Time-Efficient Quantum Entropy Estimator via Samplizer

Authors Qisheng Wang , Zhicheng Zhang



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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

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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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

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.

Subject Classification

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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References

  1. Jayadev Acharya, Ibrahim Issa, Nirmal V. Shende, and Aaron B. Wagner. Estimating quantum entropy. IEEE Journal on Selected Areas in Information Theory, 1(2):454-468, 2020. URL: https://doi.org/10.1109/JSAIT.2020.3015235.
  2. Jayadev Acharya, Alon Orlitsky, Ananda Theertha Suresh, and Himanshu Tyagi. Estimating Renyi entropy of discrete distributions. IEEE Transactions on Information Theory, 63(1):38-56, 2017. URL: https://doi.org/10.1109/TIT.2016.2620435.
  3. Dorit Aharonov, Vaughan Jones, and Zeph Landau. A polynomial quantum algorithm for approximating the Jones polynomial. Algorithmica, 55(3):395-421, 2009. URL: https://doi.org/10.1007/s00453-008-9168-0.
  4. Robert Alicki, Sławomir Rudnicki, and Sławomir Sadowski. Symmetry properties of product states for the system of n n-level atoms. Journal of Mathematical Physics, 29(5):1158-1162, 1988. URL: https://doi.org/10.1063/1.527958.
  5. Anurag Anshu, Srinivasan Arunachalam, Tomotaka Kuwahara, and Mehdi Soleimanifar. Sample-efficient learning of interacting quantum systems. Nature Physics, 17(8):931-935, 2021. URL: https://doi.org/10.1038/s41567-021-01232-0.
  6. 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.
  7. Mohammad Bavarian, Saeed Mehraban, and John Wright. Learning entropy. A manuscript on von Neumann entropy estimation, private communication, 2016. Google Scholar
  8. Christian Beck and Friedrich Schögl. Thermodynamics of Chaotic Systems: An Introduction. Cambridge University Press, 1993. Google Scholar
  9. Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M. Svore, and Xiaodi Wu. Quantum SDP solvers: large speed-ups, optimality, and applications to quantum learning. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming, pages 27:1-27:14, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.27.
  10. 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.
  11. Andrew M. Childs, Aram W. Harrow, and Paweł Wocjan. Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem. In Proceedings of the 24th Annual Symposium on Theoretical Aspects of Computer Science, pages 598-609, 2007. URL: https://doi.org/10.1007/978-3-540-70918-3_51.
  12. Anirban N. Chowdhury, Guang Hao Low, and Nathan Wiebe. A variational quantum algorithm for preparing quantum Gibbs states. ArXiv e-prints, 2020. URL: https://arxiv.org/abs/2002.00055.
  13. 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.
  14. William Feller. An Introduction to Probability Theory and Its Applications, Volume 1. John Wiley & Sons, 1968. Google Scholar
  15. F. Franchini, A. R. Its, and V. E. Korepin. Renyi entropy of the XY spin chain. Journal of Physics A: Mathematical and Theoretical, 41(2):025302, 2008. URL: https://doi.org/10.1088/1751-8113/41/2/025302.
  16. Alexandru Gheorghiu and Matty J. Hoban. Estimating the entropy of shallow circuit outputs is hard. ArXiv e-prints, 2020. URL: https://arxiv.org/abs/2002.12814.
  17. András Gilyén and Tongyang Li. Distributional property testing in a quantum world. In Proceedings of the 11th Innovations in Theoretical Computer Science Conference, pages 25:1-25:19, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.25.
  18. András Gilyén and Alexander Poremba. Improved quantum algorithms for fidelity estimation. ArXiv e-prints, 2022. URL: https://arxiv.org/abs/2203.15993.
  19. 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.
  20. Tom Gur, Min-Hsiu Hsieh, and Sathyawageeswar Subramanian. Sublinear quantum algorithms for estimating von Neumann entropy. ArXiv e-prints, 2021. URL: https://arxiv.org/abs/2111.11139.
  21. Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Physical Review Letters, 103(15):150502, 2009. URL: https://doi.org/10.1103/PhysRevLett.103.150502.
  22. Matthew B. Hastings, Iván González, Ann B. Kallin, and Roger G. Melko. Measuring Renyi entanglement entropy in quantum Monte Carlo simulations. Physical Review Letters, 104(15):157201, 2010. URL: https://doi.org/10.1103/PhysRevLett.104.157201.
  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. Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58(301):13-30, 1963. URL: https://doi.org/10.1080/01621459.1963.10500830.
  25. 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.
  26. Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Reviews of Modern Physics, 81(2):865, 2009. URL: https://doi.org/10.1103/RevModPhys.81.865.
  27. Rajibul Islam, Ruichao Ma, Philipp M. Preiss, M. Eric Tai, Alexander Lukin, Matthew Rispoli, and Markus Greiner. Measuring entanglement entropy in a quantum many-body system. Nature, 528(7580):77-83, 2015. URL: https://doi.org/10.1038/nature15750.
  28. Richard Jozsa and Benjamin Schumacher. A new proof of the quantum noiseless coding theorem. Journal of Modern Optics, 41(12):2343-2349, 1994. URL: https://doi.org/10.1080/09500349414552191.
  29. Yasuhito Kawano and Hiroshi Sekigawa. Quantum Fourier transform over symmetric groups - improved result. Journal of Symbolic Computation, 75:219-243, 2016. URL: https://doi.org/10.1016/j.jsc.2015.11.016.
  30. M. Keyl and R. F. Werner. Estimating the spectrum of a density operator. Physical Review A, 64(5):052311, 2001. URL: https://doi.org/10.1103/PhysRevA.64.052311.
  31. 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.
  32. Eugenia-Maria Kontopoulou, Gregory-Paul Dexter, Wojciech Szpankowski, Ananth Grama, and Petros Drineas. Randomized linear algebra approaches to estimate the von Neumann entropy of density matrices. IEEE Transactions on Information Theory, 66(8):5003-5021, 2020. URL: https://doi.org/10.1109/TIT.2020.2971991.
  33. Nicolas Laflorencie. Quantum entanglement in condensed matter systems. Physics Reports, 646:1-59, 2016. URL: https://doi.org/10.1016/j.physrep.2016.06.008.
  34. Tongyang Li and Xiaodi Wu. Quantum query complexity of entropy estimation. IEEE Transactions on Information Theory, 65(5):2899-2921, 2019. URL: https://doi.org/10.1109/TIT.2018.2883306.
  35. 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.
  36. Hoi-Kwong Lo. Quantum coding theorem for mixed states. Optics Communications, 119(5-6):552-556, 1995. URL: https://doi.org/10.1016/0030-4018(95)00406-X.
  37. Guang Hao Low and Isaac L. Chuang. Hamiltonian simulation by qubitization. Quantum, 3:163, 2019. URL: https://doi.org/10.22331/q-2019-07-12-163.
  38. Ashley Montanaro and Ronald de Wolf. A survey of quantum property testing. In Theory of Computing Library, number 7 in Graduate Surveys, pages 1-81. University of Chicago, 2016. URL: https://doi.org/10.4086/toc.gs.2016.007.
  39. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2010. Google Scholar
  40. Ryan O'Donnell and John Wright. Efficient quantum tomography II. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing, pages 962-974, 2017. URL: https://doi.org/10.1145/3055399.3055454.
  41. 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.
  42. Alfréd Rényi. On measures of entropy and information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, pages 547-561, 1961. URL: https://projecteuclid.org/ebook/Download?urlid=bsmsp/1200512181&isFullBook=false.
  43. Benjamin Schumacher. Quantum coding. Physical Review A, 51(4):2738, 1995. URL: https://doi.org/10.1103/PhysRevA.51.2738.
  44. Sathyawageeswar Subramanian and Min-Hsiu Hsieh. Quantum algorithm for estimating α-Renyi entropies of quantum states. Physical Review A, 104(2):022428, 2021. URL: https://doi.org/10.1103/PhysRevA.104.022428.
  45. Gregory Valiant and Paul Valiant. Estimating the unseen: an n/log(n)-sample estimator for entropy and support size, shown optimal via new CLTs. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, pages 685-694, 2011. URL: https://doi.org/10.1145/1993636.1993727.
  46. John von Neumann. Mathematische Grundlagen der Quantenmechanik (Mathematical Foundations of Quantum Mechanics). Springer, 1932. Google Scholar
  47. 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.
  48. Qisheng Wang and Zhicheng Zhang. Quantum lower bounds by sample-to-query lifting. ArXiv e-prints, 2023. URL: https://arxiv.org/abs/2308.01794.
  49. 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.
  50. Qisheng Wang and Zhicheng Zhang. Time-efficient quantum entropy estimator via samplizer. ArXiv e-prints, 2024. The full version of this paper also includes references [Beck and Schögl, 1993; Brandão et al., 2019; Feller, 1968; Harrow et al., 2009; Hoeffding, 1963; Wang et al., 2023; Watrous, 2002; Watrous, 2018; Wilde, 2013]. URL: https://arxiv.org/abs/2401.09947.
  51. 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.
  52. Xinzhao Wang, Shengyu Zhang, and Tongyang Li. A quantum algorithm framework for discrete probability distributions with applications to Rényi entropy estimation. IEEE Transactions on Information Theory, 70(5):3399-3426, 2024. URL: https://doi.org/10.1109/TIT.2024.3382037.
  53. Youle Wang, Guangxi Li, and Xin Wang. Variational quantum Gibbs state preparation with a truncated Taylor series. Physical Review Applied, 16(5):054035, 2021. URL: https://doi.org/10.1103/PhysRevApplied.16.054035.
  54. Youle Wang, Benchi Zhao, and Xin Wang. Quantum algorithms for estimating quantum entropies. Physical Review Applied, 19(4):044041, 2023. URL: https://doi.org/10.1103/PhysRevApplied.19.044041.
  55. John Watrous. Limits on the power of quantum statistical zero-knowledge. In Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, pages 459-468, 2002. URL: https://doi.org/10.1109/SFCS.2002.1181970.
  56. John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. Google Scholar
  57. Mark M. Wilde. Quantum Information Theory. Cambridge University Press, 2013. Google Scholar
  58. John Wright. Private communication, 2022. Google Scholar
  59. Jingxiang Wu and Timothy H. Hsieh. Variational thermal quantum simulation via thermofield double states. Physical Review Letters, 123(22):220502, 2019. URL: https://doi.org/10.1103/PhysRevLett.123.220502.
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