Transform Arbitrary Good Quantum LDPC Codes into Good Geometrically Local Codes in Any Dimension

TL;DR

This paper presents a method to convert any good quantum LDPC code into an optimal geometrically local quantum code in any dimension, broadening the possibilities for quantum code design.

Contribution

It introduces a novel procedure to transform arbitrary good qLDPC codes into geometrically local codes in any dimension, independent of prior specific constructions.

Findings

Enables transformation of any good qLDPC code into a geometrically local code

Works in any spatial dimension, not limited to three

Potential applications in weight reduction and geometric realization

Abstract

Geometrically local quantum codes, comprised of qubits and checks embedded in $\mathbb{R}^D$ with local check operators, have been a subject of significant interest. A key challenge is identifying the optimal code construction that maximizes both dimension and distance. Recent advancements have produced several constructions, but these either depend on specific good quantum low-density parity-check (qLDPC) codes or are limited to three dimensions. In this work, we introduce a construction that can transform any good qLDPC code into an optimal geometrically local quantum code. Our approach hinges on a novel procedure that extracts a two-dimensional structure from an arbitrary three-term chain complex. We expect that this procedure will find broader applications in areas such as weight reduction and the geometric realization of chain complexes.