Deciding Whether a C-Q Channel Preserves a Bit is QCMA-Complete

2025-08-14

AI Summary

This work addresses the decision problem of whether a classical-quantum (C-Q) channel can perfectly preserve a classical bit. The authors formulate it as a biquadratic optimization problem subject to orthogonality constraints. They provide the first complete characterization of optimal witness states for the bit-preservation problem: the minimum is attained on computational basis states, while the maximum corresponds to the |+⟩/|−⟩ states within a single basis. By integrating matrix analysis with quantum information theory—and employing the Hilbert–Schmidt inner product to quantify output-state similarity—they rigorously prove that both the decision problem and its associated optimization task are QCMA-complete. This establishes the first known QCMA-complete problem featuring orthogonality constraints and introduces a novel analytical framework and completeness toolkit for structurally constrained optimization problems in quantum complexity theory.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We prove that deciding whether a classical-quantum (C-Q) channel can exactly preserve a single classical bit is QCMA-complete. This "bit-preservation" problem is a special case of orthogonality-constrained optimization tasks over C-Q channels, in which one seeks orthogonal input states whose outputs have small or large Hilbert-Schmidt overlap after passing through the channel. Both problems can be cast as biquadratic optimization with orthogonality constraints. Our main technical contribution uses tools from matrix analysis to give a complete characterization of the optimal witnesses: computational basis states for the minimum, and |+>, |-> over a single basis pair for the maximum. Using this characterization, we give concise proofs of QCMA-completeness for both problems.

Problem

• Decide if C-Q channel preserves a classical bit
• Solve orthogonality-constrained C-Q channel optimization
• Characterize optimal witnesses for bit-preservation problem

Innovation

• Characterizes optimal witnesses using matrix analysis
• Formulates problems as biquadratic optimization tasks
• Proves QCMA-completeness for bit-preservation problems