Abstract
Since Grover’s seminal work which provides a way to speed up combinatorial search, quantum search has been studied in great detail. We propose a new method for designing quantum search algorithms for finding a marked element in the state space of a graph. The algorithm is based on a local adiabatic evolution of the Hamiltonian associated with the graph. The main new idea is to apply some techniques such as Krylov subspace projection methods, Lanczos algorithm and spectral distribution methods. Indeed, using these techniques together with the second-order perturbation theory, we give a systematic method for calculating the approximate search time at which the marked state can be reached. That is, for any undirected regular connected graph which is considered as the state space of the database, the introduced algorithm provides a systematic and programmable way for evaluation of the search time, in terms of the corresponding graph polynomials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325 (1997)
Ambainis, A.: Quantum search algorithms. SIGACT News, 35, 22–35 (2004). Also quant-ph/0504012
Aaronson, S., Ambainis, A.: Quantum search of spatial regions, quant-ph/0303041. In: Proceedings of 44th IEEE Symposium on Foundations of Computer Science, 200, (2003)
Childs, A.M., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A 70, 022314 (2004)
Farhi, E., Gutmann, S.: Analog analogue of a digital quantum computation. Phys. Rev. A 57, 2403 (1998)
Farhi E., Goldstone J., Gutmann S., Sipser M.: Quantum computation by adiabatic evolution, quant- ph/0001106.
Childs, A.M., Deotto, E., Farhi, E., Goldstone, J., Gutmann, S., Landahl, A.J.: Quantum search by measurement. Phys. Rev. A 66, 032314 (2002)
Childs, A., Deotto E., Cleve, R., Farhi, E., Gutmann S., Spielman D.: In: Proceedings of the 35th Annual Symposium on Theory of Computing. ACM Press, 59, (2003)
Roland, J., Cerf, N.J.: Quantum search by local adiabatic evolution. Phys. Rev. A 65, 042308 (2002)
Rezakhani, A.T., Pimachev, A.K., Lidar, D.A.: Accuracy versus run time in an adiabatic quantum search. Phys. Rev. A 82, 052305 (2010)
Parlett, B.: The Symmetric Eigenvalue Problem. Prentice-Hall Inc., Englewood Cliffs, NJ (1980)
Wilkkinson, J.: The Algebraic Eigenvalue Problem. Clarendon Press, Oxford (1965)
Trefethen, L., Bau, D.: Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIMA), Philadelphia, PA (1997)
Cullum, J., Willoughby, R.: L\(\acute{a}\)nczos Algorithems for Large Symmetric Eigenvalue Computations, Vol. I: Theory. Birkh\(\ddot{a}\)user Boston Inc., Boston, MA (1985)
Jafarizadeh, M.A., Sufiani, R., Salimi, S., Jafarizadeh, S.: Investigation of continuous-time quantum walk by using Krylov subspace-Lanczos algorithm. Eur. Phys. J. B 59, 199–216 (2007)
Shohat, J.A., Tamarkin, J.D.: The Problem of Moments. American Mathematical Society, Providence, RI (1943)
Hislop, P.D., Sigal, I.M.: Introduction to spectral theory: with applications to schrodinger operators (1995)
Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach Science Publishers Inc., London (1978)
Alipour, Z., Jafarizadeh, M.A., Fooladi, N., Sufiani, R: Int. J. Modern Phys. E 1465–1476 (2011)
Schiff, L.I.: Quantum Mechanics. McGraw-Hill, Singapore (1955)
Bransden, B.H., Joachain, C.J.: Quantum Mechanics. Pearson, Education, London (2000)
Jafarizadeh, M.A., Salimi, S.: Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix. Ann. Phys. 322, 1005–1033 (2007)
Jafarizadeh, M., Salimi, S.: Investigation of continuous-time quantum walk via modules of Bose–Mesner and Terwilliger algebras. J. Phys. A: Math. Gen. 39, 13295–13323 (2006)
Bailey, R.A.: Association Schemes: Designed Experiments, Algebra and Combinatorics. Cambridge University Press, Cambridge (2004)
Jafarizadeh, M.A., Sufiani, R., Taghavi, S.F., Barati, E.: J. Stat. Mech. 04004, (2009)
Gordon James and Martin Liebeck: Representations and characters of groups. Cambridge University Press, Cambridge (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sufiani, R., Bahari, N. Quantum search in structured database using local adiabatic evolution and spectral methods. Quantum Inf Process 12, 2813–2831 (2013). https://doi.org/10.1007/s11128-013-0563-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11128-013-0563-3