- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Measurement-based time evolution for quantum simulation of fermionic systems

Woo-Ram Lee1, Zhangjie Qin1, Robert Raussendorf2, Eran Sela3, and V. W. Scarola1,*

  • 1Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA
  • 2Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada
  • 3Department of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel

  • *scarola@vt.edu
PDF

Phys. Rev. Research 4, L032013 – Published 25 July, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L032013

Abstract

Quantum simulation using time evolution in phase-estimation-based quantum algorithms can yield unbiased solutions of classically intractable models. However, long runtimes open such algorithms to decoherence. We show how measurement-based quantum simulation uses effective time evolution via measurement to allow runtime advantages over conventional circuit-based algorithms that use real-time evolution with quantum gates. We construct a hybrid algorithm to find energy eigenvalues in fermionic models using only measurements on graph states. We apply the algorithm to the Kitaev and Hubbard chains. Resource estimates show a runtime advantage if measurements can be performed faster than gates, and graph states compactification is fully used. In this letter, we set the stage to allow advances in measurement precision to improve quantum simulation.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (61)

  1. R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  2. I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Rev. Mod. Phys. 86, 153 (2014).
  3. D. S. Abrams and S. Lloyd, Simulation of Many-Body Fermi Systems on a Universal Quantum Computer, Phys. Rev. Lett. 79, 2586 (1997).
  4. D. S. Abrams and S. Lloyd, Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors, Phys. Rev. Lett. 83, 5162 (1999).
  5. G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Quantum algorithms for fermionic simulations, Phys. Rev. A 64, 022319 (2001).
  6. R. Somma, G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Simulating physical phenomena by quantum networks, Phys. Rev. A 65, 042323 (2002).
  7. A. Aspuru-Guzik, Simulated quantum computation of molecular energies, Science 309, 1704 (2005).
  8. I. Kassal, J. D. Whitfield, A. Perdomo-Ortiz, M.-H. Yung, and A. Aspuru-Guzik, Simulating chemistry using quantum computers, Annu. Rev. Phys. Chem. 62, 185 (2011).
  9. M.-H. Yung, J. D. Whitfield, S. Boixo, D. G. Tempel, and A. Aspuru-Guzik, Introduction to quantum algorithms for physics and chemistry, in Quantum Information and Computation for Chemistry, edited by S. A. Rice, A. R. Dinner, and S. Kais (John Wiley and Sons, Inc., Hoboken, 2014), Vol. 154, pp. 67–106.
  10. D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings, and M. Troyer, Gate-count estimates for performing quantum chemistry on small quantum computers, Phys. Rev. A 90, 022305 (2014).
  11. J. R. McClean, R. Babbush, P. J. Love, and A. Aspuru-Guzik, Exploiting locality in quantum computation for quantum chemistry, J. Phys. Chem. Lett. 5, 4368 (2014).
  12. D. Wecker, M. B. Hastings, N. Wiebe, B. K. Clark, C. Nayak, and M. Troyer, Solving strongly correlated electron models on a quantum computer, Phys. Rev. A 92, 062318 (2015).
  13. B. Bauer, D. Wecker, A. J. Millis, M. B. Hastings, and M. Troyer, Hybrid Quantum-Classical Approach to Correlated Materials, Phys. Rev. X 6, 031045 (2016).
  14. J. R. McClean, M. E. Kimchi-Schwartz, J. Carter, and W. A. de Jong, Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states, Phys. Rev. A 95, 042308 (2017).
  15. R. Babbush, N. Wiebe, J. McClean, J. McClain, H. Neven, and G. K.-L. Chan, Low-Depth Quantum Simulation of Materials, Phys. Rev. X 8, 011044 (2018).
  16. D. Poulin, A. Kitaev, D. S. Steiger, M. B. Hastings, and M. Troyer, Quantum Algorithm for Spectral Measurement with a Lower Gate Count, Phys. Rev. Lett. 121, 010501 (2018).
  17. S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Quantum computational chemistry, Rev. Mod. Phys. 92, 015003 (2020).
  18. M. Troyer and U.-J. Wiese, Computational Complexity and Fundamental Limitations to Fermionic Quantum Monte Carlo Simulations, Phys. Rev. Lett. 94, 170201 (2005).
  19. N. Moll, P. Barkoutsos, L. S. Bishop, J. M. Chow, A. Cross, D. J. Egger, S. Filipp, A. Fuhrer, J. M. Gambetta, M. Ganzhorn, A. Kandala, A. Mezzacapo, P. Müller, W. Riess, G. Salis, J. Smolin, I. Tavernelli, and K. Temme, Quantum optimization using variational algorithms on near-term quantum devices, Quantum Sci. Technol. 3, 030503 (2018).
  20. R. R. Ferguson, L. Dellantonio, A. A. Balushi, K. Jansen, W. Dür, and C. A. Muschik, Measurement-Based Variational Quantum Eigensolver, Phys. Rev. Lett. 126, 220501 (2021).
  21. R. Raussendorf and H. J. Briegel, A One-Way Quantum Computer, Phys. Rev. Lett. 86, 5188 (2001).
  22. R. Raussendorf, D. E. Browne, and H. J. Briegel, Measurement-based quantum computation on cluster states, Phys. Rev. A 68, 022312 (2003).
  23. A. Yu. Kitaev, Quantum measurements and the Abelian stabilizer problem, arXiv:quant-ph/9511026 (1995).
  24. S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996).
  25. D. W. Berry and H. M. Wiseman, Optimal States and Almost Optimal Adaptive Measurements for Quantum Interferometry, Phys. Rev. Lett. 85, 5098 (2000).
  26. B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, Entanglement-free Heisenberg-limited phase estimation, Nature (London) 450, 393 (2007).
  27. K. M. Svore, M. B. Hastings, and M. Freedman, Faster phase estimation, Quantum Inf. Comput. 14, 306 (2014).
  28. A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'Brien, A variational eigenvalue solver on a photonic quantum processor, Nat. Commun. 5, 4213 (2014).
  29. R. D. Somma, Quantum eigenvalue estimation via time series analysis, New J. Phys. 21, 123025 (2019).
  30. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.4.L032013 for the detailed proof of the main results.
  31. A. Yu. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Usp. 44, 131 (2001).
  32. A. Kitaev and C. Laumann, Topological phases and quantum computation, in Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing, edited by J. Jacobsen, S. Ouvry, V. Pasquier, D. Serban, and L. Cugliandolo, Lecture Notes of the Les Houches Summer School (Oxford University Press, Oxford, 2008), Vol. 89, pp. 101–125.
  33. F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümper, and V. E. Korepin, The One-Dimensional Hubbard Model (Cambridge University Press, Cambridge, 2005), p. 302.
  34. M. Hein, J. Eisert, and H. J. Briegel, Multiparty entanglement in graph states, Phys. Rev. A 69, 062311 (2004).
  35. C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
  36. H. F. Trotter, On the product of semi-groups of operators, Proc. Am. Math. Soc. 10, 545 (1959).
  37. M. Suzuki, Improved Trotter-like formula, Phys. Lett. A 180, 232 (1993).
  38. Specifically, we have ||exp[−i(H1+H2)t]−[exp(−iH1t/M)exp(−iH2t/M)]M||<δT, where ||A|| is the spectral norm, i.e., the largest singular value of A.
  39. P. Jordan and E. Wigner, Über das paulische äquivalenzverbot, Z. Phys. 47, 631 (1928).
  40. D. Gottesman, The Heisenberg representation of quantum computers, in Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, edited by S. P. Corney, R. Delbourgo, and P. D. Jarvis (International Press, Cambridge, MA, 1999), pp. 32–43.
  41. S. Anders and H. J. Briegel, Fast simulation of stabilizer circuits using a graph-state representation, Phys. Rev. A 73, 022334 (2006).
  42. W. J. Camp, Note on the one-dimensional Hubbard model, Phys. Rev. B 10, 2903 (1974).
  43. A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of Trotter Error with Commutator Scaling, Phys. Rev. X 11, 011020 (2021).
  44. Here, δT is a strict upper bound on Trotter error, and estimates of energies with lower M are possible. Furthermore, M will decrease substantially for higher-order Trotter-Suzuki decompositions: M∼O[(Ntn)1+1/pδT−1/p] for the order of p.
  45. M. B. Hastings, D. Wecker, B. Bauer, and M. Troyer, Improving quantum algorithms for quantum chemistry, Quantum Inf. Comput. 15, 1 (2015).
  46. D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Exponential improvement in precision for simulating sparse Hamiltonians, in Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing (ACM, New York, 2014), pp. 283–292.
  47. D. Berry and L. Novo, Corrected quantum walk for optimal Hamiltonian simulation, Quantum Inf. Comput. 16, 1295 (2016).
  48. A. M. Childs, A. Ostrander, and Y. Su, Faster quantum simulation by randomization, Quantum 3, 182 (2019).
  49. G. H. Low and I. L. Chuang, Optimal Hamiltonian Simulation by Quantum Signal Processing, Phys. Rev. Lett. 118, 010501 (2017).
  50. H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, Measurement-based quantum computation, Nat. Phys. 5, 19 (2009).
  51. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear optical quantum computing with photonic qubits, Rev. Mod. Phys. 79, 135 (2007).
  52. A. Pick, E. S. Matekole, Z. Aqua, G. Guendelman, O. Firstenberg, J. P. Dowling, and B. Dayan, Boosting Photonic Quantum Computation with Moderate Nonlinearity, Phys. Rev. Appl. 15, 054054 (2021).
  53. O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hänsch, and I. Bloch, Controlled collisions for multi-particle entanglement of optically trapped atoms, Nature (London) 425, 937 (2003).
  54. Y. Wang, X. Zhang, T. A. Corcovilos, A. Kumar, and D. S. Weiss, Coherent Addressing of Individual Neutral Atoms in a 3D Optical Lattice, Phys. Rev. Lett. 115, 043003 (2015).
  55. Y. Wang, A. Kumar, T.-Y. Wu, and D. S. Weiss, Single-qubit gates based on targeted phase shifts in a 3D neutral atom array, Science 352, 1562 (2016).
  56. P. Aliferis and D. W. Leung, Simple proof of fault tolerance in the graph-state model, Phys. Rev. A 73, 032308 (2006).
  57. C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, Noise Thresholds for Optical Quantum Computers, Phys. Rev. Lett. 96, 020501 (2006).
  58. R. Raussendorf, J. Harrington, and K. Goyal, A fault-tolerant one-way quantum computer, Ann. Phys. 321, 2242 (2006).
  59. R. Raussendorf and J. Harrington, Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions, Phys. Rev. Lett. 98, 190504 (2007).
  60. N. Wiebe and C. Granade, Efficient Bayesian Phase Estimation, Phys. Rev. Lett. 117, 010503 (2016).
  61. T. E. O'Brien, B. Tarasinski, and B. M. Terhal, Quantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments, New J. Phys. 21, 023022 (2019).

Outline

Information

Phys. Rev. Research 4, L032013– Published 25 July, 2022
  • Received 3 November 2021
  • Revised 17 May 2022
  • Accepted 31 May 2022
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation