Time-Efficient Quantum Entropy Estimator via Samplizer
Qisheng Wang, Zhicheng Zhang
TL;DR
This work introduces samplizer, a unifying framework that converts quantum query algorithms into quantum sample algorithms, enabling time-efficient entropy estimation from independent samples of a quantum state. The authors present a von Neumann entropy estimator with time complexity $\tilde O(N^2)$ (and rank-$r$-dependent $\tilde O(r^2)$) and Rényi entropy estimators with complexities $\tilde O(N^{4/\alpha-2})$ for $0<\alpha<1$ and $\tilde O(N^{4-2/\alpha})$ for $\alpha>1$, at the cost of increased sample complexity relative to prior work. The samplizer achieves optimality up to polylogarithmic factors and is backed by lower bounds for entropy estimation, including a refined bound for $0<\alpha<1$. The framework also extends naturally to low-rank states and connects to density-matrix exponentiation, block-encoding, and polynomial approximations, offering a versatile toolkit for quantum entropy estimation and potentially other quantum property estimation tasks.
Abstract
Entropy is a measure of the randomness of a system. Estimating the entropy of a quantum state is a basic problem in quantum information. In this paper, we introduce a time-efficient quantum approach to estimating the von Neumann entropy $S(ρ)$ and Rényi entropy $S_α(ρ)$ of an $N$-dimensional quantum state $ρ$, given access to independent samples of $ρ$. Specifically, we provide the following: 1. A quantum estimator for $S(ρ)$ with time complexity $\tilde O(N^2)$, improving the prior best time complexity $\tilde O(N^6)$ by Acharya, Issa, Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016). 2. A quantum estimator for $S_α(ρ)$ with time complexity $\tilde O(N^{4/α-2})$ for $0<α<1$ and $\tilde O(N^{4-2/α})$ for $α>1$, improving the prior best time complexity $\tilde O(N^{6/α})$ for $0<α<1$ and $\tilde O(N^6)$ for $α>1$ by Acharya, Issa, Shende, and Wagner (2020), though at a cost of a slightly larger sample complexity. Moreover, these estimators are naturally extensible to the low-rank case. We also provide a sample lower bound for estimating $S_α(ρ)$. Technically, our method is quite different from the previous ones that are based on weak Schur sampling and Young diagrams. At the heart of our construction, is a novel tool called samplizer, which can "samplize" a quantum query algorithm to a quantum algorithm with similar behavior using only samples of quantum states; this suggests a unified framework for estimating quantum entropies. Specifically, when a quantum oracle $U$ block-encodes a mixed quantum state $ρ$, any quantum query algorithm using $Q$ queries to $U$ can be samplized to a $δ$-close (in the diamond norm) quantum algorithm using $\tildeΘ(Q^2/δ)$ samples of $ρ$. Moreover, this samplization is proven to be optimal, up to a polylogarithmic factor.