitted 2026-05-19 · 🪐 quant-ph

Forced Gap Post-Selection for Quantum LDPC Codes and their Operations

Adam Wills , Theodore J. Yoder , Isaac Chuang This is my paper

Pith reviewed 2026-05-21 07:27 UTC · model grok-4.3

classification 🪐 quant-ph
keywords post-selectionquantum LDPC codesbivariate bicycle codesbelief propagationlogical error ratequantum error correctionRelay-BP decoder
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The pith

A forced-gap post-selection strategy improves logical error rates by over a factor of 4 for quantum LDPC codes using only belief propagation decoding.

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

The paper develops a post-selection method for high-rate quantum codes in which a decoder is first run normally on a circuit and then re-run once per logical observable while forced to output the complementary logical outcome. Shots are rejected when the likelihood scores for the baseline and forced-complementary solutions are similar. When benchmarked with the Relay-BP decoder on 72-qubit and 144-qubit bivariate bicycle codes plus associated surgery gadgets, the approach yields more than fourfold lower logical error rates than prior post-selection techniques at the same rejection fraction. The method is lightweight because it requires only repeated belief-propagation passes rather than higher-latency decoders such as BP-OSD.

Core claim

By forcing the decoder to produce a complementary logical solution for each observable after the baseline run and rejecting shots whose likelihood scores for the two solutions are close, logical error rates on the 72-qubit and 144-qubit bivariate bicycle codes and their surgery gadgets fall by more than a factor of four compared with previous post-selection methods, while using only FPGA-friendly belief propagation at the same post-selection rate.

What carries the argument

The forced-gap rejection rule that discards a shot when the decoder likelihood for the baseline solution is comparable to the likelihood obtained when the decoder is forced to return the logically complementary outcome.

If this is right

  • Logical error rates drop by more than a factor of four on the tested 72-qubit and 144-qubit bivariate bicycle codes and surgery gadgets at fixed post-selection rate.
  • The strategy applies across different decoders and to other high-rate quantum LDPC codes.
  • Only repeated belief-propagation decoding passes are required, avoiding the computational cost of BP-OSD.
  • The same improvement is observed for both memory and logical surgery operations.

Where Pith is reading between the lines

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

  • If likelihood calibration holds for other decoders, the method could be combined with existing surface-code or other LDPC constructions to lower overall qubit overhead.
  • The rejection criterion might be generalized to multiple logical observables simultaneously or to analog syndrome data.
  • Hardware implementations could exploit the FPGA-friendly nature of belief propagation to perform the extra decoding rounds with low added latency.

Load-bearing premise

The decoder likelihood scores correctly rank the relative probabilities of logically complementary solutions.

What would settle it

An experiment in which the decoder likelihood scores are deliberately miscalibrated or biased and the logical error rate is then measured with and without the gap-based rejection rule.

Figures

Figures reproduced from arXiv: 2605.20346 by Adam Wills, Isaac Chuang, Theodore J. Yoder.

Figure 1
Figure 1. Figure 1: Overview of the Forced Gap post-selection strategy. Phase 1 (blue): the decoder is run on syndrome σ to produce a baseline correction e (0) with logical class L (0) = A·e (0). If the decoder declares erasure the instance is immediately rejected. Phase 2 (amber): for each of the K logical observables (i = 1, . . . , K), a forced run solves the modified problem (H(i) , σ(i) ), where an extra row appended to … view at source ↗
Figure 2
Figure 2. Figure 2: Simulated error rates per round for the idling [[72, 12, 6]] code, and the [[144, 12, 12]] code under post-selection, at physical noise p = 10−3 and 2.5 × 10−3 , respectively. The x-axis shows “Post￾selection rate”, that is, the fraction of shots that are rejected, where a curve is generated by varying the threshold T. Both codes are simulated for 6 and 12 rounds of syndrome extraction, respectively, and t… view at source ↗
Figure 3
Figure 3. Figure 3: Logical error rates of various surgery bicycle instructions [Yod+25] subject to post-selection under the forced gap strategy, all at physical noise p = 2.5 × 10−3 . We present the logical error rate for the entire circuit, without normalising by the number of rounds. The number of syndrome extraction rounds that we used in our simulations of each gadget are shown above. These were chosen by optimality of t… view at source ↗
read the original abstract

We develop a simple and general post-selection strategy for high-rate quantum codes that is transferrable across decoders. After an initial baseline run, the decoder is re-run once per logical observable, and forced in these latter runs to provide a solution where the given observable has the complementary outcome. Shots are rejected that find logically complementary solutions with similar likelihoods compared to the baseline. Using the Relay-BP decoder, we benchmark the strategy on the $72$-qubit and $144$-qubit bivariate bicycle codes, as well as surgery gadgets for the latter. In comparison to previous post-selection strategies, our results offer an improved logical error rate by over a factor of $4$ on the same circuit and physical error rate, and at the same rate of post-selection. Our strategies are also lightweight, relying only on FPGA-friendly belief propagation, whereas the previous best used repeated rounds of a high-latency BP-OSD decoder.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript presents a forced-gap post-selection strategy for quantum low-density parity-check (LDPC) codes. After a baseline decoder run, the decoder is re-run with each logical observable forced to its complementary outcome; shots are rejected when the likelihoods of the baseline and forced-complementary solutions are similar. Using the Relay-BP decoder, the strategy is benchmarked on the 72-qubit and 144-qubit bivariate bicycle codes as well as surgery gadgets for the latter, claiming an improvement in logical error rate by more than a factor of 4 relative to prior post-selection methods at identical post-selection rate and physical error rate. The approach is positioned as lightweight and transferable because it relies only on belief propagation rather than repeated BP-OSD rounds.

Significance. If the central empirical claim holds, the work supplies a practical, decoder-agnostic post-selection technique that materially lowers logical error rates for high-rate quantum LDPC codes without increasing post-selection overhead. The explicit use of FPGA-friendly belief propagation (rather than higher-latency BP-OSD) constitutes a concrete implementation advantage. The direct benchmarking on concrete 72- and 144-qubit codes and surgery gadgets supplies reproducible evidence of the performance gain when the decoder likelihoods behave as assumed.

major comments (1)
  1. [Benchmarking results on the 72-qubit and 144-qubit bivariate bicycle codes] Benchmarking results on the 72-qubit and 144-qubit bivariate bicycle codes: the reported factor-of-4 logical-error-rate reduction at fixed post-selection rate depends on Relay-BP likelihood scores being at least monotonically related to the true posterior probabilities of logically complementary error patterns. No calibration diagnostics (likelihood histograms conditioned on actual logical outcome, or direct comparison against BP-OSD or exact enumeration on small instances) are supplied to substantiate this assumption. Systematic bias in the approximate decoder could therefore cause the gap-rejection rule either to discard useful shots or to retain erroneous ones, erasing the claimed improvement.
minor comments (2)
  1. [Abstract and benchmarking sections] The abstract and benchmarking sections omit statistical error bars, the total number of shots collected, and any description of decoder calibration procedures. Inclusion of these details would allow readers to assess the statistical significance of the factor-of-4 claim.
  2. [Figures] Figure captions and axis labels should explicitly state the physical error rate and post-selection rate at which each curve is evaluated so that the constant-rate comparison is immediately verifiable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our work. We address the major comment in detail below, providing clarifications and committing to revisions where appropriate to strengthen the manuscript.

read point-by-point responses

  1. Referee: Benchmarking results on the 72-qubit and 144-qubit bivariate bicycle codes: the reported factor-of-4 logical-error-rate reduction at fixed post-selection rate depends on Relay-BP likelihood scores being at least monotonically related to the true posterior probabilities of logically complementary error patterns. No calibration diagnostics (likelihood histograms conditioned on actual logical outcome, or direct comparison against BP-OSD or exact enumeration on small instances) are supplied to substantiate this assumption. Systematic bias in the approximate decoder could therefore cause the gap-rejection rule either to discard useful shots or to retain erroneous ones, erasing the claimed improvement.

    Authors: We acknowledge the referee's concern regarding the lack of explicit calibration for the Relay-BP likelihood scores. However, we note that the post-selection strategy is designed to exploit relative likelihood differences between the baseline and forced-complementary runs, rather than relying on absolute probability values. This relative approach reduces sensitivity to systematic biases in the decoder's approximations. The empirical results demonstrate consistent improvements across different code sizes and operations, which would be unlikely if the likelihoods were not sufficiently correlated with the true posteriors. To further substantiate this, in the revised manuscript we will include additional diagnostics: specifically, likelihood ratio histograms for cases where the logical outcome is known (e.g., via exact methods on smaller analogs or by injecting known errors), and a comparison of post-selection performance using Relay-BP versus BP-OSD on the 72-qubit code. We believe these additions will confirm the validity of our assumptions while maintaining the lightweight nature of the method. revision: yes

Circularity Check

0 steps flagged

No circularity: strategy and benchmarks are defined independently of results

full rationale

The paper defines the forced-gap post-selection rule directly from baseline and forced-complementary decoder runs, then reports empirical logical error rates from direct benchmarking on 72- and 144-qubit bivariate bicycle codes. No equation or claim reduces to a fitted parameter renamed as prediction, no self-citation chain justifies the central performance gain, and the improvement is measured against prior post-selection methods on identical circuits. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The strategy assumes standard quantum error correction noise models and that belief-propagation likelihoods are sufficiently informative; no new free parameters or invented entities are introduced beyond the post-selection threshold choice.

axioms (1)
  • domain assumption Belief-propagation decoder likelihoods provide a reliable ranking of logical error probabilities under the physical noise model used.
    Invoked when the gap-based rejection rule is defined and when comparing to baseline runs.

pith-pipeline@v0.9.0 · 5688 in / 1316 out tokens · 34262 ms · 2026-05-21T07:27:40.441805+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Concatenating Algebraic Codes over High-Rate Quantum LDPC Codes

    quant-ph 2026-05 unverdicted novelty 6.0

    Concatenating quantum Reed-Solomon outer codes over the gross code using Galois qudits reaches teraquop regime at 10^{-3} physical noise with lower overhead than prior two-gross-code constructions.

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