Hardness of decoding quantum stabilizer codes

TL;DR

This paper proves that optimal decoding of quantum stabilizer codes is computationally much harder than classical decoding, being #P-complete, which has significant implications for quantum error correction.

Contribution

It establishes that quantum stabilizer code decoding is #P-hard, highlighting a fundamental computational difficulty not present in classical decoding.

Findings

Optimal quantum decoding is #P-complete.

Classical decoding is NP-complete, but quantum decoding is even harder.

Decoding complexity impacts quantum error correction strategies.

Abstract

In this article we address the computational hardness of optimally decoding a quantum stabilizer code. Much like classical linear codes, errors are detected by measuring certain check operators which yield an error syndrome, and the decoding problem consists of determining the most likely recovery given the syndrome. The corresponding classical problem is known to be NP-complete, and a similar decoding problem for quantum codes is also known to be NP-complete. However, this decoding strategy is not optimal in the quantum setting as it does not take into account error degeneracy, which causes distinct errors to have the same effect on the code. Here, we show that optimal decoding of stabilizer codes is computationally much harder than optimal decoding of classical linear codes, it is #P.