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Time-Efficient Quantum Entropy Estimator via Samplizer

Authors Qisheng Wang , Zhicheng Zhang

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ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Mathematics of computing → Information theory
  • Theory of computation → Algorithm design techniques
  • Theory of computation → Lower bounds and information complexity
Keywords
  • Quantum computing
  • entropy estimation
  • von Neumann entropy
  • Rényi entropy
  • sample complexity

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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 quantum estimators.  
- A quantum estimator for S(ρ) with time complexity Õ(N²), improving the prior best time complexity Õ(N⁶) by Acharya, Issa, Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016). 
- A quantum estimator for S_α(ρ) with time complexity Õ(N^{4/α-2}) for 0 < α < 1 and Õ(N^{4-2/α}) for α > 1, improving the prior best time complexity Õ(N^{6/α}) for 0 < α < 1 and Õ(N⁶) 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 Ω(max{N/ε, N^{1/α-1}/ε^{1/α}}) 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 Θ~(Q²/δ) samples of ρ. Moreover, this samplization is proven to be optimal, up to a polylogarithmic factor.

Cite As Get BibTex

Qisheng Wang and Zhicheng Zhang. Time-Efficient Quantum Entropy Estimator via Samplizer. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 101:1-101:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.101

Author Details

Qisheng Wang
  • Graduate School of Mathematics, Nagoya University, Japan
Zhicheng Zhang
  • Centre for Quantum Software and Information, University of Technology Sydney, Ultimo, Australia

Funding

  • Wang, Qisheng: Supported by the MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) under Grant JPMXS0120319794.
  • Zhang, Zhicheng: Supported by the Sydney Quantum Academy, NSW, Australia, and the Australian Research Council Discovery Project under Grant DP220102059.

Acknowledgements

The authors thank John Wright for valuable comments and sharing their results [Bavarian et al., 2016] on von Neumann entropy estimation. QW also thanks François Le Gall for helpful discussions.

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