On estimating operator norm distance, with optimal trace distance estimation when one state is pure

2026-07-04

AI Summary

This work addresses the efficient estimation of the operator norm distance between two n-qubit quantum states prepared by polynomial-size quantum circuits. When one state is pure, the authors exploit its structural overlap with the leading eigenvector of the difference matrix to construct an optimal estimator, achieving—for the first time—a rank-independent query complexity of Θ(1/ε). For general mixed states, they propose an improved algorithm that reduces the query complexity to Õ(1/ε^{3/2}) and establish that the problem is BQP-complete, thereby tightening the previously known QMA upper bound to BQP. Their approach combines black-box state preparation, spectral analysis, and quantum amplitude estimation, enabling estimation with constant error in poly(n) time for the pure-state case.

Abstract

We investigate the computational complexity of estimating the operator norm distance ${\rm T}_{\infty}(ρ_0,ρ_1)$, defined via the operator norm $\|A\|_{\infty} = σ_{\max}(A)$, given ${\rm poly}(n)$-size state-preparation circuits of $n$-qubit quantum states $ρ_0$ and $ρ_1$. We provide efficient quantum estimators for the operator norm distance whose complexity is independent of the rank (and thus the dimension) of the states: 1. When one state is pure, we establish an optimal quantum estimator using $Θ(1/ε)$ queries to the state-preparation circuits. Consequently, for constant additive error, say $ε=1/5$, our estimator runs in ${\rm poly}(n)$ time. Since the operator norm distance ${\rm T}_{\infty}(|ψ\rangle\!\langleψ|,ρ)$ is exactly half of the trace distance ${\rm T}(|ψ\rangle\!\langleψ|,ρ)$, our result also gives rank-independent query complexity for estimating both quantities, whereas the approaches due to van Apeldoorn, Cornelissen, Gily{é}n, and Nannicini (SODA 2023) and Wang and Zhang (TIT 2024) have query complexity scaling at least linearly with ${\rm rank}(ρ)$, which can be $\exp(n)$ in general. 2. For general quantum states, we also provide a quantum estimator using $\widetilde{O}(1/ε^{3/2})$ queries to the state-preparation circuits, which shows that the corresponding promise problem is ${\sf BQP}$-complete and improves the ${\sf QMA}$ upper bound sketched by Liu and Wang (ESA 2025). Together with an $Ω(1/ε)$ quantum query complexity lower bound, this leaves only square-root room for improvement. The key intuition behind our estimators is that, when one state is pure, the pure state $|ψ\rangle$ has overlap at least $1/2$ with the top unit eigenvector of $|ψ\rangle\!\langleψ|-ρ$, reflecting a structural feature specific to the operator norm distance.

Problem

• operator norm distance
• quantum state estimation
• trace distance
• pure quantum state
• query complexity

Innovation

• operator norm distance
• quantum state estimation
• rank-independent complexity
• trace distance
• BQP-completeness