Quantum error correcting codes based on privacy amplification

TL;DR

This paper introduces a new construction of CSS quantum error-correcting codes using classical codes and two-universal hash functions, achieving rates close to the hashing bound for large block lengths.

Contribution

It proposes a novel CSS code construction combining classical codes with hash functions, improving code rate bounds for quantum error correction.

Findings

Communication rates approach the hashing bound for large block-lengths.

Bit-flip errors can be efficiently decoded using classical methods.

Efficient decoding of phase-flip errors remains an open problem.

Abstract

Calderbank-Shor-Steane (CSS) quantum error-correcting codes are based on pairs of classical codes which are mutually dual containing. Explicit constructions of such codes for large blocklengths and with good error correcting properties are not easy to find. In this paper we propose a construction of CSS codes which combines a classical code with a two-universal hash function. We show, using the results of Renner and Koenig, that the communication rates of such codes approach the hashing bound on tensor powers of Pauli channels in the limit of large block-length. While the bit-flip errors can be decoded as efficiently as the classical code used, the problem of efficiently decoding the phase-flip errors remains open.