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Efficient Tomography of Non-Interacting-Fermion States

Authors Scott Aaronson, Sabee Grewal

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Author Details

Scott Aaronson
  • The University of Texas at Austin, TX, USA
Sabee Grewal
  • The University of Texas at Austin, TX, USA

Funding

  • Aaronson, Scott: Supported by a Vannevar Bush Fellowship from the US Department of Defense, the Berkeley NSF-QLCI CIQC Center, a Simons Investigator Award, and the Simons "It from Qubit" collaboration.
  • Grewal, Sabee: Supported by Scott Aaronson’s Vannevar Bush Fellowship from the US Department of Defense, Berkeley NSF-QLCI CIQC Center, Simons Investigator Award, and Simons "It from Qubit" collaboration.

Acknowledgements

We thank Andrew Zhao for notifying us that the previous version of this manuscript contained an error and providing other insightful comments. We also thank Yuxuan Zhang, Alex Kulesza, Ankur Moitra, William Kretschmer, Dax Enshan Koh, Andrea Rocchetto, and Patrick Rall for helpful discussions, and Alex Arkhipov, William Kretschmer, and Daniel Liang for helpful comments on a previous version of this manuscript.

Cite AsGet BibTex

Scott Aaronson and Sabee Grewal. Efficient Tomography of Non-Interacting-Fermion States. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 12:1-12:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.TQC.2023.12

Abstract

We give an efficient algorithm that learns a non-interacting-fermion state, given copies of the state. For a system of n non-interacting fermions and m modes, we show that O(m³ n² log(1/δ) / ε⁴) copies of the input state and O(m⁴ n² log(1/δ)/ ε⁴) time are sufficient to learn the state to trace distance at most ε with probability at least 1 - δ. Our algorithm empirically estimates one-mode correlations in O(m) different measurement bases and uses them to reconstruct a succinct description of the entire state efficiently.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Quantum information theory
  • Mathematics of computing → Probabilistic inference problems
  • Theory of computation → Quantum complexity theory
Keywords
  • free-fermions
  • Gaussian fermions
  • non-interacting fermions
  • quantum state tomography
  • efficient tomography

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References

  1. Scott Aaronson. The learnability of quantum states. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463(2088):3089-3114, 2007. URL: https://doi.org/10.1098/rspa.2007.0113.
  2. Scott Aaronson. Shadow Tomography of Quantum States. SIAM Journal on Computing, 49(5):STOC18-368-STOC18-394, 2020. URL: https://doi.org/10.1145/3188745.3188802.
  3. Scott Aaronson. Introduction to Quantum Information Science II Lecture Notes, 2022. URL: https://scottaaronson.com/qisii.pdf.
  4. Scott Aaronson and Alex Arkhipov. BosonSampling is Far From Uniform. Quantum Information and Computation, 14(15-16):1383-1423, 2014. Google Scholar
  5. Scott Aaronson and Sabee Grewal. Efficient Learning of Non-Interacting Fermion Distributions. arXiv preprint arXiv:2102.10458v2, 2021. Google Scholar
  6. Srinivasan Arunachalam, Sergey Bravyi, Arkopal Dutt, and Theodore J. Yoder. Optimal algorithms for learning quantum phase states, 2022. URL: https://doi.org/10.48550/arxiv.2208.07851.
  7. Costin Bădescu and Ryan O'Donnell. Improved Quantum Data Analysis. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 1398-1411, 2021. URL: https://doi.org/10.1145/3406325.3451109.
  8. Clément L. Canonne. A short note on learning discrete distributions. arXiv preprint arXiv:2002.11457, 2020. URL: https://doi.org/10.48550/arXiv.2002.11457.
  9. Marcus Cramer, Martin B. Plenio, Steven T. Flammia, Rolando Somma, David Gross, Stephen D. Bartlett, Olivier Landon-Cardinal, David Poulin, and Yi-Kai Liu. Efficient quantum state tomography. Nature communications, 1(1):1-7, 2010. URL: https://doi.org/10.1038/ncomms1147.
  10. Sabee Grewal, Vishnu Iyer, William Kretschmer, and Daniel Liang. Low-Stabilizer-Complexity Quantum States Are Not Pseudorandom. arXiv preprint arXiv:2209.14530, 2022. URL: https://doi.org/10.48550/arXiv.2209.14530.
  11. Madalin Guţă, Jonas Kahn, Richard Kueng, and Joel A. Tropp. Fast state tomography with optimal error bounds. Journal of Physics A: Mathematical and Theoretical, 53(20):204001, 2020. URL: https://doi.org/10.1088/1751-8121/ab8111.
  12. 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.
  13. Pierre Hohenberg and Walter Kohn. Inhomogeneous Electron Gas. Physical Review, 136(3B):B864, 1964. URL: https://doi.org/10.1103/PhysRev.136.B864.
  14. Roger A. Horn and Charles R. Johnson. Topics in Matrix Analysis. Cambridge University Press, 1994. URL: https://doi.org/10.1017/CBO9780511840371.
  15. 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.
  16. Emanuel Knill. Fermionic Linear Optics and Matchgates. arXiv preprint quant-ph/0108033, 2001. URL: https://doi.org/10.48550/arXiv.quant-ph/0108033.
  17. Walter Kohn and Lu Jeu Sham. Self-Consistent Equations Including Exchange and Correlation Effects. Physical Review, 140(4A):A1133, 1965. URL: https://doi.org/10.1103/PhysRev.140.A1133.
  18. Richard Kueng, Holger Rauhut, and Ulrich Terstiege. Low Rank Matrix Recovery From Rank One Measurements. Applied and Computational Harmonic Analysis, 42(1):88-116, 2017. URL: https://doi.org/10.1016/j.acha.2015.07.007.
  19. Guang Hao Low. Classical shadows of fermions with particle number symmetry, 2022. URL: https://doi.org/10.48550/arxiv.2208.08964.
  20. Odile Macchi. The Coincidence Approach to Stochastic Point Processes. Advances in Applied Probability, 7(1):83-122, 1975. URL: https://doi.org/10.2307/1425855.
  21. Ashley Montanaro. Learning stabilizer states by Bell sampling. arXiv:1707.04012, 2017. URL: https://doi.org/10.48550/arXiv.1707.04012.
  22. Ryan O'Donnell and John Wright. Efficient Quantum Tomography. In Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, pages 899-912, 2016. URL: https://doi.org/10.1145/2897518.2897544.
  23. Bryan O'Gorman. Fermionic tomography and learning, 2022. URL: https://doi.org/10.48550/arxiv.2207.14787.
  24. Michał Oszmaniec, Ninnat Dangniam, Mauro E.S. Morales, and Zoltán Zimborás. Fermion Sampling: A Robust Quantum Computational Advantage Scheme Using Fermionic Linear Optics and Magic Input States. PRX Quantum, 3:020328, 2022. URL: https://doi.org/10.1103/PRXQuantum.3.020328.
  25. Google AI Quantum and Collaborators. Hartree-Fock on a superconducting qubit quantum computer. Science, 369(6507):1084-1089, 2020. URL: https://doi.org/10.1126/science.abb9811.
  26. Michael Reck, Anton Zeilinger, Herbert J. Bernstein, and Philip Bertani. Experimental realization of any discrete unitary operator. Phys. Rev. Lett., 73:58-61, 1994. URL: https://doi.org/10.1103/PhysRevLett.73.58.
  27. Justin Rising, Alex Kulesza, and Ben Taskar. An Efficient Algorithm for the Symmetric Principal Minor Assignment Problem. Linear Algebra and its Applications, 473:126-144, 2015. URL: https://doi.org/10.1016/j.laa.2014.04.019.
  28. Barbara M. Terhal and David P. DiVincenzo. Classical simulation of noninteracting-fermion quantum circuits. Physical Review A, 65(3):032325, 2002. URL: https://doi.org/10.1103/PhysRevA.65.032325.
  29. John Urschel, Victor-Emmanuel Brunel, Ankur Moitra, and Philippe Rigollet. Learning Determinantal Point Processes with Moments and Cycles. In International Conference on Machine Learning, pages 3511-3520. PMLR, 2017. Google Scholar
  30. Leslie G. Valiant. Quantum Circuits That Can Be Simulated Classically in Polynomial Time. SIAM Journal on Computing, 31(4):1229-1254, 2002. URL: https://doi.org/10.1137/S0097539700377025.
  31. Kianna Wan, William J. Huggins, Joonho Lee, and Ryan Babbush. Matchgate Shadows for Fermionic Quantum Simulation, 2022. URL: https://doi.org/10.48550/arxiv.2207.13723.
  32. Hermann Weyl. Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Mathematische Annalen, 71(4):441-479, 1912. URL: https://doi.org/10.1007/BF01456804.
  33. Gian-Carlo Wick. The Evaluation of the Collision Matrix. Physical Review, 80(2):268, 1950. URL: https://doi.org/10.1103/PhysRev.80.268.
  34. Andrew Zhao, Nicholas C. Rubin, and Akimasa Miyake. Fermionic Partial Tomography via Classical Shadows. Phys. Rev. Lett., 127:110504, 2021. URL: https://doi.org/10.1103/PhysRevLett.127.110504.
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