Almost Linear Decoder for Optimal Geometrically Local Quantum Codes

TL;DR

This paper introduces an almost linear time decoder for optimal three-dimensional geometrically local quantum codes, combining existing decoding techniques to achieve efficient error correction with a finite threshold.

Contribution

It presents the first decoder for an optimal 3D geometrically local quantum code using subdivision, Union-Find, and minimum weight perfect matching methods.

Findings

Decoder operates in almost linear time.

Achieves a finite threshold error rate.

Applicable to code capacity noise model.

Abstract

Geometrically local quantum codes, which are error correction codes embedded in $\mathbb{R}^D$ with checks acting only on qubits within a fixed spatial distance, have garnered significant interest. Recently, it has been demonstrated how to achieve geometrically local codes that maximize both the dimension and the distance, as well as the energy barrier of the code. In this work, we focus on the constructions involving subdivision and show that they have an almost linear time decoder, obtained by combining the decoder of the outer good qLDPC code and a generalized version of the Union-Find decoder. This provides the first decoder for an optimal geometrically local three-dimensional code. We demonstrate the existence of a finite threshold error rate under the code capacity noise model using a minimum weight perfect matching decoder. Furthermore, we argue that this threshold is also applicable to the decoder based on the generalized Union-Find algorithm.