Quantum LDPC Codes of Almost Linear Distance via Homological Products

TL;DR

This paper introduces new quantum LDPC codes with almost linear distance and dimension, constructed via homological products of codes with product-expansion properties, advancing quantum error correction capabilities.

Contribution

It demonstrates that homological products can produce quantum codes with near-linear distance, overcoming previous limitations, using a novel approach based on product-expansion and subsystem codes.

Findings

Constructed asymptotically good quantum codes with linear or near-linear distance.

Developed near-linear distance quantum LDPC codes with constant stabilizer weight.

Showed homological products can preserve good code distance under certain conditions.

Abstract

We present new constructions of quantum codes of linear or close-to-linear distance and dimension with low-weight stabilizers. Only a few constructions of such codes were previously known, and were primarily based on a specific operation from homological algebra, namely the balanced product. In contrast, our constructions are based on a more basic and widely used product, namely the homological product (i.e. the tensor product of chain complexes). Our results help address the natural question: When do homological products preserve good code distance? Our first main result constructs asymptotically good $[[N,\Theta(N),\Theta(N)]]$ quantum codes with small polynomial stabilizer weight from homological products of codes with a property called product-expansion. This notion was recently introduced and used to bound the distance of balanced product quantum codes; we apply it instead to homological products. For every $\epsilon>0$, our second main result constructs close-to-linear distance $[[N,N^{1-\epsilon},N^{1-\epsilon}]]$ (subsystem) quantum LDPC codes with constant stabilizer weight from iterated homological products of a constant-sized quantum locally testable code. The key insight here is that by using subsystem codes (but still with constant-weight stabilizers), we can circumvent a particular obstruction that limited the distance of many prior product code constructions to at most $\tilde{O}(\sqrt{N})$.