Renormalization group decoder for a four-dimensional toric code

TL;DR

This paper introduces an efficient renormalization-group heuristic algorithm for decoding four-dimensional toric codes, achieving high error thresholds and advancing topological quantum error correction methods.

Contribution

The paper presents a novel renormalization-group based decoding algorithm for 4D toric codes, improving error thresholds and handling measurement errors effectively.

Findings

Achieves a 4.35% threshold with measurement errors in 5D algorithm.

Attains a 7.3% threshold without measurement errors in 4D.

Finds a 0.31% threshold for depolarizing errors, lower than 2D toric code.

Abstract

We describe a computationally-efficient heuristic algorithm based on a renormalization-group procedure which aims at solving the problem of finding minimal surface given its boundary (curve) in any hypercubic lattice of dimension $D>2$. We use this algorithm to correct errors occurring in a four-dimensional variant of the toric code, having open as opposed to periodic boundaries. For a phenomenological error model which includes measurement errors we use a five-dimensional version of our algorithm, achieving a threshold of $4.35\pm0.1\%$. For this error model, this is the highest known threshold of any topological code. Without measurement errors, a four-dimensional version of our algorithm can be used and we find a threshold of $7.3\pm0.1\%$. For the gate-based depolarizing error model we find a threshold of $0.31\pm0.01\%$ which is below the threshold found for the two-dimensional toric code.