Affine Filtering Measurements and Their Applications to Quantum Decoding

2026-06-05

AI Summary

This work addresses the problem of efficient decoding for linear codes over pure-state classical-quantum channels by introducing affine-filter measurements—a structured approach to unambiguous state discrimination. The method identifies an affine subspace containing the transmitted codeword and models inconclusive outcomes as erasures, thereby establishing a quantum decoding framework tailored to locally structured codes. By integrating group-covariant pure-state indexing, the local structure of LDPC codes, and semidefinite programming techniques based on character diagonalization, the design of optimal measurements is reduced to a tractable linear program, enabling code-aware, fine-grained measurement strategies. Numerical simulations demonstrate that, over i.i.d. pure-state channels, the proposed decoder outperforms both symbol-wise unambiguous state discrimination and near-optimal measurement schemes when applied to regular LDPC codes.

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.

Problem

• affine filtering
• unambiguous state discrimination
• quantum decoding
• classical-quantum channels
• linear codes

Innovation

• affine filtering measurements
• quantum decoding
• semidefinite programming
• character-based diagonalization
• group-covariant codes