Technology Category
Machine Learning: Quantum Machine LearningKnowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability
The color code, the surface code, and the transversal CNOT: NP-hardness of minimum-weight decoding
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
minimum-weight decoding
NP-hardness
color code
surface code
transversal CNOT
Innovation
Methods, ideas, or system contributions that make the work stand out.
minimum-weight decoding
NP-hardness
color code
surface code
transversal CNOT
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