Locally Decodable Quantum Codes

TL;DR

This paper explores quantum analogues of locally decodable error-correcting codes, demonstrating that quantum codes with a small number of queries do not significantly outperform classical codes in terms of decoding efficiency and noise tolerance.

Contribution

It establishes a transformation from quantum to classical locally decodable codes, showing quantum codes offer limited advantage over classical ones for small query numbers.

Findings

Quantum locally decodable codes can be converted into classical codes with similar properties.

Quantum codes do not significantly outperform classical codes for small query numbers.

The transformation preserves decoding success probability and noise tolerance to a certain extent.

Abstract

We study a quantum analogue of locally decodable error-correcting codes. A q-query locally decodable quantum code encodes n classical bits in an m-qubit state, in such a way that each of the encoded bits can be recovered with high probability by a measurement on at most q qubits of the quantum code, even if a constant fraction of its qubits have been corrupted adversarially. We show that such a quantum code can be transformed into a classical q-query locally decodable code of the same length that can be decoded well on average (albeit with smaller success probability and noise-tolerance). This shows, roughly speaking, that q-query quantum codes are not significantly better than q-query classical codes, at least for constant or small q.