General Construction of Quantum Error-Correcting Codes from Multiple Classical Codes

TL;DR

This paper introduces a general construction method for quantum error-correcting codes from multiple classical codes, unifying and extending existing models, and enabling the design of codes with improved parameters.

Contribution

It presents a universal recipe for constructing QECCs from D classical codes, encompassing known models and introducing new three-dimensional lattice code frameworks.

Findings

Recovers the hypergraph product construction for D=2.

Introduces four types of D=3 constructions, including new models.

Achieves codes with a trade-off between distance and dimension.

Abstract

The hypergraph product (HGP) construction of quantum error-correcting codes (QECC) offers a general and explicit method for building a QECC from two classical codes, thereby paving the way for the discovery of good quantum low-density parity-check codes. In this letter, we propose a general and explicit construction recipe for QECCs from a total of D classical codes for arbitrary D. Following this recipe guarantees the obtainment of a QECC within the stabilizer formalism and nearly exhausts all possible constructions. As examples, we demonstrate that our construction recovers the HGP construction when D = 2 and leads to four distinct types of constructions for D = 3, including a previously studied case as one of them. When the input classical codes are repetition codes, our D = 3 constructions unify various three-dimensional lattice models into a single framework, encompassing the three-dimensional toric code model, a fracton model, and two other intriguing models not previously investigated. Among these, two types of constructions exhibit a trade-off between code distance and code dimension for a fixed number of qubits by adjusting the lengths of the different classical codes, and the optimal choice can simultaneously achieve relatively large values for both code distance and code dimension. Our general construction protocol provides another perspective for enriching the structure of QECCs and enables the exploration of richer possibilities for good codes.