Almost optimal geometrically local quantum LDPC codes in any dimension

TL;DR

This paper presents a method to create almost optimal quantum codes that are geometrically local in any dimension.

Contribution

A new procedure to transform any good quantum LDPC code into an almost optimal geometrically local quantum code in any dimension.

Findings

The method generalizes previous constructions by removing specialized prerequisites.

It builds a connection between geometric operations and code constructions.

The approach works for arbitrary three-term chain complexes.

Abstract

Geometrically local quantum codes, comprised of qubits and checks embedded in \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${{\mathbb{R}}}^{D}$$\end{document}RD with local check operators, have been a subject of significant interest. A key challenge is identifying the optimal code construction that maximizes both code dimension and distance under the geometric constraints. In this work, we introduce a construction that can transform any good quantum LDPC code into an almost optimal geometrically local quantum code. Our approach hinges on a novel yet simple procedure that extracts a two-dimensional structure from an arbitrary three-term chain complex, building a connection between geometric operations and code constructions. We expect that this procedure will find broader applications in areas such as weight reduction and the geometric realization of chain complexes.

The authors provide a method to realise almost optimal geometrically local quantum codes from quantum LDPC codes, which generalizes previous near-optimal locality constructions by removing specialized prerequisites and working in any dimension.

Figures