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Fast and Robust Quantum State Tomography from Few Basis Measurements

Authors Daniel Stilck França , Fernando G.S L. Brandão , Richard Kueng

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Author Details

Daniel Stilck França
  • QMATH, Department of Mathematical Sciences, University of Copenhagen, Denmark
Fernando G.S L. Brandão
  • AWS Center for Quantum Computing, Pasadena, CA, USA
  • Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA
Richard Kueng
  • Institute for Integrated Circuits, Johannes Kepler University Linz, Austria

Funding

  • França, Daniel Stilck: D.S.F. acknowledges financial support from VILLUM FONDEN via the QMATH Centre of Excellence (Grant no. 10059).
  • Brandão, Fernando G.S L.: F.B. acknowledges funding from the US National Science Foundation (PHY1733907). The Institute for Quantum Information and Matter is an NSF Physics Frontiers Center.
  • Kueng, Richard: R.K. acknowledges funding from the US National Science Foundation (PHY1733907). The Institute for Quantum Information and Matter is an NSF Physics Frontiers Center.

Acknowledgements

We thank Chris Ferrie, David Gross, Thomas Grurl, Cécilia Lancien, Robert König, Oliver H. Schwarze and Joel Tropp for valuable input and helpful discussions.

Cite AsGet BibTex

Daniel Stilck França, Fernando G.S L. Brandão, and Richard Kueng. Fast and Robust Quantum State Tomography from Few Basis Measurements. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.TQC.2021.7

Abstract

Quantum state tomography is a powerful but resource-intensive, general solution for numerous quantum information processing tasks. This motivates the design of robust tomography procedures that use relevant resources as sparingly as possible. Important cost factors include the number of state copies and measurement settings, as well as classical postprocessing time and memory. In this work, we present and analyze an online tomography algorithm designed to optimize all the aforementioned resources at the cost of a worse dependence on accuracy. The protocol is the first to give provably optimal performance in terms of rank and dimension for state copies, measurement settings and memory. Classical runtime is also reduced substantially and numerical experiments demonstrate a favorable comparison with other state-of-the-art techniques. Further improvements are possible by executing the algorithm on a quantum computer, giving a quantum speedup for quantum state tomography.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Hardware → Quantum technologies
  • Theory of computation → Quantum information theory
  • Mathematics of computing → Probabilistic inference problems
Keywords
  • quantum tomography
  • low-rank tomography
  • Gibbs states
  • random measurements

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References

  1. Scott Aaronson. Shadow tomography of quantum states. In STOC'18 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 325-338. ACM, New York, 2018. URL: https://doi.org/10.1145/3188745.3188802.
  2. Scott Aaronson, Xinyi Chen, Elad Hazan, Satyen Kale, and Ashwin Nayak. Online learning of quantum states. Journal of Statistical Mechanics: Theory and Experiment, 2019(12):124019, 2019. URL: https://doi.org/10.1088/1742-5468/ab3988.
  3. Andris Ambainis and Joseph Emerson. Quantum t-designs: t-wise independence in the quantum world. In 22nd Annual IEEE Conference on Computational Complexity (CCC 2007), 13-16 June 2007, San Diego, California, USA, pages 129-140. IEEE Computer Society, 2007. URL: https://doi.org/10.1109/CCC.2007.26.
  4. Sanjeev Arora and Satyen Kale. A combinatorial, primal-dual approach to semidefinite programs. J. ACM, 63(2):12:1-12:35, 2016. URL: https://doi.org/10.1145/2837020.
  5. K Banaszek, M Cramer, and D Gross. Focus on quantum tomography. New J. Phys, 15(12):125020, 2013. URL: https://doi.org/10.1088/1367-2630/15/12/125020.
  6. 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 46th International Colloquium on Automata, Languages, and Programming, volume 132 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 27, 14. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2019. Google Scholar
  7. Fernando G. S. L. Brandão, Richard Kueng, and Daniel Stilck França. Faster quantum and classical SDP approximations for quadratic binary optimization. preprint arXiv:1909.04613, 2019. Google Scholar
  8. Fernando G. S. L. Brandão and Krysta M. Svore. Quantum speed-ups for solving semidefinite programs. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 415-426. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.45.
  9. Fernando G.S.L. Brandão, Richard Kueng, and Daniel Stilck França. Fast and robust quantum state tomography from few basis measurements, 2020. arXiv:2009.08216v2. URL: http://arxiv.org/abs/2009.08216.
  10. Sébastien Bubeck. Convex optimization: Algorithms and complexity. Found. Trends Mach. Learn., 8(3-4):231-357, 2015. URL: https://doi.org/10.1561/2200000050.
  11. Robert J. Chapman, Christopher Ferrie, and Alberto Peruzzo. Experimental demonstration of self-guided quantum tomography. Phys. Rev. Lett., 117:040402, July 2016. URL: https://doi.org/10.1103/PhysRevLett.117.040402.
  12. Inc. CVX Research. CVX: Matlab software for disciplined convex programming, version 2.0. http://cvxr.com/cvx, 2012.
  13. Christopher Ferrie. Self-guided quantum tomography. Phys. Rev. Lett., 113:190404, November 2014. URL: https://doi.org/10.1103/PhysRevLett.113.190404.
  14. Steven T Flammia, David Gross, Yi-Kai Liu, and Jens Eisert. Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators. New J. Phys., 14(9):095022, 2012. URL: https://doi.org/10.1088/1367-2630/14/9/095022.
  15. Daniel Stilck Franca. Hamiltonian updates tomography. https://github.com/dsfranca/hamiltonian_updates_tomography, 2020.
  16. Christopher Granade, Christopher Ferrie, and Steven T Flammia. Practical adaptive quantum tomography. New J. Phys, 19(11):113017, November 2017. URL: https://doi.org/10.1088/1367-2630/aa8fe6.
  17. David Gross, Yi-Kai Liu, Steven T. Flammia, Stephen Becker, and Jens Eisert. Quantum state tomography via compressed sensing. Phys. Rev. Lett., 105:150401, 2010. URL: https://doi.org/10.1103/PhysRevLett.105.150401.
  18. Madalin Guţă, Jonas Kahn, Richard Kueng, and Joel A Tropp. Fast state tomography with optimal error bounds. J. Phys. A, 53(20):204001, 2020. URL: https://doi.org/10.1088/1751-8121/ab8111.
  19. Jeongwan Haah, Aram Wettroth Harrow, Zhengfeng Ji, Xiaodi Wu, and Nengkun Yu. Sample-optimal tomography of quantum states. IEEE Trans. Inf. Theory, 63(9):5628-5641, 2017. URL: https://doi.org/10.1109/TIT.2017.2719044.
  20. Elad Hazan. Efficient algorithms for online convex optimization and their applications. PhD thesis, Princeton University, 2006. Google Scholar
  21. Teiko Heinosaari, Luca Mazzarella, and Michael M. Wolf. Quantum tomography under prior information. Commun. Math. Phys, 318(2):355-374, 2013. URL: https://doi.org/10.1007/s00220-013-1671-8.
  22. Carl W. Helstrom. Quantum detection and estimation theory. J. Statist. Phys., 1:231-252, 1969. URL: https://doi.org/10.1007/BF01007479.
  23. Alexander S. Holevo. Statistical decision theory for quantum systems. J. Multivariate Anal., 3:337-394, 1973. URL: https://doi.org/10.1016/0047-259X(73)90028-6.
  24. Zhibo Hou, Jun-Feng Tang, Christopher Ferrie, Guo-Yong Xiang, Chuan-Feng Li, and Guang-Can Guo. Experimental realization of self-guided quantum process tomography. Phys. Rev. A, 101:022317, February 2020. URL: https://doi.org/10.1103/PhysRevA.101.022317.
  25. Michael Kech and Michael M. Wolf. Constrained quantum tomography of semi-algebraic sets with applications to low-rank matrix recovery. Inf. Inference, 6(2):171-195, 2017. URL: https://doi.org/10.1093/imaiai/iaw019.
  26. Iordanis Kerenidis and Anupam Prakash. A Quantum Interior Point Method for LPs and SDPs. ACM Transactions on Quantum Computing, 1(1):1-32, December 2020. URL: https://doi.org/10.1145/3406306.
  27. Richard Kueng. Low rank matrix recovery from few orthonormal basis measurements. In 2015 International Conference on Sampling Theory and Applications (SampTA), pages 402-406, 2015. Google Scholar
  28. Richard Kueng, Holger Rauhut, and Ulrich Terstiege. Low rank matrix recovery from rank one measurements. Appl. Comput. Harmon. Anal., 42(1):88-116, 2017. URL: https://doi.org/10.1016/j.acha.2015.07.007.
  29. Richard Kueng, Huangjun Zhu, and David Gross. Distinguishing quantum states using Clifford orbits. preprint arXiv:1609.08595, 2016. Google Scholar
  30. Cécilia Lancien and Andreas Winter. Distinguishing multi-partite states by local measurements. Comm. Math. Phys., 323(2):555-573, 2013. URL: https://doi.org/10.1007/s00220-013-1779-x.
  31. James R. Lee, Prasad Raghavendra, and David Steurer. Lower bounds on the size of semidefinite programming relaxations. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 567-576. ACM, 2015. URL: https://doi.org/10.1145/2746539.2746599.
  32. Yi-Kai Liu. Universal low-rank matrix recovery from Pauli measurements. In J. Shawe-Taylor, R. S. Zemel, P. L. Bartlett, F. Pereira, and K. Q. Weinberger, editors, Advances in Neural Information Processing Systems 24, pages 1638-1646. Curran Associates, Inc., 2011. URL: http://papers.nips.cc/paper/4222-universal-low-rank-matrix-recovery-from-pauli-measurements.pdf.
  33. William Matthews, Stephanie Wehner, and Andreas Winter. Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding. Comm. Math. Phys., 291(3):813-843, 2009. URL: https://doi.org/10.1007/s00220-009-0890-5.
  34. Ryan O'Donnell and John Wright. Efficient quantum tomography. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 899-912. ACM, 2016. URL: https://doi.org/10.1145/2897518.2897544.
  35. John Preskill. Quantum Computing in the NISQ era and beyond. Quantum, 2:79, 2018. URL: https://doi.org/10.22331/q-2018-08-06-79.
  36. Carlos A. Riofrio, David Gross, Steven T. Flammia, Thomas Monz, Daniel Nigg, Rainer Blatt, and Jens Eisert. Experimental quantum compressed sensing for a seven-qubit system. Nat. Commun., 8(1), 2017. URL: https://doi.org/10.1038/ncomms15305.
  37. Takanori Sugiyama, Peter S. Turner, and Mio Murao. Precision-guaranteed quantum tomography. Phys. Rev. Lett., 111:160406, 2013. URL: https://doi.org/10.1103/PhysRevLett.111.160406.
  38. Koji Tsuda, Gunnar Rätsch, and Manfred K. Warmuth. Matrix exponentiated gradient updates for on-line learning and Bregman projection. J. Mach. Learn. Res., 6:995-1018, 2005. URL: http://jmlr.org/papers/v6/tsuda05a.html.
  39. Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. Quantum SDP-solvers: Better upper and lower bounds. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 403-414. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.44.
  40. Vladislav Voroninski. Quantum tomography from few full-rank observables. preprint arXiv:1309.7669, 2013. Google Scholar
  41. Akram Youssry, Christopher Ferrie, and Marco Tomamichel. Efficient online quantum state estimation using a matrix-exponentiated gradient method. New J. Phys., 21(3):033006, 2019. URL: https://doi.org/10.1088/1367-2630/ab0438.
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