Breaking the Orthogonality Barrier in Quantum LDPC Codes

TL;DR

This paper introduces a novel method for constructing quantum LDPC codes that achieve large girth and high minimum distance by overcoming orthogonality constraints, resulting in improved decoding performance.

Contribution

It proposes a new construction technique for quantum LDPC codes using permutation matrices with controlled commutativity, enabling large girth and better error correction.

Findings

Constructed a girth-8, (3,12)-regular quantum LDPC code with 9216 qubits.

Achieved a frame error rate as low as 10^{-8} under BP decoding on the depolarizing channel.

Overcame structural limitations of orthogonality in quantum LDPC code design.

Abstract

Classical low-density parity-check (LDPC) codes are a widely deployed and well-established technology, forming the backbone of modern communication and storage systems. It is well known that, in this classical setting, increasing the girth of the Tanner graph while maintaining regular degree distributions leads simultaneously to good belief-propagation (BP) decoding performance and large minimum distance. In the quantum setting, however, this principle does not directly apply because quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. When one enforces both orthogonality and regularity in a straightforward manner, the girth is typically reduced and the minimum distance becomes structurally upper bounded. In this work, we overcome this limitation by using permutation matrices with controlled commutativity and by restricting the orthogonality constraints to only the active part of the construction, while preserving regular check-matrix structures. This design circumvents conventional structural distance limitations induced by parent-matrix orthogonality, and enables the construction of quantum LDPC codes with large girth while avoiding latent low-weight logical operators. As a concrete demonstration, we construct a girth-8, (3,12)-regular $[[9216,4612, \leq 48]]$ quantum LDPC code and show that, under BP decoding combined with a low-complexity post-processing algorithm, it achieves a frame error rate as low as $10^{-8}$ on the depolarizing channel with error probability $4 \%$.