The color code, the surface code, and the transversal CNOT: NP-hardness of minimum-weight decoding

TL;DR

This paper proves that minimum-weight decoding in key quantum error correction codes is NP-hard, revealing fundamental computational limits in fault-tolerant quantum computing and impacting practical decoding strategies.

Contribution

It establishes NP-hardness of minimum-weight decoding for the color code and surface code in various error models, a significant complexity result in quantum error correction.

Findings

Decoding NP-hardness applies to color and surface codes with different error types.

Computational intractability exists even in basic, practically relevant decoding problems.

Highlights a complexity gap between exact decoding and approximate methods.

Abstract

The decoding problem is a ubiquitous algorithmic task in fault-tolerant quantum computing, and solving it efficiently is essential for scalable quantum computing. Here, we prove that minimum-weight decoding is NP-hard in three quintessential settings: (i) the color code with Pauli $Z$ errors, (ii) the surface code with Pauli $X$, $Y$ and $Z$ errors, and (iii) the surface code with a transversal CNOT gate, Pauli $Z$ and measurement bit-flip errors. Our results show that computational intractability already arises in basic and practically relevant decoding problems central to both quantum memories and logical circuit implementations, highlighting a sharp computational complexity separation between minimum-weight decoding and its approximate realizations.