Stabilizer-Code Channel Transforms Beyond Repetition Codes for Improved Hashing Bounds

TL;DR

This paper generalizes stabilizer-code channel transforms to improve quantum hashing bounds by analyzing logical error distributions, achieving better rates for certain Pauli channels.

Contribution

It introduces a method to use arbitrary stabilizer codes as channel transforms and computes induced logical error distributions to surpass baseline hashing bounds.

Findings

Identified stabilizer code transforms that improve hashing bounds for specific Pauli channels.

Developed a systematic approach to analyze logical error distributions under stabilizer code transforms.

Reported instances where the method outperforms traditional hashing bounds for skewed, independent errors.

Abstract

The quantum hashing bound guarantees that rates up to $1-H(p_I, p_X, p_Y, p_Z)$ are achievable for memoryless Pauli channels, but it is not generally tight. A known way to improve achievable rates for certain asymmetric Pauli channels is to apply a small inner stabilizer code to a few channel uses, decode, and treat the resulting logical noise as an induced Pauli channel; reapplying the hashing argument to this induced channel can beat the baseline hashing bound. We generalize this induced-channel viewpoint to arbitrary stabilizer codes used purely as channel transforms. Given any $ [\![ n, k ]\!] $ stabilizer generator set, we construct a full symplectic tableau, compute the induced joint distribution of logical Pauli errors and syndromes under the physical Pauli channel, and obtain an achievable rate via a hashing bound with decoder side information. We perform a structured search over small transforms and report instances that improve the baseline hashing bound for a family of Pauli channels with skewed and independent errors studied in prior work.