Quantum LDPC Codes with Almost Linear Minimum Distance

TL;DR

This paper introduces new quantum LDPC code constructions with near-linear minimum distance and dimension, utilizing chain complex products and a novel lifted product operation, also yielding classical LDPC codes with optimal circulant size.

Contribution

It presents a new construction of quantum LDPC codes with almost linear minimum distance and introduces the lifted product operation, expanding the toolkit for quantum code design.

Findings

Quantum LDPC codes with distance $ heta(N/ ext{log} N)$ and dimension $ heta( ext{log} N)$.

Family of quantum LDPC codes with distance $ ilde{ heta}(N^{1-rac{eta}{2}})$ and dimension $ ilde{ heta}(N^{eta})$ for $0 o eta < 1$.

Existence of classical quasi-cyclic LDPC codes with rate at least $R$ and optimal circulant size $ ilde{ heta}(N/ ext{log} N)$.

Abstract

We give a construction of quantum LDPC codes of dimension $\Theta(\log N)$ and distance $\Theta(N/\log N)$ as the code length $N\to\infty$. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance $\Omega(N^{1-\alpha/2}/\log N)$ and dimension $\Omega(N^\alpha \log N)$, where $0 \le \alpha < 1$. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixed $R < 1$ there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least $R$ with, in some sense, optimal circulant size $\Omega(N/\log N)$ as the code length $N\to\infty$.