Communications in Mathematical Physics

, Volume 270, Issue 2, pp 359–371 | Cite as

Efficient Quantum Algorithms for Simulating Sparse Hamiltonians

  • Dominic W. Berry
  • Graeme Ahokas
  • Richard Cleve
  • Barry C. Sanders


We present an efficient quantum algorithm for simulating the evolution of a quantum state for a sparse Hamiltonian H over a given time t in terms of a procedure for computing the matrix entries of H. In particular, when H acts on n qubits, has at most a constant number of nonzero entries in each row/column, and ||H|| is bounded by a constant, we may select any positive integer k such that the simulation requires O((log* n)t 1+1/2k ) accesses to matrix entries of H. We also show that the temporal scaling cannot be significantly improved beyond this, because sublinear time scaling is not possible.


Quantum Algorithm Quantum Walk Trace Distance Nonzero Matrix Element Tensor Product Structure 
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  1. 1.
    Shor, P. W.: Algorithms for quantum computation: Discrete logarithms and factoring. In: Proc. 35th Symp. on Foundations of Computer Science, Los Alamitos, CA:IEEE, 1994, pp. 124–134Google Scholar
  2. 2.
    Grover L. (1997) Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325–328CrossRefADSGoogle Scholar
  3. 3.
    Kempe J., Kitaev A., Regev O. (2006) The complexity of the local Hamiltonian problem. SIAM J. Computing 35: 1070–1097CrossRefMathSciNetGoogle Scholar
  4. 4.
    Feynman R.P. (1982) Simulating physics with computers. Int. J. Theoret. Phys. 21, 467–488MathSciNetGoogle Scholar
  5. 5.
    Lloyd S. (1996) Universal quantum simulators. Science 273: 1073–1078CrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Aharonov, D., Ta-Shma, A.: Adiabatic quantum state generation and statistical zero knowledge. In: Proc. 35th Annual ACM Symp. on Theory of Computing, New York:ACM, 2003, pp. 20–29Google Scholar
  7. 7.
    Childs A., Farhi E., Gutmann S. (2002) An example of the difference between quantum and classical random walks. J. Quant. Inf. Proc. 1, 35–43CrossRefMathSciNetGoogle Scholar
  8. 8.
    Shenvi N., Kempe J., Whaley K.B. (2003) Quantum random-walk search algorithm. Phys. Rev. A 67: 052307CrossRefADSGoogle Scholar
  9. 9.
    Childs A., Goldstone J. (2004) Spatial search by quantum walk. Phys. Rev. A 70: 022314CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Ambainis, A.: Quantum walk algorithm for element distinctness. In: Proc. 45th Symp. on Foundations of Computer Science, Los Alamitos, CA: IEEE, 2004, pp. 22–31Google Scholar
  11. 11.
    Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proc. 16th ACM-SIAM SODA, Philadelphia, PA:SIAM, 2005, pp. 1099–1108Google Scholar
  12. 12.
    Suzuki M. (1990) Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A 146, 319–323CrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Suzuki M. (1991) General theory of fractal path integrals with applications to many-body theories and statistical physics. J. Math. Phys. 32, 400–407CrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Childs A.M. Quantum information processing in continuous time. Ph.D. Thesis, Massachusetts Institute of Technology, 2004Google Scholar
  15. 15.
    Cole R., Vishkin U. (1986) Deterministic coin tossing with applications to optimal parallel list ranking. Inform. and Control 70, 32–53CrossRefMathSciNetGoogle Scholar
  16. 16.
    Linial N. (1992) Locality in distributed graph algorithms. SIAM J. Comput. 21, 193–201CrossRefMathSciNetGoogle Scholar
  17. 17.
    Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Guttman, S., Spielman, D.A.: Exponential algorithmic speedup by quantum walk. In: Proc. 35th Annual ACM Symp. on Theory of Computing, New York: ACM, 2003, pp. 59–68Google Scholar
  18. 18.
    Ahokas, G.: Improved algorithms for approximate quantum Fourier transforms and sparse Hamiltonian simulations. M.Sc. Thesis, University of Calgary, 2004Google Scholar
  19. 19.
    Beals R., Buhrman H., Cleve R., Mosca M., de Wolf R. (2001) Quantum lower bounds by polynomials. J. ACM 48, 778–797CrossRefMathSciNetGoogle Scholar
  20. 20.
    Farhi E., Goldstone J., Gutmann S., Sipser M. (1998) Limit on the speed of quantum computation in determining parity. Phys. Rev. Lett. 81: 5442–5444CrossRefADSGoogle Scholar
  21. 21.
    Nielsen M.A., Chuang I.L., (2000) Quantum Computation and Quantum Information. Cambridge, Cambridge University PresszbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Dominic W. Berry
    • 1
    • 2
  • Graeme Ahokas
    • 2
    • 3
  • Richard Cleve
    • 2
    • 3
    • 4
    • 5
  • Barry C. Sanders
    • 2
    • 6
  1. 1.Department of PhysicsThe University of QueenslandQueenslandAustralia
  2. 2.Institute for Quantum Information ScienceUniversity of CalgaryAlbertaCanada
  3. 3.Department of Computer ScienceUniversity of CalgaryAlbertaCanada
  4. 4.School of Computer ScienceUniversity of WaterlooOntarioCanada
  5. 5.Institute for Quantum ComputingUniversity of WaterlooOntarioCanada
  6. 6.Centre for Quantum Computer TechnologyMacquarie UniversitySydneyAustralia

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