The Need for Structure in Quantum LDPC Codes

TL;DR

This paper investigates the limitations of high-dimensional expander-based quantum LDPC codes, showing that their structure constraints prevent achieving linear minimal distance, and provides bounds on related topological properties.

Contribution

It demonstrates that quantum codes derived from certain high-dimensional expanders cannot attain linear distance and establishes bounds on systoles and discrepancy in these complexes.

Findings

Quantum LDPC codes from non-random complexes have small minimal distance.

Provides an upper bound on systole of high-dimensional expanders with small discrepancy.

Establishes a lower bound on discrepancy of Ramanujan complex skeletons.

Abstract

Existence of quantum low-density parity-check (LDPC) codes whose minimal distance scales linearly with the number of qubits is a major open problem in quantum information. Its practical interest stems from the need to protect information in a future quantum computer, and its theoretical appeal relates to a deep "global-to-local" notion in quantum mechanics: whether we can constrain long-range entanglement using local checks. Given the inability of lattice-based quantum LDPC codes to achieve linear distance, research has recently shifted to the other extreme end of topologies, so called high-dimensional expanders. In this work we show that trying to leverage the mere "random-like" property of these expanders to find good quantum codes may be futile: quantum CSS codes of $n$ quits built from $d$-complexes that are $\varepsilon$-far from perfectly random, in a well-known sense called discrepancy, have a small minimal distance. Quantum codes aside, our work places a first upper-bound on the systole of high-dimensional expanders with small discrepancy, and a lower-bound on the discrepancy of skeletons of Ramanujan complexes due to Lubotzky.