Randomized Algorithms for Hamiltonian Simulation

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)

Abstract

We consider randomized algorithms for simulating the evolution of a Hamiltonian \(H ={ \sum \nolimits }_{j=1}^{m}{H}_{j}\) for time t. The evolution is simulated by a product of exponentials of H j in a random sequence, and random evolution times. Hence the final state of the system is approximated by a mixed quantum state. First we provide a scheme to bound the error of the final quantum state in a randomized algorithm. Then we obtain randomized algorithms which have the same efficiency as certain deterministic algorithms but which are simpler to implement.

Notes

Acknowledgements

We are grateful to Anargyros Papageorgiou, Joseph F. Traub, Henryk Wozniakowski, Columbia University and Zhengfeng Ji, Perimeter Institute for Theoretical Physics, for their very helpful discussions and comments.

References

  1. 1.
    R. P. Feynman, Simulating Physics with computers, Int. J. Theoret. Phys. 21, 467–488 (1982)Google Scholar
  2. 2.
    S. Lloyd, Universal quantum simulators, Science 273, 1073–1078 (1996)Google Scholar
  3. 3.
    C. Zalka, Simulating Quantum Systems on a Quantum Computer, Proc. R. Soc. Lond. A, 454, 313–323 (1998)Google Scholar
  4. 4.
    D. W. Berry, G. Ahokas, R. Cleve, B. C. Sanders, Efficient quantum algorithms for simulating sparse Hamiltonians, Communications in Mathematical Physics 270, 359 (2007)Google Scholar
  5. 5.
    I. Kassal, S. P. Jordan, P. J. Love, M. Mohseni, A. Aspuru-Guzik, Polynomial-time quantum algorithm for the simulation of chemical dynamics, Proc. Natl. Acad. Sci. 105, 18681(2008)Google Scholar
  6. 6.
    D. Aharonov, A. Ta-Shma, Adiabatic quantum state generation and statistical zero knowledge, Proc. 35th Annual ACM Symp. on Theoty of Computing, 20–29 (2003)Google Scholar
  7. 7.
    E. Farhi, J. Goldstone, S. Gutmann, M. Sipser, Quantum computation by adiabatic evolution, quant-ph/0001106 (2000)Google Scholar
  8. 8.
    E. Farhi, S. Gutmann, Analog analogue of a digital quantum computation. Phys. Rev. A 57(4), 24032406 (1998)Google Scholar
  9. 9.
    A. M. Childs, E. Farhi, S. Gutmann, An example of the difference between quantum and classical random walks, J. Quant. Inf. Proc. 1, 35–43 (2002)Google Scholar
  10. 10.
    E. Farhi, J. Goldstone, S. Gutmann, A Quantum Algorithm for the Hamiltonian NAND Tree, quant-ph/0702144 (2007)Google Scholar
  11. 11.
    A. M. Childs, Universal computation by quantum walk, Phys. Rev. Lett, 102, 180501 (2009)Google Scholar
  12. 12.
    A. M. Childs, On the relationship between continuous- and discrete-time quantum walk, quant-ph/0810.0312 (2008)Google Scholar
  13. 13.
    D. W. Berry, A. M. Childs, The quantum query complexity of implementing black-box unitary transformations, quant-ph/0910. 4157 (2009)Google Scholar
  14. 14.
    M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000)Google Scholar
  15. 15.
    G. Strang, On the construction and comparison of difference schemes, SIAM J. Num. Analysis, 506–517 (1968)Google Scholar
  16. 16.
    M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations, Phys. Lett. A 146, 319–323 (1990)Google Scholar
  17. 17.
    M. Suzuki, General theory of fractal path integrals with application to many-body theories and statistical physics, J. Math. Phys, 32, 400–407 (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Computer ScienceColumbia UniversityNew YorkUSA

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