Communications in Mathematical Physics

, Volume 294, Issue 2, pp 581–603

On the Relationship Between Continuous- and Discrete-Time Quantum Walk

Article

Abstract

Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantum walks can be defined either in continuous or discrete time. But whereas a continuous-time random walk can be obtained as the limit of a sequence of discrete-time random walks, the two types of quantum walk appear fundamentally different, owing to the need for extra degrees of freedom in the discrete-time case.

In this article, I describe a precise correspondence between continuous- and discrete- time quantum walks on arbitrary graphs. Using this correspondence, I show that continuous-time quantum walk can be obtained as an appropriate limit of discrete-time quantum walks. The correspondence also leads to a new technique for simulating Hamiltonian dynamics, giving efficient simulations even in cases where the Hamiltonian is not sparse. The complexity of the simulation is linear in the total evolution time, an improvement over simulations based on high-order approximations of the Lie product formula. As applications, I describe a continuous-time quantum walk algorithm for element distinctness and show how to optimally simulate continuous-time query algorithms of a certain form in the conventional quantum query model. Finally, I discuss limitations of the method for simulating Hamiltonians with negative matrix elements, and present two problems that motivate attempting to circumvent these limitations.

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References

  1. 1.
    Aaronson, S., Shi, Y.: Quantum lower bounds for the collision and the element distinctness problems. J. ACM 51 (4) 595–605 (2004), preliminary versions in STOC 2002 and FOCS 2002Google Scholar
  2. 2.
    Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proc. 33rd ACM Symposium on Theory of Computing, pp. 50–59, 2001, available at http://arxiv.org/abs/quant-ph/0012090, 2000
  3. 3.
    Aharonov, D., Ta-Shma, A.: Adiabatic quantum state generation and statistical zero knowledge. In: Proc. 35th ACM Symposium on Theory of Computing, pp. 20–29, 2003, available at http://arxiv.org/abs/quant-ph/0301023, 2003
  4. 4.
    Aldous, D., Fill, J.A.: Reversible Markov chains and random walks on graphs (in preparation), http://www.stat.berkeley.edu/~aldous/RWG/book.html
  5. 5.
    Ambainis A.: Quantum walk algorithm for element distinctness. SIAM J. Comput. 37(1), 210–239 (2007)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. Proc. 33rd ACM Symposium on Theory of Computing, pp. 37–49, 2001, available at http://arxiv.org/abs/quant-ph/0010117, 2000
  7. 7.
    Ambainis, A., Childs, A.M., Reichardt, B.W., Špalek, R., Zhang S.: Any AND-OR formula of size N can be evaluated in time N 1/2+o(1) on a quantum computer. In: Proc. 48th IEEE Symposium on Foundations of Computer Science, pp. 363–372, 2007, available at http://arxiv.org/abs/quant-ph/0703015 and http://arxiv.org/abs/0704.3628, 2007
  8. 8.
    Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proc. 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 1099–1108, 2005, available at http://arxiv.org/abs/quant-ph/0402107, 2004
  9. 9.
    Berry D.W., Ahokas G., Cleve R., Sanders B.C.: Efficient quantum algorithms for simulating sparse Hamiltonians. Commun. Math. Phys. 270(2), 359–371 (2007)MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Buhrman, H., Špalek, R.: Quantum verification of matrix products. In: Proc. 17th ACM-SIAM Symposium on Discrete Algorithms, pp. 880–889, 2006, available at http://arxiv.org/abs/quant-ph/0409035, 2004
  11. 11.
    Bužek V., Derka R., Massar S.: Optimal quantum clocks. Phys. Rev. Lett. 82(10), 2207–2210 (1999)CrossRefADSGoogle Scholar
  12. 12.
    Childs, A.M.: Quantum information processing in continuous time. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 2004Google Scholar
  13. 13.
    Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.A.: Exponential algorithmic speedup by quantum walk. In: Proc. 35th ACM Symposium on Theory of Computing, pp. 59–68, 2003, available at http://arxiv.org/abs/quant-ph/0209131, 2002
  14. 14.
    Childs A.M., Eisenberg J.M.: Quantum algorithms for subset finding. Quant. Inf. Comp. 5(7), 593–604 (2005)MATHGoogle Scholar
  15. 15.
    Childs A.M., Farhi E., Gutmann S.: An example of the difference between quantum and classical random walks. Quant. Inf. Proc. 1(1-2), 35–43 (2002)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Childs A.M., Goldstone J.: Spatial search by quantum walk. Phys. Rev. A 70(2), 022314 (2004)CrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Childs A.M., Goldstone J.: Spatial search and the Dirac equation. Phys. Rev. A 70(4), 042312 (2004)CrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Childs, A.M., Schulman, L.J., Vazirani, U.V.: Quantum algorithms for hidden nonlinear structures. In: Proc. 48th IEEE Symposium on Foundations of Computer Science, pp. 395–404, 2007, available at http://arxiv.org/abs/0705.2784, 2007
  19. 19.
    Cleve, R., Gottesman, D., Mosca, M., Somma, R.D., Yonge-Mallo, D.L.: Efficient discrete-time simulations of continuous-time quantum query algorithms. In: Proc. 41st ACM Symposium on Theory of Computing, pp. 409–416, 2009, available at http://arxiv.org/abs/0811.4428, 2008
  20. 20.
    van Dam W., D’Ariano G.M., Ekert A., Macchiavello C., Mosca M.: Optimal phase estimation in quantum networks. J. Phys. A 40(28), 7971–7984 (2007)MATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    van Dam, W., Hallgren, S., Ip, L.: Quantum algorithms for some hidden shift problems. In: Proc. 14th ACM-SIAM Symposium on Discrete Algorithms, pp. 489–498, 2002, available at http://arxiv.org/abs/quant-ph/0211140, 2002
  22. 22.
    van Dam, W., Mosca, M., Vazirani, U.: How powerful is adiabatic quantum computation?. In: Proc. 42nd IEEE Symposium on Foundations of Computer Science, pp. 279–287, 2001, available at http://arxiv.org/abs/quant-ph/0206003, 2002
  23. 23.
    van Dam, W., Seroussi, G.: Quantum algorithms for estimating Gauss sums and calculating discrete logarithms. Manuscript, 2003, available at http://www.cs.ucsb.edu/~vandam/gausssumdlog.pdf
  24. 24.
    Damgård, I.B.: On the randomness of Legendre and Jacobi sequences. Advances in Cryptology - CRYPTO ’88, Lecture Notes in Computer Science, vol. 403, New York: Springer, 1990, pp. 163–172Google Scholar
  25. 25.
    Farhi E., Goldstone J., Gutmann S.: A quantum algorithm for the Hamiltonian NAND tree. Theory of Computing 4(1), 169–190 (2008)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Farhi E., Gutmann S.: Analog analogue of a digital quantum computation. Phys. Rev. A 57(4), 2403–2406 (1998)CrossRefMathSciNetADSGoogle Scholar
  27. 27.
    Farhi E., Gutmann S.: Quantum computation and decision trees. Phys. Rev. A 58(2), 915–928 (1998)CrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Godsil, C.: Association schemes. Lecture notes, available at http://quoll.uwaterloo.ca/pstuff/assoc.pdf, 2004
  29. 29.
    Grover, L., Rudolph, T.: Creating superpositions that correspond to efficiently integrable probability distributions, available at http://arxiv.org/abs/quant-ph/0208112v1, 2002
  30. 30.
    Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79(2), 325–328 (1997), preliminary version in STOC 1996Google Scholar
  31. 31.
    Kedlaya K.S.: Quantum computation of zeta functions of curves. Comput. Complex. 15(1), 1–19 (2006)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Linial N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Lloyd S.: Universal quantum simulators. Science 273(5278), 1073–1078 (1996)CrossRefMathSciNetADSGoogle Scholar
  34. 34.
    Luis A., Peřina J.: Optimum phase-shift estimation and the quantum description of the phase difference. Phys. Rev. A 54(5), 4564–4570 (1996)CrossRefADSGoogle Scholar
  35. 35.
    Magniez, F., Nayak, A.: Quantum complexity of testing group commutativity. Algorithmica 48(3), 221–232 (2007), preliminary version in ICALP 2005Google Scholar
  36. 36.
    Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. In: Proc. 39th ACM Symposium on Theory of Computing, pp. 575–584, 2007, available at http://arvix.org/abs/quant-ph/0608026, 2006
  37. 37.
    Magniez, F., Santha, M., Szegedy, M.: Quantum algorithms for the triangle problem. In: Proc. 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 1109–1117, 2005, available at http://arxiv.org/abs/quant-ph/0310134, 2003
  38. 38.
    Meyer D.A.: From quantum cellular automata to quantum lattice gasses. J. Stat. Phys. 85(5/6), 551–574 (1996)MATHCrossRefADSGoogle Scholar
  39. 39.
    Meyer D.A.: On the absence of homogeneous scalar unitary cellular automata. Phys. Lett. A 223, 337–340 (1996)MATHCrossRefMathSciNetADSGoogle Scholar
  40. 40.
    Mochon C.: Hamiltonian oracles. Phys. Rev. A 75(4), 042313 (2007)CrossRefADSGoogle Scholar
  41. 41.
    Moore, C., Russell, A.: Quantum walks on the hypercube. In: Proc. 6th International Workshop on Randomization and Approximation Techniques in Computer Science, Lecture Notes in Computer Science, Vol. 2483, Berlin: Springer, 2002 pp. 164–178Google Scholar
  42. 42.
    Regev, O.: Witness-preserving amplification of QMA. Lecture notes, http://www.cs.tau.ac.il/~odedr/teaching/quantum_fall_2005/ln/qma.pdf, 2006
  43. 43.
    Reichardt, B.W., Špalek, R.: Span-program-based quantum algorithm for evaluating formulas. In: Proc. 40th ACM Symposium on Theory of Computing, pp. 103–112, 2008, available at http://arxiv.org/abs/0710.2630, 2007
  44. 44.
    Roland J., Cerf N.J.: Quantum search by local adiabatic evolution. Phys. Rev. A 65(4), 042308 (2002)CrossRefMathSciNetADSGoogle Scholar
  45. 45.
    Roland J., Cerf N.J.: Quantum-circuit model of Hamiltonian search algorithms. Phys. Rev. A 68(6), 062311 (2003)CrossRefADSGoogle Scholar
  46. 46.
    Schmidt W.M.: Equations over Finite Fields: An Elementary Approach. 2nd ed. Kendrick Press, Hebercity, UT (2004)MATHGoogle Scholar
  47. 47.
    Severini S.: On the digraph of a unitary matrix. SIAM J. Matrix Anal. Appl. 25(1), 295–300 (2003)MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Shenvi N., Kempe J., Whaley K.B.: A quantum random walk search algorithm. Phys. Rev. A 67(5), 052307 (2003)CrossRefADSGoogle Scholar
  49. 49.
    Shor P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Strauch F.W.: Connecting the discrete- and continuous-time quantum walks. Phys. Rev. A 74(3), 030301 (2006)CrossRefMathSciNetADSGoogle Scholar
  51. 51.
    Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: Proc. 45th IEEE Symposium on Foundations of Computer Science, pp. 32–41, 2004, available at http://arxiv.org/abs/quant-ph/0401053, 2004
  52. 52.
    Tulsi A.: Faster quantum-walk algorithm for the two-dimensional spatial search. Phys. Rev. A 78(1), 012310 (2008)CrossRefADSGoogle Scholar
  53. 53.
    Watrous J.: Quantum simulations of classical random walks and undirected graph connectivity. J. Comput. System Sci. 62(2), 376–391 (2001)MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Weil A.: On some exponential sums. Proc. Natl. Acad. Sci. 34(5), 204–207 (1948)MATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Combinatorics & Optimization and Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada

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