Simulating Sparse Hamiltonians with Star Decompositions

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


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.


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  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

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