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Optimal Quantum Algorithm for Estimating Fidelity to a Pure State

Authors Wang Fang , Qisheng Wang

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Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • Theory of computation → Quantum information theory
Keywords
  • Quantum computing
  • fidelity estimation
  • quantum algorithms
  • quantum query complexity

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Abstract

We present an optimal quantum algorithm for fidelity estimation between two quantum states when one of them is pure. In particular, the (square root) fidelity of a mixed state to a pure state can be estimated to within additive error ε by using Θ(1/ε) queries to their state-preparation circuits, achieving a quadratic speedup over the folklore O(1/ε²). Our approach is technically simple, and can moreover estimate the quantity √{tr(ρσ²)} that is not common in the literature. To the best of our knowledge, this is the first query-optimal approach to fidelity estimation involving mixed states.

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Wang Fang and Qisheng Wang. Optimal Quantum Algorithm for Estimating Fidelity to a Pure State. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.4

Author Details

Wang Fang
  • School of Informatics, University of Edinburgh, UK
Qisheng Wang
  • School of Informatics, University of Edinburgh, UK

Funding

  • Fang, Wang: Supported by the Engineering and Physical Sciences Research Council under Grant EP/X025551/1.
  • Wang, Qisheng: Supported by the Engineering and Physical Sciences Research Council under Grant EP/X026167/1.

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