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Beam search over GF(2) row operations reduces CNOT counts in EA quantum QC-LDPC encoders by 7.3-34 percent.

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2026-06-27 12:58 UTC pith:4CLMPQZQ

load-bearing objection Beam search with Hamming heuristic cuts CNOT counts 7-34% below SKG baseline on the tested EA QC-LDPC families and beats PMH synthesis, with stabilizer verification. the 1 major comments →

arxiv 2606.11468 v1 pith:4CLMPQZQ submitted 2026-06-09 quant-ph cs.ITmath.IT

Optimizing Encoder Circuits of Entanglement-Assisted Quantum LDPC Codes via Beam Search

Aditya Sodhani (1) , Pavan Kumar (2) , Shayan Srinivasa Garani (2) , Keshab K. Parhi (1) ((1) University of Minnesota , Minneapolis , USA , (2) Indian Institute of Science , Bengaluru
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India)
This is my paper
classification quant-ph cs.ITmath.IT
keywords entanglement-assisted quantum codesquantum LDPC codesencoder circuit optimizationbeam searchCNOT count reductionstabilizer formalismquantum error correction
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 formulates encoder optimization for entanglement-assisted quantum quasi-cyclic LDPC codes as a search problem over sequences of GF(2) row operations applied to the binary matrix that encodes the CNOT sub-sequence of the SKG construction. It solves this search with a beam search algorithm directed by a Hamming-distance heuristic. On the families examined, the resulting circuits require fewer CNOT gates than the direct SKG construction and also fewer than those produced by Patel-Markov-Hayes synthesis. The circuits are confirmed to implement the intended encoders through stabilizer-tableau simulation. The work therefore shows that the structured matrices arising in these codes admit substantial CNOT-count reductions through targeted search.

Core claim

Encoder optimization for EA quantum QC-LDPC codes is cast as a search over GF(2) row operations that decompose the binary matrix derived from the CNOT sub-sequence of the SKG construction. This search is performed by a beam search algorithm guided by a Hamming-distance heuristic. For the tested code families the search produces circuits whose CNOT counts are 7.3-34.0 percent lower than the SKG baseline and lower than those obtained from Patel-Markov-Hayes synthesis; the circuits are verified by stabilizer-tableau simulation.

What carries the argument

Beam search over GF(2) row operations on the binary matrix from the CNOT sub-sequence, guided by a Hamming-distance heuristic.

Load-bearing premise

The beam search guided by the Hamming-distance heuristic will locate GF(2) row-operation sequences that produce strictly lower CNOT counts than the direct SKG construction for the matrices of the tested code families.

What would settle it

An instance from one of the tested EA quantum QC-LDPC families in which the beam search returns a circuit whose CNOT count is not lower than the SKG baseline, or whose stabilizer-tableau simulation fails to match the target encoder.

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

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If this is right

  • Encoders for the examined EA quantum QC-LDPC families can be realized with fewer CNOT gates than the standard SKG construction.
  • The same optimized circuits also require fewer CNOT gates than circuits obtained from Patel-Markov-Hayes synthesis.
  • Stabilizer-tableau simulation confirms that the reduced-CNOT circuits implement the correct encoders.
  • Substantial simplification of the encoder is possible whenever the parity-check matrices possess the structure of the tested QC-LDPC families.

Where Pith is reading between the lines

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

  • The same search formulation could be applied to encoder circuits of other stabilizer codes whose parity-check matrices admit a comparable binary decomposition.
  • Lower CNOT counts may reduce the total gate resources needed when these codes are compiled to a specific quantum hardware gate set.
  • The effectiveness of the Hamming-distance heuristic may vary with the density or quasi-cyclic structure of the underlying matrices.

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

1 major / 2 minor

Summary. The manuscript proposes using beam search over GF(2) row operations, guided by a Hamming-distance heuristic, to optimize the CNOT count in the encoder circuits obtained from the Sharma-Kumar-Garani (SKG) construction for entanglement-assisted quantum quasi-cyclic LDPC codes. On the tested EA QC-LDPC families the method yields CNOT-count reductions of 7.3–34.0 % relative to the direct SKG encoder, lower counts than Patel–Markov–Hayes synthesis on every instance, and correctness is confirmed by stabilizer-tableau simulation.

Significance. If the reported reductions hold, the work supplies concrete evidence that a simple heuristic search can produce substantially simpler encoders for structured EA QC-LDPC codes, where CNOT count is the dominant cost. The independent stabilizer-tableau verification step adds reliability to the empirical claim and the comparison against two external baselines (SKG and PMH) is clearly presented.

major comments (1)
  1. [§4 and results tables] §4 (Experimental Setup) and the results tables: the manuscript does not state the beam width employed, the precise quasi-cyclic parameters (n, k, ebits) of each tested family, or the exact definition and tie-breaking rule of the Hamming-distance heuristic. These omissions prevent independent reproduction of the 7.3–34.0 % reductions that constitute the central empirical claim.
minor comments (2)
  1. [Figure 1 and §3.2] Figure 1 caption and §3.2: the notation for the binary matrix derived from the CNOT sub-sequence is introduced without an explicit equation reference, making the mapping from SKG encoder to the search problem harder to follow on first reading.
  2. [Table 2] Table 2: the column headings for CNOT counts should explicitly indicate whether they count only the data-qubit CNOTs or include the ebit-assisted operations, to avoid ambiguity when comparing with PMH.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the reproducibility issues in §4 and the results tables. We agree that the omitted details are necessary for independent verification of the reported CNOT-count reductions and will revise the manuscript to supply them.

read point-by-point responses

  1. Referee: [§4 and results tables] §4 (Experimental Setup) and the results tables: the manuscript does not state the beam width employed, the precise quasi-cyclic parameters (n, k, ebits) of each tested family, or the exact definition and tie-breaking rule of the Hamming-distance heuristic. These omissions prevent independent reproduction of the 7.3–34.0 % reductions that constitute the central empirical claim.

    Authors: We agree that these parameters and definitions were not stated explicitly. In the revised manuscript we will expand §4 with a new subsection that (i) specifies the beam width used throughout the experiments, (ii) tabulates the exact quasi-cyclic parameters (n, k, ebits) for every tested EA QC-LDPC family, and (iii) gives the precise mathematical definition of the Hamming-distance heuristic together with the tie-breaking rule employed when multiple rows yield the same distance. These additions will allow full reproduction of the 7.3–34.0 % reductions and the comparisons against the SKG and PMH baselines. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical optimization results independent of any self-referential inputs

full rationale

The paper's central claim consists of measured CNOT-count reductions (7.3-34.0%) obtained by running a beam-search algorithm on concrete EA QC-LDPC matrices, with correctness checked by independent stabilizer-tableau simulation and comparison against the SKG construction and PMH synthesis. No equations, predictions, or uniqueness arguments are presented that reduce the reported outcomes to quantities defined by the method itself. Adoption of the SKG baseline is used only for comparison and does not justify the optimization results. The work is therefore self-contained against external benchmarks with no load-bearing self-citation or self-definitional steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract introduces no free parameters, additional axioms, or invented entities beyond the standard stabilizer formalism and the SKG encoder construction already present in the cited literature.

pith-pipeline@v0.9.1-grok · 5770 in / 1172 out tokens · 30476 ms · 2026-06-27T12:58:37.680069+00:00 · methodology

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Cite this review

Pith. "Pith review of Optimizing Encoder Circuits of Entanglement-Assisted Quantum LDPC Codes via Beam Search." pith.science (2026). https://pith.science/paper/4CLMPQZQ

@misc{pith2026260611468,
  author       = {Pith},
  title        = {Pith review of: Optimizing Encoder Circuits of Entanglement-Assisted Quantum LDPC Codes via Beam Search},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4CLMPQZQ}},
  note         = {Machine review of arXiv:2606.11468}
}
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read the original abstract

Entanglement-assisted (EA) quantum QC-LDPC codes offer strong error-correction capabilities with structured parity-check matrices, but their practical use depends on efficient encoder circuits and the availability of pre-shared Bell pairs (ebits). In all encoder implementations based on the stabilizer formalism, the dominant contribution to this complexity comes from the use of controlled gates. In this paper, we adopt the Sharma-Kumar-Garani (SKG) encoder construction. We formulate the encoder optimization as a search over GF(2) row operations that decompose the binary matrix derived from its CNOT sub-sequence. We solve this problem using a beam search algorithm guided by a Hamming-distance heuristic. For the tested EA quantum QC-LDPC code families, the proposed method achieves CNOT-count reductions of 7.3-34.0% relative to the SKG baseline encoder. The optimized circuits also yield lower CNOT counts than Patel-Markov-Hayes synthesis on all tested instances and are verified by stabilizer-tableau simulation. These results show that substantial encoder simplification is possible for structured EA QC-LDPC codes.

Figures

Figures reproduced from arXiv: 2606.11468 by (2) Indian Institute of Science, Aditya Sodhani (1), Bengaluru, India), Keshab K. Parhi (1) ((1) University of Minnesota, Minneapolis, Pavan Kumar (2), Shayan Srinivasa Garani (2), USA.

Figure 1
Figure 1. Figure 1: Architecture of the proposed CNOT-count optimization flow. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Beam-search-optimized encoder circuit for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: SKG encoder circuit (baseline) for [[9, 4; 1]]: 2 H, 17 CNOTs. A. Gate-Count Comparison Table I presents the CNOT gate counts and gate-reduction percentages across all methods. The “SKG” column is the SKG construction count LSKG. The “PMH” column shows the Patel–Markov–Hayes [19] result. The “Beam” column reports the best beam search result obtained over the tested beam widths. The reduction column is meas… view at source ↗
Figure 4
Figure 4. Figure 4: Two CNOT circuits implementing the same encoder matrix [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: SKG-baseline encoder for [[8, 2; 2]]: 3 Hadamards, 13 CNOTs, ASAP two-qubit depth 9 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Beam-search-optimized encoder for [[8, 2; 2]]: 3 Hadamards, 10 CNOTs, ASAP two-qubit depth 6. CODE [[25, 8; 1]] The encoder for [[25, 8; 1]] acts on nt = 26 qubits. The SKG baseline uses 8 Hadamards and 89 CNOTs at ASAP depth 26 ( [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: SKG-baseline encoder for [[25, 8; 1]]: 8 Hadamards, 89 CNOTs, ASAP two-qubit depth 26 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Beam-search-optimized encoder for [[25, 8; 1]]: 8 Hadamards, 60 CNOTs, ASAP two-qubit depth 20 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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