Minimum Weight Decoding in the Colour Code is NP-hard

TL;DR

This paper proves that exact decoding of the colour code in quantum error correction is NP-hard, indicating no efficient polynomial-time algorithm exists unless P=NP, and emphasizes the need for heuristic approaches.

Contribution

It establishes the computational complexity of colour code decoding as NP-hard, a significant theoretical result contrasting with the polynomial-time decoding of surface codes.

Findings

Decoding the colour code is NP-hard.

Exact decoding cannot be efficiently computed unless P=NP.

Heuristic methods are necessary for practical colour code decoding.

Abstract

All utility-scale quantum computers will require some form of Quantum Error Correction in which logical qubits are encoded in a larger number of physical qubits. One promising encoding is known as the colour code which has broad applicability across all qubit types and can decisively reduce the overhead of certain logical operations when compared to other two-dimensional topological codes such as the surface code. However, whereas the surface code decoding problem can be solved exactly in polynomial time by finding minimum weight matchings in a graph, prior to this work, it was not known whether exact and efficient colour code decoding was possible. Optimism in this area, stemming from the colour code's significant structure and well understood similarities to the surface code, fanned this uncertainty. In this paper we resolve this, proving that exact decoding of the colour code is NP-hard -- that is, there does not exist a polynomial time algorithm unless P=NP. This highlights a notable contrast to some of the colour code's key competitors, such as the surface code, and motivates continued work in the narrower space of heuristic and approximate algorithms for fast, accurate and scalable colour code decoding.