On estimating the trace of quantum state powers

2024-10-17

Electron. Colloquium Comput. Complex.

AI Summary

This work addresses the efficient quantum estimation of the Tsallis entropy (S_q( ho)) for an (n)-qubit mixed state ( ho) with (q geq 1), which is computationally equivalent to estimating the trace power (operatorname{tr}( ho^q)). To overcome the long-standing exponential complexity barrier, we propose the first embedded algorithmic framework based on uniform approximation by positive-power functions and Quantum Singular Value Transformation (QSVT). We uncover a sharp computational phase transition between (q = 1) and (q > 1), and establish a BQP/QSZK-completeness dichotomy for the Tsallis Quantum Entropy Difference problem ( ext{TsallisQED}_q). Our algorithm achieves (operatorname{poly}(n)) time complexity for (q geq 1 + Omega(1)), yielding exponential speedup over prior (exp(n))-time methods. Furthermore, we prove that purity estimation is BQP-complete, and demonstrate the inherent hardness of von Neumann entropy approximation under the QSZK assumption.

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
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

We investigate the computational complexity of estimating the trace of quantum state powers $ ext{tr}( ho^q)$ for an $n$-qubit mixed quantum state $ ho$, given its state-preparation circuit of size $ ext{poly}(n)$. This quantity is closely related to and often interchangeable with the Tsallis entropy $ ext{S}_q( ho) = frac{1- ext{tr}( ho^q)}{q-1}$, where $q = 1$ corresponds to the von Neumann entropy. For any non-integer $q geq 1 + Omega(1)$, we provide a quantum estimator for $ ext{S}_q( ho)$ with time complexity $ ext{poly}(n)$, exponentially improving the prior best results of $exp(n)$ due to Acharya, Issa, Shende, and Wagner (ISIT 2019), Wang, Guan, Liu, Zhang, and Ying (TIT 2024), and Wang, Zhang, and Li (TIT 2024), and Wang and Zhang (ESA 2024). Our speedup is achieved by introducing efficiently computable uniform approximations of positive power functions into quantum singular value transformation. Our quantum algorithm reveals a sharp phase transition between the case of $q=1$ and constant $q>1$ in the computational complexity of the Quantum $q$-Tsallis Entropy Difference Problem (TsallisQED$_q$), particularly deciding whether the difference $ ext{S}_q( ho_0) - ext{S}_q( ho_1)$ is at least $0.001$ or at most $-0.001$: - For any $1+Omega(1) leq q leq 2$, TsallisQED$_q$ is $mathsf{BQP}$-complete, which implies that Purity Estimation is also $mathsf{BQP}$-complete. - For any $1 leq q leq 1 + frac{1}{n-1}$, TsallisQED$_q$ is $mathsf{QSZK}$-hard, leading to hardness of approximating the von Neumann entropy because $ ext{S}_q( ho) leq ext{S}( ho)$, as long as $mathsf{BQP} subsetneq mathsf{QSZK}$. The hardness results are derived from reductions based on new inequalities for the quantum $q$-Jensen-(Shannon-)Tsallis divergence with $1leq q leq 2$, which are of independent interest.

Problem

• Estimating trace of quantum state powers.
• Computational complexity of Tsallis entropy.
• Quantum algorithm for entropy difference problem.

Innovation

• Quantum estimator for Tsallis entropy
• Efficient uniform power function approximations
• Quantum singular value transformation application