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

TL;DR

The paper addresses the non-tightness of the quantum hashing bound for memoryless Pauli channels by introducing stabilizer-code channel transforms that induce a logical Pauli channel with side information. It develops a complete pipeline to compute the induced distribution $p(a,b,s)$ via a full binary symplectic tableau and evaluates the induced hashing rate $R_{ ext{ind}}=rac{k-H(L|S)}{n}$ under a decoder-side-information setting. A structured search over small transforms, using Gottesman standard form and depth-first/random generation strategies, yields evidence that high-rate, non-repetition stabilizers (notably all-Z codes) can outperform the baseline hashing bound for skewed independent Pauli channels, particularly at lower noise levels. The work aims to illuminate how conditioning on syndrome information affects logical entropy and to guide future capacity-achieving constructions by focusing on the evolution of relevant information quantities under channel transforms.

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.

Figures (2)

• Figure 1: Achievable rates for the skewed independent channel \ref{['eq:indepXZ']} with bias $\eta=9$. Horizontal axis: $p=1-p_I=p_X+p_Z+p_Y$. We plot the baseline hashing bound, the envelope of induced rates obtained from the high-rate stabilizer transforms found in our search, as well as envelope of the induced rates obtained from the $Z$-repetition code from SmithSmolin2007DegeneratePauli. Low-rate repetition-type transforms help at larger $p$, while high-rate single-check "all-$Z$" transforms yield gains at smaller $p$. Note that the currently best known upper bounds are fairly loose in this region. We therefore did not include them in this figure.
• Figure 2: For the $[\![ n,n-1]\!]$ all-$Z$ family, the smallest $p$ at which $R_{\mathrm{ind}}$ exceeds the physical hashing bound, as a function of $n$ (bias $\eta=9$). Increasing $n$ pushes the improvement to smaller $p$, and even $n$ performs better than neighboring odd lengths in this regime.

Theorems & Definitions (1)

• Lemma 1: Hashing bound with decoder side information