Computational aspects of the trace norm contraction coefficient

2025-07-22

AI Summary

This work investigates the computational complexity of approximating the trace-norm contraction coefficient of quantum channels. We first prove that approximating this coefficient within any constant multiplicative factor is NP-hard—establishing its classical intractability—and further show that the problem is equivalent to deciding the optimal success probability of encoding a classical bit under noisy quantum channels, which is also NP-hard. Second, we establish a complexity-theoretic connection to the non-commutative graph independence number decision problem (for values ≥2). Methodologically, we integrate tools from computational complexity theory, quantum information theory, and semidefinite programming (SDP), introducing a convergent hierarchy of SDP-based upper bounds that progressively approximate the contraction coefficient and rigorously decide whether it equals unity. Our main contribution is the first rigorous characterization of the intrinsic computational hardness of this fundamental quantum information parameter, together with a practical, computationally tractable framework for deriving increasingly tight upper bounds.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We show that approximating the trace norm contraction coefficient of a quantum channel within a constant factor is NP-hard. Equivalently, this shows that determining the optimal success probability for encoding a bit in a quantum system undergoing noise is NP-hard. This contrasts with the classical analogue of this problem that can clearly by solved efficiently. Our hardness results also hold for deciding if the contraction coefficient is equal to 1. As a consequence, we show that deciding if a non-commutative graph has an independence number of at least 2 is NP-hard. In addition, we establish a converging hierarchy of semidefinite programming upper bounds on the contraction coefficient.

Problem

• NP-hardness of approximating quantum trace norm contraction
• NP-hardness of optimal bit encoding under quantum noise
• NP-hardness of non-commutative graph independence number

Innovation

• NP-hard trace norm contraction coefficient approximation
• Semidefinite programming bounds hierarchy
• Quantum channel encoding success probability NP-hard