Bias-Aware BP Decoding of Quantum Codes via Directional Degeneracy

TL;DR

The work addresses finite-length decoding of quantum CSS codes under biased, anisotropic noise by introducing directionally annotated Tanner graphs and a single bias parameter $β$ that yields site-dependent LLRs for BP→OSD decoding. It defines a directional degeneracy enumerator, derives bounds relating directional to Hamming distances, and provides a MacWilliams-type expression linking to the dual code. The proposed anisotropic BP+OSD decoder uses directional weights to tilt priors and coset posteriors, achieving substantial logical-error-rate reductions in code-capacity simulations without altering the codes. The results demonstrate that modest spatial anisotropy aligned with hardware or scheduling can yield practical gains, with a clear pathway to hardware-aware decoding optimizations and extensions to more complex noise models.

Abstract

We study directionally informed belief propagation (BP) decoding for quantum CSS codes, where anisotropic Tanner-graph structure and biased noise concentrate degeneracy along preferred directions. We formalize this by placing orientation weights on Tanner-graph edges, aggregating them into per-qubit directional weights, and defining a \emph{directional degeneracy enumerator} that summarizes how degeneracy concentrates along those directions. A single bias parameter~$β$ maps these weights into site-dependent log-likelihood ratios (LLRs), yielding anisotropic priors that plug directly into standard BP$\rightarrow$OSD decoders without changing the code construction. We derive bounds relating directional and Hamming distances, upper bound the number of degenerate error classes per syndrome as a function of distance, rate, and directional bias, and give a MacWilliams-type expression for the directional enumerator. Finite-length simulations under code-capacity noise show significant logical error-rate reductions -- often an order of magnitude at moderate physical error rates -- confirming that modest anisotropy is a simple and effective route to hardware-aware decoding gains.

Figures (4)

• Figure 1: Directionally annotated Tanner graph. Edge thickness encodes the magnitude of orientation weights $D_X,D_Z$. Summing over incident edges yields per-qubit directional weights $w_i$ that define the cost $\Delta_{\bm w}(E)$.
• Figure 2: Logical error rate vs. physical error rate for the $\llbracket 162,2,9\rrbracket$ Toric code
• Figure 3: $P_L$ versus anisotropy strength $\beta$ for the $\llbracket 162,2,9\rrbracket$ Toric code
• Figure 4: Logical error rate vs. physical error rate for the $\llbracket 36,4\rrbracket$ NE3N code

Theorems & Definitions (18)

• Definition 1: Degeneracy classes
• Definition 2: Directional annotation
• Proposition 1: Edge-to-qubit reduction
• proof
• Definition 3: Class score
• Definition 4: Directional degeneracy enumerator
• Lemma 1: Tail bound for low-cost classes
• proof
• Proposition 2: Comparison with Hamming distance
• proof