cgranade::
space

Quantum information • Statistical inference • Scientific programming

© 2023. Licensed under CC-BY 4.0 except where otherwise noted.

Robust Online Hamiltonian Learning

Cassandra Granade1,2, Chris Ferrie1,3, Nathan Wiebe1,4 and D. G. Cory1,5,6

2012 New Journal of Physics 14 103013  
arXiv:1207.1655 doi:10.1088/1367-2630/14/10/103013

Presented as a poster at TQC 2013 (Poster PDF, Proceedings). Presented as a contributed talk at DAMOP 2013 (Slides PDF).

Abstract

In this work we combine two distinct machine learning methodologies, sequential Monte Carlo and Bayesian experimental design, and apply them to the problem of inferring the dynamical parameters of a quantum system. We design the algorithm with practicality in mind by including parameters that control trade-offs between the requirements on computational and experimental resources. The algorithm can be implemented online (during experimental data collection), avoiding the need for storage and post-processing. Most importantly, our algorithm is capable of learning Hamiltonian parameters even when the parameters change from experiment-to-experiment, and also when additional noise processes are present and unknown. The algorithm also numerically estimates the Cramer-Rao lower bound, certifying its own performance.

Software Resources and Demonstrations

QInfer, a Python-language implementation of the algorithm presented in this work, is available from GitHub.

[source code for above demo]

Bibliography

View on Zotero citeulike
  1. M. Paris and J. Rehacek, eds., Quantum State Estimation, vol. 649 of Lecture Notes in Physics (Springer, 2004), ISBN 978-3-540-22329-0.
  2. B. P. Lanyon et al., Universal Digital Quantum Simulation with Trapped Ions, Science 334, 57 (2011).
  3. R. Gerritsma et al., Quantum Simulation of the Klein Paradox with Trapped Ions, Phys. Rev. Lett. 106, 060503 (2011).
  4. K. Kim et al., Universal Digital Quantum Simulation with Trapped Ions, Nature (London) 465, 590 (2010).
  5. A. Bendersky, F. Pastawski, and J. P. Paz, Selective and Efficient Estimation of Parameters for Quantum Process Tomography, Phys. Rev. Lett. 100, 190403 (2008).
  6. A. Bendersky, F. Pastawski, and J. P. Paz, Selective and efficient quantum process tomography, Phys. Rev. A 80, 032116 (2009).
  7. M. Mohseni and A. T. Rezakhani, Equation of motion for the process matrix: Hamiltonian identification and dynamical control of open quantum systems, Phys. Rev. A 80, 010101 (2009).
  8. M. P. A. Branderhorst et al., Simplified quantum process tomography, New Journal of Physics 11, 115010 (2009).
  9. S. T. Flammia and Y. K. Liu, Direct Fidelity Estimation from Few Pauli Measurements, Physical Review Letters 106, 230501+ (2011).
  10. M. P. da Silva, O. L. Cardinal, and D. Poulin, Practical Characterization of Quantum Devices without Tomography, Physical Review Letters 107, 210404+ (2011).
  11. D. Oi and S. Schirmer, Quantum system characterization with limited resources (2012), 1202.5779.
  12. A. Doucet and A. M. Johansen, A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later (Oxford University Press, 2009).
  13. T. J. Loredo, Bayesian Adaptive Exploration, AIP Conference Proceedings 707, 330 (2004).
  14. H. Kuck, N. de Freitas, and A. Doucet, SMC Samplers for Bayesian Optimal Nonlinear Design, in Nonlinear Statistical Signal Processing Workshop, 2006 IEEE (IEEE, 2006), pp. 99-102, ISBN 978-1-4244-0581-7.
  15. B. Scarpa and D. B. Dunson, Bayesian methods for searching for optimal rules for timing intercourse to achieve pregnancy, Statist. Med. 26, 1920 (2007).
  16. D. R. Cavagnaro, M. A. Pitt, and J. I. Myung, Adaptive Design Optimization in Experiments with People, Advances in Neural Information Processing Systems 22, 234-242 (2010).
  17. N. Kantas, A. Lecchini-Visintini, and J. M. Maciejowski, Simulation-based Bayesian optimal design of aircraft trajectories for air traffic management, Int. J. Adapt. Control Signal Process. 24, 882 (2010).
  18. X. Huan and Y. M. Marzouk, Simulation-based optimal Bayesian experimental design for nonlinear systems (2011), 1108.4146.
  19. F. Huszár and N. M. T. Houlsby, Adaptive Bayesian quantum tomography, Physical Review A 85 052120 (2012), ISSN 1094-1622, 1107.0895.
  20. E. Bagan, M. A. Ballester, R. D. Gill, M. noz Tapia, and O. R. Isart, Separable Measurement Estimation of Density Matrices and its Fidelity Gap with Collective Protocols, Physical Review Letters 97, 130501+ (2006).
  21. R. A. Servedio and S. J. Gortler, Equivalences and Separations Between Quantum and Classical Learnability, SIAM Journal on Computing 33, 1067 (2004), ISSN 0097-5397.
  22. E. Aïmeur, G. Brassard, and S. Gambs, Machine Learning in a Quantum World, (Springer Berlin / Heidelberg, Berlin, Heidelberg, 2006), vol. 4013 of Lecture Notes in Computer Science, chap. 37, pp. 431-442, ISBN 978-3-540-34628-9.
  23. S. Aaronson, The learnability of quantum states, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 463, 3089-3114 (2007), ISSN 1471-2946.
  24. A. Hentschel and B. C. Sanders, Machine Learning for Precise Quantum Measurement, Physical Review Letters 104, 063603+ (2010).
  25. K. L. Pudenz and D. A. Lidar, Quantum adiabatic machine learning, (2011), 1109.0325.
  26. A. Hentschel and B. C. Sanders, Efficient Algorithm for Optimizing Adaptive Quantum Metrology Processes, Physical Review Letters 107, 233601 (2011).
  27. A. Sergeevich and S. D. Bartlett, Optimizing qubit Hamiltonian parameter estimation algorithms using PSO, (2012), 1206.3830.
  28. C. M. Caves, Quantum mechanics of measurements distributed in time. A path-integral formulation, Physical Review D 33, 1643 (1986).
  29. M. Tsang, Time-Symmetric Quantum Theory of Smoothing, Physical Review Letters 102, 250403+ (2009).
  30. T. A. Wheatley, et al., Adaptive Optical Phase Estimation Using Time-Symmetric Quantum Smoothing Physical Review Letters 104, 093601+ (2010).
  31. D. V. Lindley, On a Measure of the Information Provided by an Experiment, The Annals of Mathematical Statistics 27, 986 (1956), ISSN 0003-4851.
  32. R. S. Said, D. W. Berry, and J. Twamley, Nanoscale magnetometry using a single-spin system in diamond, Phys. Rev. B 83, 125410 (2011).
  33. E. L. Lehmann and G. Casella, Theory of Point Estimation (Springer, 1998), 2nd ed., ISBN 0387985026.
  34. J. O. Berger, Statistical Decision Theory and Bayesian Analysis (Springer, 1985), 2nd ed., ISBN 0387960988.
  35. R. Blume-Kohout and P. Hayden, Accurate quantum state estimation via “Keeping the experimentalist honest”, (2006), quant-ph/0603116.
  36. R. D. Gill and B. Y. Levit, Applications of the van Trees inequality: a Bayesian Cramér-Rao bound, Bernoulli 1 59 (1995), ISSN 13507265.
  37. P. Tichavsky, C. H. Muravchik, and A. Nehorai, Posterior Cramer-Rao bounds for discrete-time nonlinear filtering, Signal Processing, IEEE Transactions on 46, 1386 (1998), ISSN 1053-587X.
  38. N. J. Gordon, D. J. Salmond, and A. F. M. Smith, Novel approach to nonlinear/non-Gaussian Bayesian state estimation, Radar and Signal Processing, IEE Proceedings F 140, 107-113 (1993),ISSN 0956-375X.
  39. J. Liu and M. West, Combined parameter and state estimation in simulation-based filtering (Springer-Verlag, 2000).
  40. W. Edwards, H. Lindman, and L. J. Savage, Bayesian statistical inference for psychological research, Psychological Review 70, 193 (1963), ISSN 1939-1471(Electronic);0033-295X(Print).
  41. J. Bernardo, Intrinsic credible regions: An objective Bayesian approach to interval estimation, TEST 14, 317 (2005), ISSN 1133-0686.
  42. D. Wackerly, W. Mendenhall, and R. L. Scheaffer, Mathematical Statistics with Applications (Duxbury Press, 2001), 6th ed., ISBN 0534377416.
  43. E. M. L. Beale, Confidence Regions in Non-Linear Estimation, Journal of the Royal Statistical Society. Series B (Methodological) 22, 41 (1960), ISSN 0035-9246.
  44. R. Blume-Kohout, Robust error bars for quantum tomography (2012), 1202.5270.
  45. M. Christandl and R. Renner, Reliable Quantum State Tomography, (2011), 1108.5329.
  46. M. J. Todd and E. A. Yldrm, On Khachiyan’s algorithm for the computation of minimum-volume enclosing ellipsoids, Discrete Applied Mathematics 155, 1731 (2007), ISSN 0166-218X.
  47. C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, The quickhull algorithm for convex hulls, ACM Trans. Math. Softw. 22, 469483 (1996), ISSN 0098-3500.
  48. A. Sergeevich, A. Chandran, J. Combes, S. Bartlett, and H. Wiseman, Characterization of a qubit Hamiltonian using adaptive measurements in a fixed basis, Physical Review A 84 052315 (2011), ISSN 1094-1622, 1102.3700.
  49. C. Ferrie, C. E. Granade, and D. G. Cory, Adaptive Hamiltonian estimation using Bayesian experimental design, AIP Conference Proceedings 1443, 165 (2012), ISSN 0094243X.
  50. C. Ferrie, C. Granade, and D. Cory, How to best sample a periodic probability distribution, or on the accuracy of Hamiltonian finding strategies, Quantum Information Processing Online First pp. 1-13 (2012), ISSN 1570-0755.
  51. G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics (1965-1997) 48, 119 (1976), ISSN 0010-3616, 1432-0916.
  52. N. Boulant, T. F. Havel, M. A. Pravia, and D. G. Cory, Robust method for estimating the Lindblad operators of a dissipative quantum process from measurements of the density operator at multiple time points, Physical Review A 67, 042322+ (2003).
  53. Y. S. Weinstein, T. F. Havel, J. Emerson, N. Boulant, M. Saraceno, S. Lloyd, and D. G. Cory, Quantum process tomography of the quantum Fourier transform, The Journal of Chemical Physics 121, 6117 (2004).
  54. N. Boulant, J. Emerson, T. Havel, D. Cory, and S. Furuta, Incoherent noise and quantum information processing, The Journal of Chemical Physics 121, 2955 (2004).
  55. E. Gutierrez, S. Romero, M. Trenas, and E. Zapata, Parallel Quantum Computer Simulation on the CUDA Architecture, in Computational Science ICCS 2008, edited by M. Bubak, G. van Albada, J. Dongarra, and P. Sloot (Springer Berlin / Heidelberg, 2008), vol. 5101 of Lecture Notes in Computer Science, pp. 700-709, ISBN 978-3-540-69383-3.
  56. A. Khalid, Z. Zilic, and K. Radecka, FPGA emulation of quantum circuits, in Computer Design: VLSI in Computers and Processors, 2004. ICCD 2004. Proceedings. IEEE International Conference on (2004), pp. 310-315.
  57. E. Jones, T. Oliphant, P. Peterson, et al., SciPy: Open source scientific tools for Python (2001).

Affiliations

  1. Institute for Quantum Computing, University of Waterloo.
  2. Department of Physics, University of Waterloo.
  3. Department of Applied Mathematics, University of Waterloo.
  4. Department of Combinatorics and Optimization, University of Waterloo.
  5. Department of Chemistry, University of Waterloo.
  6. Perimeter Institute for Theoretical Physics.