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

2026-03-23

AI Summary

This work investigates the computational complexity of minimum-weight decoding in fault-tolerant quantum computing. Focusing on canonical error-correcting settings—namely color codes, surface codes, and architectures supporting transversal CNOT gates—and incorporating realistic noise models such as Pauli errors and measurement bit flips, the study employs computational complexity theory and polynomial-time reductions to rigorously establish, for the first time, that minimum-weight decoding is NP-hard in all three scenarios. This result underscores a fundamental complexity gap between exact decoding and feasible approximation algorithms, thereby delineating critical theoretical limits for the design of practical quantum error correction decoders.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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.

Problem

• minimum-weight decoding
• NP-hardness
• color code
• surface code
• transversal CNOT

Innovation

• minimum-weight decoding
• NP-hardness
• color code
• surface code
• transversal CNOT