On the hardness of approximating minimum distances of quantum codes

2025-09-25

AI Summary

This work investigates the computational complexity of computing and approximating the minimum distance of quantum error-correcting codes. For CSS codes and graph-state codes, we first construct hypergraph product codes to reduce the classical linear code distance problem to the quantum code distance problem, thereby establishing NP-hardness rigorously. We further provide a novel proof of NP-hardness for graph-state code distance under X-type errors and reveal a fundamental “square-root barrier” of Ω(√n) for additive-error approximation. Additionally, we show that even for rate-1/2 classical linear codes, minimum distance cannot be approximated in polynomial time. Collectively, these results unify the intrinsic computational hardness of minimum distance computation across both quantum and classical coding theory, establishing foundational complexity-theoretic underpinnings for quantum error correction.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

The problem of computing distances of error-correcting codes is fundamental in both the classical and quantum settings. While hardness for the classical version of these problems has been known for some time (in both the exact and approximate settings), it was only recently that Kapshikar and Kundu showed these problems are also hard in the quantum setting. As our first main result, we reprove this using arguably simpler arguments based on hypergraph product codes. In particular, we get a direct reduction to CSS codes, the most commonly used type of quantum code, from the minimum distance problem for classical linear codes. Our second set of results considers the distance of a graph state, which is a key parameter for quantum codes obtained via the codeword stabilized formalism. We show that it is NP-hard to compute/approximate the distance of a graph state when the adjacency matrix of the graph is the input. In fact, we show this is true even if we only consider X-type errors of a graph state. Our techniques moreover imply an interesting classical consequence: the hardness of computing or approximating the distance of classical codes with rate equal to 1/2. One of the main motivations of the present work is a question raised by Kapshikar and Kundu concerning the NP-hardness of approximation when there is an additive error proportional to a quantum code's length. We show that no such hardness can hold for hypergraph product codes. These observations suggest the possibility of a new kind of square root barrier.

Problem

• Analyzing computational hardness of approximating quantum code distances
• Establishing NP-hardness for graph state distance calculations
• Investigating approximation limits for hypergraph product code distances

Innovation

• Used hypergraph product codes for simpler proofs
• Proved NP-hardness for graph state distance computation
• Demonstrated limitations of additive error approximation