Quantum "hyperbicycle" low-density parity check codes with finite rate

TL;DR

This paper introduces hyperbicycle quantum LDPC codes that generalize existing quantum codes, offering finite rate and scalable minimum distance, with potential for improved parameters and error thresholds.

Contribution

The authors propose a new hyperbicycle ansatz for quantum codes that unifies and extends previous hypergraph-product and bicycle codes, enabling better parameter flexibility.

Findings

Codes have minimum distance scaling as square root of block length

Many codes possess finite rate and low-weight stabilizers

Hyperbicycle codes can achieve higher rate while maintaining error threshold

Abstract

We introduce a "hyperbicycle" ansatz for quantum codes which gives the hypergraph-product (generalized toric) codes by Tillich and Z\'emor and generalized bicycle codes by MacKay et al. as limiting cases. The construction allows for both the lower and the upper bounds on the minimum distance; they scale as a square root of the block length. Many of thus defined codes have finite rate and a limited-weight stabilizer generators, an analog of classical low-density parity check (LDPC) codes. Compared to the hypergraph-product codes, hyperbicycle codes generally have wider range of parameters; in particular, they can have higher rate while preserving the (estimated) error threshold.