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arxiv: 2606.07852 · v1 · pith:YL5ZOPNFnew · submitted 2026-06-05 · 🪐 quant-ph · cs.IT· math.IT

Affine Filtering Measurements and Their Applications to Quantum Decoding

Avijit Mandal , Noah Shutty , Henry D. Pfister , Stephen P. Jordan This is my paper

Pith reviewed 2026-06-27 21:27 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords affine filtering measurementsunambiguous state discriminationquantum decodinglinear codesLDPC codespure-state channelssemidefinite programminggroup-covariant indexing
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The pith

For group-covariant pure-state codeword indexing, optimal affine filtering measurements reduce to a linear program and outperform symbol-wise decoding on LDPC codes.

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

The paper establishes a method for designing affine filtering measurements to decode classical linear codes over pure-state classical-quantum channels. A conclusive outcome identifies an affine subspace containing the transmitted codeword, with inconclusive results treated as erasures. When the indexing of codewords is group-covariant, character-based diagonalization reduces the optimal design from a semidefinite program to a linear program. Simulations on regular LDPC codes from Gallager ensembles show this yields better performance than symbol-wise unambiguous state discrimination or pretty good measurement decoding on i.i.d. pure-state channels.

Core claim

For a group-covariant indexing of pure-state codewords, the optimal design of affine filtering measurements is a semidefinite program that can be reduced to a linear program via character-based diagonalization. The resulting measurement builds a quantum decoding framework for local codes that outperforms symbol-wise USD and symbol-wise pretty good measurement based decoding on i.i.d. pure-state channels in simulations on regular LDPC codes from Gallager ensembles.

What carries the argument

Affine filtering measurements that return an affine subspace containing the codeword on conclusive outcomes, with optimal design obtained by reducing the semidefinite program to a linear program through character-based diagonalization when the indexing is group-covariant.

If this is right

  • The linear program yields efficient computation of measurements for decoding local codes such as those with single parity-check constraints.
  • Performance gains appear on i.i.d. pure-state channels for the tested LDPC ensembles.
  • The construction is code-aware and produces fine-grained USD measurements tailored to the linear code structure.
  • The framework applies directly to decoding classical linear codes transmitted over pure-state channels.

Where Pith is reading between the lines

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

  • The diagonalization technique could be adapted to other symmetric families of states if analogous group structures exist.
  • Combining affine filtering with classical message-passing algorithms on the same LDPC graph might produce hybrid decoders.
  • Performance on non-i.i.d. or non-pure-state channels remains untested and could limit or extend the observed gains.
  • The subspace-identification property may connect to erasure decoding techniques already used in classical coding theory.

Load-bearing premise

The indexing of the pure-state codewords must be group-covariant to permit character-based diagonalization that reduces the semidefinite program to a linear program.

What would settle it

A simulation on regular LDPC codes from Gallager ensembles in which affine filtering decoding shows no performance gain over symbol-wise USD and pretty good measurement methods on i.i.d. pure-state channels, or a group-covariant indexing where the character diagonalization fails to solve the optimal design problem.

Figures

Figures reproduced from arXiv: 2606.07852 by Avijit Mandal, Henry D. Pfister, Noah Shutty, Stephen P. Jordan.

Figure 1
Figure 1. Figure 1: Performance Comparison for (3,4) Regular LDPC Codes on F2 and r = 1 with N = 1600 5.2 Affine Filtering+GE Decoder Performance for LDPC Codes with q > 2 Similarly to the binary case, we simulate the affine filtering+GE decoder for q > 2. In this case, we fix r and compare the performance of affine filtering+GE decoder with qudit PGM+BP and qudit USD+GE decoders. In [PITH_FULL_IMAGE:figures/full_fig_p038_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Performance Comparison for (4,5) Regular LDPC Codes on F2 and r = 1 with N = 1600 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 ® 0.0 0.2 0.4 0.6 0.8 1.0 Ps u c c e s s (6, 7) LDPC Code F2 r1 ® USD ® Affine ¡ Filtering ® Shannon ® Holevo ® BPQM Affine-Filtering+GE (N=1400) Affine-Filtering+GE (N=2100) Affine-Filtering+GE (N=4200) Qubit-USD+GE (N=1400) Qubit-USD+GE (N=2100) Qubit-USD+GE (N=4200) [PITH_FULL_… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between affine-filtering+GE and qubit USD+GE decoding for (6,7) regular LDPC codes 39 [PITH_FULL_IMAGE:figures/full_fig_p039_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Threshold plot for affine-filtering+GE decoding for (6,7) regular LDPC codes 0.2 0.4 0.6 0.8 1.0 ® 0.0 0.2 0.4 0.6 0.8 1.0 Ps u c c e s s (3, 4) LDPC Code F3 r2 N1600 ® Shannon max ® Holevo max ® Shannon min ® Holevo min Affine-Filtering+GE Qudit-USD+GE Qudit-PGM+BP [PITH_FULL_IMAGE:figures/full_fig_p040_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Performance Comparison for (3,4) Regular LDPC Codes on F3 and r = 2 with N = 1600 40 [PITH_FULL_IMAGE:figures/full_fig_p040_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Performance Comparison for (4,5) Regular LDPC Codes on F3 and r = 2 with N = 1600 6 Conclusion We studied affine filtering measurements, a structured class of unambiguous measurements whose conclusive outcomes filter affine subspaces containing the transmitted codeword of a linear code. For symmetric codeword indexed states, we showed that the optimal affine filtering design problem reduces from an SDP to … view at source ↗
read the original abstract

Unambiguous state discrimination (USD) measurements are attractive because outcomes are either marked as conclusive (i.e., error free) or inconclusive (i.e., erased). We study affine filtering measurements, a structured variant of USD for decoding classical linear codes over pure-state classical-quantum channels, where a conclusive outcome identifies an affine subspace containing the transmitted codeword and an inconclusive outcome is treated as an erasure. For a group-covariant indexing of pure-state codewords, we show that the optimal design of affine filtering measurements is a semidefinite program that can be reduced to a linear program via character-based diagonalization. We use the resulting measurement to build a quantum decoding framework for local codes, and we demonstrate (via simulations on regular LDPC codes from Gallager ensembles using single parity check local constraints) that affine filtering based decoding can outperform symbol-wise USD and symbol-wise pretty good measurement based decoding methods on i.i.d. pure-state channels. In an independent and concurrent work, Buzet and Chailloux study similar fine-grained USD measurements for symmetric families of states. Their focus is on the code-agnostic setting whereas our focus is on code-aware constructions and decoding.

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

3 major / 1 minor

Summary. The paper introduces affine filtering measurements, a structured form of unambiguous state discrimination (USD) tailored to decoding classical linear codes transmitted over pure-state classical-quantum channels. Under the assumption of group-covariant indexing of the pure-state codewords, the optimal design of these measurements is formulated as a semidefinite program (SDP) that reduces to a linear program (LP) via character-based diagonalization. The resulting measurements are used to construct a quantum decoding framework for local codes. Simulations on regular LDPC codes drawn from Gallager ensembles with single-parity-check constraints are reported to show that this affine-filtering decoder outperforms both symbol-wise USD and symbol-wise pretty-good-measurement decoding on i.i.d. pure-state channels. The work is positioned as code-aware, in contrast to concurrent code-agnostic work by Buzet and Chailloux.

Significance. If the SDP-to-LP reduction is valid and the simulation results are statistically robust, the paper supplies a computationally tractable, code-aware method for designing fine-grained USD measurements that can improve decoding of linear codes on quantum channels. The explicit reduction via character theory is a concrete technical contribution that turns an otherwise general SDP into an efficiently solvable LP, which is a strength when the group-covariant indexing condition can be met.

major comments (3)
  1. [Abstract, optimal design paragraph; derivation section] Abstract (paragraph on optimal design) and the section deriving the measurement: The central claim that the optimal affine filtering measurement reduces from an SDP to an LP rests entirely on the group-covariant indexing assumption. No explicit construction or proof is supplied showing that a group-covariant indexing of the codewords can always be chosen for arbitrary local codes (or for the specific Gallager LDPC ensembles used in the simulations) without altering the underlying channel or code structure. This assumption is load-bearing for both the efficient construction and the subsequent performance claims.
  2. [Simulations section] Simulations section (results on Gallager LDPC ensembles): The reported outperformance over symbol-wise USD and PGM lacks error bars, the number of Monte-Carlo trials, dataset size, or any correction for multiple testing. Without these, it is impossible to assess whether the observed gains are statistically significant or reproducible, directly undermining the empirical support for the decoding framework.
  3. [SDP formulation and reduction section] Section stating the SDP formulation and its reduction: The character-based diagonalization step that converts the SDP into an LP is asserted but the explicit algebraic steps, the relevant representation theory, or a self-contained proof are not provided in sufficient detail to allow independent verification of the reduction.
minor comments (1)
  1. [Abstract and introduction] The abstract and introduction should state the group-covariant indexing assumption more prominently and earlier, so that readers immediately understand the scope of the claimed LP reduction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract, optimal design paragraph; derivation section] Abstract (paragraph on optimal design) and the section deriving the measurement: The central claim that the optimal affine filtering measurement reduces from an SDP to an LP rests entirely on the group-covariant indexing assumption. No explicit construction or proof is supplied showing that a group-covariant indexing of the codewords can always be chosen for arbitrary local codes (or for the specific Gallager LDPC ensembles used in the simulations) without altering the underlying channel or code structure. This assumption is load-bearing for both the efficient construction and the subsequent performance claims.

    Authors: The manuscript presents the SDP-to-LP reduction explicitly under the assumption of group-covariant indexing and does not claim or prove that such an indexing can always be chosen for arbitrary local codes. For the Gallager LDPC ensembles in the simulations, the single-parity-check structure permits the required indexing. We will revise the text to clarify the scope of the assumption and the conditions under which the method applies. revision: partial

  2. Referee: [Simulations section] Simulations section (results on Gallager LDPC ensembles): The reported outperformance over symbol-wise USD and PGM lacks error bars, the number of Monte-Carlo trials, dataset size, or any correction for multiple testing. Without these, it is impossible to assess whether the observed gains are statistically significant or reproducible, directly undermining the empirical support for the decoding framework.

    Authors: We agree that the simulations require additional statistical details for proper evaluation. The revised manuscript will report the number of Monte-Carlo trials, include error bars on the performance curves, specify dataset sizes, and address any relevant statistical considerations. revision: yes

  3. Referee: [SDP formulation and reduction section] Section stating the SDP formulation and its reduction: The character-based diagonalization step that converts the SDP into an LP is asserted but the explicit algebraic steps, the relevant representation theory, or a self-contained proof are not provided in sufficient detail to allow independent verification of the reduction.

    Authors: We will expand the derivation to include the explicit algebraic steps and a self-contained outline of the representation-theoretic argument underlying the character-based diagonalization, either in the main text or an appendix. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained under explicit assumption

full rationale

The paper formulates affine filtering measurements as an optimization problem (SDP) whose reduction to LP is derived from the group-covariant indexing assumption via character-based diagonalization; this is stated as a conditional result rather than a self-definition or fitted prediction. Simulations on Gallager LDPC ensembles evaluate the resulting decoder but do not feed back into the optimization or rely on self-citations. No load-bearing step reduces by construction to its own inputs, and the concurrent independent work is cited without overlap. The central claims remain independent of the evaluation data.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the group-covariance assumption and the pure-state channel model are background facts rather than paper-specific inventions.

pith-pipeline@v0.9.1-grok · 5742 in / 1259 out tokens · 23604 ms · 2026-06-27T21:27:17.788379+00:00 · methodology

discussion (0)

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Reference graph

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