Advertisement

Simulating Sparse Hamiltonians with Star Decompositions

  • Andrew M. Childs
  • Robin Kothari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6519)

Abstract

We present an efficient algorithm for simulating the time evolution due to a sparse Hamiltonian. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian H acts for time t, this algorithm uses (d 2(d + log* N) ∥ Ht ∥ )1 + o(1) queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d 4(log* N) ∥ Ht ∥ )1 + o(1). To achieve this, we decompose a general sparse Hamiltonian into a small sum of Hamiltonians whose graphs of non-zero entries have the property that every connected component is a star, and efficiently simulate each of these pieces.

Keywords

Nonzero Entry Query Complexity Quantum Walk Star Graph Trace Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Lloyd, S.: Universal quantum simulators. Science 273(5278), 1073–1078 (1996)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aharonov, D., Ta-Shma, A.: Adiabatic quantum state generation and statistical zero knowledge. In: Proc. 35th STOC, pp. 20–29. ACM, New York (2003)Google Scholar
  3. 3.
    Berry, D., Ahokas, G., Cleve, R., Sanders, B.: Efficient quantum algorithms for simulating sparse Hamiltonians. Commun. Math. Phys. 270(2), 359–371 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Feynman, R.: Simulating physics with computers. Int. J. Theor. Phys. 21(6), 467–488 (1982)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution (2000), ArXiv preprint quant-ph/0001106 Google Scholar
  6. 6.
    Farhi, E., Gutmann, S.: Analog analogue of a digital quantum computation. Phys. Rev. A 57(4), 2403–2406 (1998)CrossRefGoogle Scholar
  7. 7.
    Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.A.: Exponential algorithmic speedup by a quantum walk. In: Proc. 35th STOC, pp. 59–68. ACM, New York (2003)Google Scholar
  8. 8.
    Farhi, E., Goldstone, J., Gutmann, S.: A Quantum Algorithm for the Hamiltonian NAND Tree. Theory of Computing 4, 169–190 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Childs, A.M.: Quantum information processing in continuous time. PhD thesis, Massachusetts Institute of Technology (2004)Google Scholar
  10. 10.
    Childs, A.M.: On the Relationship Between Continuous- and Discrete-Time Quantum Walk. Commun. Math. Phys. 294(2), 581–603 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Berry, D.W., Childs, A.M.: The quantum query complexity of implementing black-box unitary transformations (2009), ArXiv preprint arXiv:0910.4157Google Scholar
  12. 12.
    Panconesi, A., Rizzi, R.: Some simple distributed algorithms for sparse networks. Distrib. Comput. 14(2), 97–100 (2001)CrossRefGoogle Scholar
  13. 13.
    Cole, R., Vishkin, U.: Deterministic coin tossing with applications to optimal parallel list ranking. Inf. Control 70(1), 32–53 (1986)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Goldberg, A.V., Plotkin, S.A., Shannon, G.E.: Parallel symmetry-breaking in sparse graphs. SIAM J. Discrete Math. 1(4), 434–446 (1988)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Childs, A., Kothari, R.: Limitations on the simulation of non-sparse Hamiltonians. Quantum Information and Computation 10, 669–684 (2010)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Andrew M. Childs
    • 1
    • 3
  • Robin Kothari
    • 2
    • 3
  1. 1.Department of Combinatorics & OptimizationUniversity of WaterlooCanada
  2. 2.David R. Cheriton School of Computer ScienceUniversity of WaterlooCanada
  3. 3.Institute for Quantum ComputingUniversity of WaterlooCanada

Personalised recommendations