On estimating the quantum $ell_{alpha}$ distance

2025-05-01

AI Summary

This work studies the efficient estimation of the quantum ℓₐ distance (T_alpha( ho_0, ho_1)) for (alpha > 1) and the computational complexity of the corresponding Quantum State Distinguishability problem (QSD_alpha). For (n)-qubit states ( ho_0, ho_1) specified by polynomial-size quantum circuits, we construct the first rank-independent quantum estimator—leveraging Quantum Singular Value Transformation (QSVT), uniform polynomial approximations to signed power functions, and Schatten-(alpha) norm analysis—to achieve a (mathrm{poly}(n))-time estimation algorithm, yielding exponential speedup over prior exponential-time methods. We rigorously prove that (QSD_alpha) is BQP-complete for any constant (alpha > 1), while (QSD_1) (i.e., trace-distance distinguishability) is QSZK-complete—establishing a fundamental complexity dichotomy between these cases. This work provides the first efficient estimation algorithm for quantum ℓₐ distances and delivers the precise complexity characterization of (QSD_alpha).

Technology Category

• Machine Learning: Quantum Machine Learning
• Intelligent Robots: State Estimation
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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 learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the computational complexity of estimating the quantum $ell_{alpha}$ distance ${mathrm{T}_alpha}( ho_0, ho_1)$, defined via the Schatten $alpha$-norm $|A|_{alpha} = mathrm{tr}(|A|^{alpha})^{1/alpha}$, given $operatorname{poly}(n)$-size state-preparation circuits of $n$-qubit quantum states $ ho_0$ and $ ho_1$. This quantity serves as a lower bound on the trace distance for $alpha>1$. For any constant $alpha>1$, we develop an efficient rank-independent quantum estimator for ${mathrm{T}_alpha}( ho_0, ho_1)$ with time complexity $operatorname{poly}(n)$, achieving an exponential speedup over the prior best results of $exp(n)$ due to Wang, Guan, Liu, Zhang, and Ying (TIT 2024). Our improvement leverages efficiently computable uniform polynomial approximations of signed positive power functions within quantum singular value transformation, thereby eliminating the dependence on the rank of the quantum states. Our quantum algorithm reveals a dichotomy in the computational complexity of the Quantum State Distinguishability Problem with Schatten $alpha$-norm (QSD$_{alpha}$), which involves deciding whether ${mathrm{T}_alpha}( ho_0, ho_1)$ is at least $2/5$ or at most $1/5$. This dichotomy arises between the cases of constant $alpha>1$ and $alpha=1$: - For any $1+Omega(1) leq alpha leq O(1)$, QSD$_{alpha}$ is $mathsf{BQP}$-complete. - For any $1 leq alpha leq 1+frac{1}{n}$, QSD$_{alpha}$ is $mathsf{QSZK}$-complete, implying that no efficient quantum estimator for $mathrm{T}_alpha( ho_0, ho_1)$ exists unless $mathsf{BQP} = mathsf{QSZK}$. The hardness results follow from reductions based on new rank-dependent inequalities for the quantum $ell_{alpha}$ distance with $1leq alpha leq infty$, which are of independent interest.

Problem

• Estimating quantum $ell_alpha$ distance complexity
• Developing efficient quantum estimator for $mathrm{T}_alpha$
• Analyzing computational dichotomy in QSD$_alpha$ problem

Innovation

• Efficient quantum estimator for Schatten α-norm
• Uniform polynomial approximations in quantum transformations
• Rank-independent quantum state distinguishability solution