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Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer

Authors Qisheng Wang , Zhicheng Zhang

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LIPIcs.ICALP.2026.154.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Algorithm design techniques
Keywords
  • Quantum algorithms
  • sample complexity
  • trace distance
  • fidelity
  • pure states
  • lower bounds
  • samplizer

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Abstract

We settle the problem of estimating the trace distance and (square root) fidelity between n-qubit pure quantum states to within additive error ε, given their independent samples, which was raised as an open question by Wang (IEEE Trans. Inf. Theory 2024). This is achieved by a quantum algorithm with optimal sample complexity Θ(1/ε²), improving the long-standing folklore with sample complexity O(1/ε⁴). At the heart of our algorithm is a samplized phase estimation of the product of two Householder reflections. This is realized by an improved (multi-)samplizer for pure states, through which any quantum query algorithm using Q queries to the reflection operator I - 2 |ψ⟩⟨ψ| can be converted to a δ-close (in the diamond norm distance) quantum sample algorithm using Θ(Q²/δ) samples of the state |ψ⟩. This samplizer for pure states is also shown to be optimal.

Cite As Get BibTex

Qisheng Wang and Zhicheng Zhang. Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 154:1-154:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.154

Author Details

Qisheng Wang
  • School of Computer Science, Shanghai Jiao Tong University, Shanghai, China
Zhicheng Zhang
  • School of Information and Communication Technology, Griffith University, Brisbane, Australia

Funding

  • Wang, Qisheng: Supported by startup funding from Shanghai Jiao Tong University.
  • Zhang, Zhicheng: Supported in part by the Australian Research Council under Grant DP250102952.

Acknowledgements

Part of the work of Zhicheng Zhang was done when the author was with the Centre for Quantum Software and Information, University of Technology Sydney, Australia.

References

  1. Anurag Anshu, Zeph Landau, and Yunchao Liu. Distributed quantum inner product estimation. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 44-51, 2022. URL: https://doi.org/10.1145/3519935.3519974.
  2. Srinivasan Arunachalam and Louis Schatzki. Generalized inner product estimation with limited quantum communication. In Proceedings of the 42nd International Symposium on Theoretical Aspects of Computer Science, pages 11:1-11:17, 2025. URL: https://doi.org/10.4230/LIPIcs.STACS.2025.11.
  3. Costin Bădescu, Ryan O'Donnell, and John Wright. Quantum state certification. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 503-514, 2019. URL: https://doi.org/10.1145/3313276.3316344.
  4. Joonwoo Bae and Leong-Chuan Kwek. Quantum state discrimination and its applications. Journal of Physics A: Mathematical and Theoretical, 48(8):083001, 2015. URL: https://doi.org/10.1088/1751-8113/48/8/083001.
  5. Adriano Barenco, André Berthiaume, David Deutsch, Artur Ekert, Richard Jozsa, and Chiara Macchiavello. Stabilization of quantum computations by symmetrization. SIAM Journal on Computing, 26(5):1541-1557, 1997. URL: https://doi.org/10.1137/S0097539796302452.
  6. Stephen M. Barnett and Sarah Croke. Quantum state discrimination. Advances in Optics and Photonics, 1(2):238-278, 2009. URL: https://doi.org/10.1364/AOP.1.000238.
  7. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. Journal of the ACM, 48(4):778-797, 2001. URL: https://doi.org/10.1145/502090.502097.
  8. Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation. In Samuel J. Lomonaco, Jr. and Howard E. Brandt, editors, Quantum Computation and Information, volume 305 of Contemporary Mathematics, pages 53-74. AMS, 2002. URL: https://doi.org/10.1090/conm/305/05215.
  9. Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. Quantum fingerprinting. Physical Review Letters, 87(16):167902, 2001. URL: https://doi.org/10.1103/PhysRevLett.87.167902.
  10. Anthony Chefles. Quantum state discrimination. Contemporary Physics, 41(6):401-424, 2000. URL: https://doi.org/10.1080/00107510010002599.
  11. Kean Chen, Qisheng Wang, and Zhicheng Zhang. Local test for unitarily invariant properties of bipartite quantum states. IEEE Transactions on Information Theory, 2026. URL: https://doi.org/10.1109/TIT.2026.3697790.
  12. Pedro C. S. Costa, Dong An, Yuval R. Sanders, Yuan Su, Ryan Babbush, and Dominic W. Berry. Optimal scaling quantum linear-systems solver via discrete adiabatic theorem. PRX Quantum, 3(4):040303, 2022. URL: https://doi.org/10.1103/PRXQuantum.3.040303.
  13. Steven T. Flammia and Yi-Kai Liu. Direct fidelity estimation from few Pauli measurements. Physical Review Letters, 106(23):230501, 2011. URL: https://doi.org/10.1103/PhysRevLett.106.230501.
  14. András Gilyén and Alexander Poremba. Improved quantum algorithms for fidelity estimation. ArXiv e-prints, 2022. URL: https://arxiv.org/abs/2203.15993.
  15. András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 193-204, 2019. URL: https://doi.org/10.1145/3313276.3316366.
  16. Byeongseon Go, Hyukjoon Kwon, Siheon Park, Dhrumil Patel, and Mark M. Wilde. Density matrix exponentiation and sample-based Hamiltonian simulation: Non-asymptotic analysis of sample complexity. ArXiv e-prints, 2024. URL: https://arxiv.org/abs/2412.02134.
  17. Weiyuan Gong, Jonas Haferkamp, Qi Ye, and Zhihan Zhang. On the sample complexity of purity and inner product estimation. ArXiv e-prints, 2024. URL: https://doi.org/10.48550/arXiv.2410.12712.
  18. David Gross, Yi-Kai Liu, Steven T. Flammia, Stephen Becker, and Jens Eisert. Quantum state tomography via compressed sensing. Physical Review Letters, 105(15):150401, 2010. URL: https://doi.org/10.1103/PhysRevLett.105.150401.
  19. Otfried Gühne, Chao-Yang Lu, Wei-Bo Gao, and Jian-Wei Pan. Toolbox for entanglement detection and fidelity estimation. Physical Review A, 76(3):030305(R), 2007. URL: https://doi.org/10.1103/PhysRevA.76.030305.
  20. Otfried Gühne and Géza Tóth. Entanglement detection. Physics Reports, 474(1-6):1-75, 2009. URL: https://doi.org/10.1016/j.physrep.2009.02.004.
  21. Jeongwan Haah, Aram W. Harrow, Zhengfeng Ji, Xiaodi Wu, and Nengkun Yu. Sample-optimal tomography of quantum states. IEEE Transactions on Information Theory, 63(9):5628-5641, 2017. URL: https://doi.org/10.1109/TIT.2017.2719044.
  22. Lisa Hales and Sean Hallgren. An improved quantum fourier transform algorithm and applications. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pages 515-525, 2000. URL: https://doi.org/10.1109/SFCS.2000.892139.
  23. Carl W. Helstrom. Detection theory and quantum mechanics. Information and Control, 10(3):254-291, 1967. URL: https://doi.org/10.1016/S0019-9958(67)90302-6.
  24. Alexander S. Holevo. Statistical decision theory for quantum systems. Journal of Multivariate Analysis, 3(4):337-394, 1973. URL: https://doi.org/10.1016/0047-259X(73)90028-6.
  25. Hsin-Yuan Huang, Richard Kueng, and John Preskill. Predicting many properties of a quantum system from very few measurements. Nature Physics, 16(10):1050-1057, 2020. URL: https://doi.org/10.1038/s41567-020-0932-7.
  26. Shelby Kimmel, Cedric Yen-Yu Lin, Guang Hao Low, Maris Ozols, and Theodore J. Yoder. Hamiltonian simulation with optimal sample complexity. npj Quantum Information, 3(1):1-7, 2017. URL: https://doi.org/10.1038/s41534-017-0013-7.
  27. A. Yu. Kitaev. Quantum measurements and the Abelian stabilizer problem. ArXiv e-prints, 1995. URL: https://arxiv.org/abs/quant-ph/9511026.
  28. Hirotada Kobayashi, Keiji Matsumoto, and Tomoyuki Yamakami. Quantum Merlin-Arthur proof systems: are multiple Merlins more helpful to Arthur? Chicago Journal of Theoretical Computer Science, 2009:3, 2009. URL: https://doi.org/10.4086/cjtcs.2009.003.
  29. François Le Gall, Yupan Liu, and Qisheng Wang. Space-bounded quantum state testing via space-efficient quantum singular value transformation. Computational Complexity, 2026. URL: https://doi.org/10.1007/s00037-025-00284-5.
  30. Nana Liu, Qisheng Wang, Mark M. Wilde, and Zhicheng Zhang. Quantum algorithms for matrix geometric means. npj Quantum Information, 11:101, 2025. URL: https://doi.org/10.1038/s41534-025-00973-7.
  31. Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quantum principal component analysis. Nature Physics, 10(9):631-633, 2014. URL: https://doi.org/10.1038/nphys3029.
  32. Ashwin Nayak and Felix Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of the 31st Annual ACM Symposium on Theory of Computing, pages 384-393, 1999. URL: https://doi.org/10.1145/301250.301349.
  33. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2010. URL: https://doi.org/10.1017/CBO9780511976667.
  34. Ryan O'Donnell and John Wright. Efficient quantum tomography. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing, pages 899-912, 2016. URL: https://doi.org/10.1145/2897518.2897544.
  35. Ryan O'Donnell and John Wright. Quantum spectrum testing. Communications in Mathematical Physics, 387(1):1-75, 2021. URL: https://doi.org/10.1007/s00220-021-04180-1.
  36. Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484-1509, 1997. URL: https://doi.org/10.1137/S0097539795293172.
  37. Yuuki Tokunaga, Takashi Yamamoto, Masato Koashi, and Nobuyuki Imoto. Fidelity estimation and entanglement verification for experimentally produced four-qubit cluster states. Physical Review A, 74(2):020301(R), 2006. URL: https://doi.org/10.1103/PhysRevA.74.020301.
  38. Gregory Valiant and Paul Valiant. Estimating the unseen: improved estimators for entropy and other properties. Journal of the ACM, 64(6):37:1-37:41, 2017. URL: https://doi.org/10.1145/3125643.
  39. Qisheng Wang. Optimal trace distance and fidelity estimations for pure quantum states. IEEE Transactions on Information Theory, 70(12):8791-8805, 2024. URL: https://doi.org/10.1109/TIT.2024.3447915.
  40. Qisheng Wang, Ji Guan, Junyi Liu, Zhicheng Zhang, and Mingsheng Ying. New quantum algorithms for computing quantum entropies and distances. IEEE Transactions on Information Theory, 70(8):5653-5680, 2024. URL: https://doi.org/10.1109/TIT.2024.3399014.
  41. Qisheng Wang and Zhicheng Zhang. Fast quantum algorithms for trace distance estimation. IEEE Transactions on Information Theory, 70(4):2720-2733, 2024. URL: https://doi.org/10.1109/TIT.2023.3321121.
  42. Qisheng Wang and Zhicheng Zhang. Sample-optimal quantum estimators for pure-state trace distance and fidelity via samplizer. ArXiv e-prints, 2024. URL: https://doi.org/10.48550/arXiv.2410.21201.
  43. Qisheng Wang and Zhicheng Zhang. Time-efficient quantum entropy estimator via samplizer. IEEE Transactions on Information Theory, 71(12):9569-9599, 2025. URL: https://doi.org/10.1109/TIT.2025.3576137.
  44. Qisheng Wang, Zhicheng Zhang, Kean Chen, Ji Guan, Wang Fang, Junyi Liu, and Mingsheng Ying. Quantum algorithm for fidelity estimation. IEEE Transactions on Information Theory, 69(1):273-282, 2023. URL: https://doi.org/10.1109/TIT.2022.3203985.
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