Biased-Noise Thresholds of Zero-Rate Holographic Codes with Tensor-Network Decoding

TL;DR

This paper analyzes zero-rate holographic quantum error-correcting codes under biased noise, demonstrating their near-optimal performance and efficient decoding, with some codes surpassing existing benchmarks in specific noise regimes.

Contribution

It provides the first comprehensive analysis of asymptotically zero-rate holographic codes with biased noise, showing their capacity to reach the hashing bound and outperform certain existing codes.

Findings

Many holographic codes reach the hashing bound under biased noise.

Codes based on the $ ext{5,1,2}$ surface code and $ ext{6,1,3}$ code outperform state-of-the-art in 2-Pauli noise.

Clifford deformations enable codes to reach the hashing bound for 1-Pauli noise.

Abstract

A crucial insight for practical quantum error correction is that different types of errors, such as single-qubit Pauli operators, typically occur with different probabilities. Finding an optimal quantum code under such biased noise is a challenging problem, related to the (generally unknown) maximum capacity of the corresponding noisy channel. A benchmark for this capacity is given by the hashing bound, which describes the performance of random stabilizer codes and leads to the matter of identifying codes that come close to the bound while also being efficiently decodable. In this work, we perform the first comprehensive analysis of asymptotically zero-rate holographic codes under biased noise. We show that many representatives from such models of this code class fulfill both the channel optimality and efficient decoding guarantees for tensor-network codes. In fact, all holographic codes tested were found to reach the hashing bound in some bias regime, while several built from the $\codepar{5,1,2}$ surface code and $\codepar{6,1,3}$ code exceed state-of-the-art code performance in the 2-Pauli noise regime. Furthermore, we consider Clifford deformations which allow all considered codes to reach the hashing bound for 1-Pauli noise as well. Our results establish that holographic codes, which were previously shown to possess efficient tensor-network decoders, also exhibit competitive thresholds under biased noise.