Andrew M. Childs
ABSTRACT
We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our black box setting. We then show how this quantum walk solves our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve the problem in subexponential time.
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).]] Google Scholar
Digital Library
- D. Aharonov, A. Ambainis, J. Kempe, and U. Vazirani. Quantum walks on graphs. In Proc. of the 33rd ACM Symp. on Theory of Computing, pages 50--59, 2001.]] Google ScholarDigital Library
- D. Aharonov and A. Ta-Shma. Adiabatic quantum state generation and statistical zero knowledge. In Proc. 35th ACM Symp. on Theory of Computing, 2003.]] Google ScholarDigital Library
- A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous. One-dimensional quantum walks. In Proc. 33rd ACM Symp. on Theory of Computing, pages 37--49, 2001.]] Google ScholarDigital Library
- J. N. de Beaudrap, R. Cleve, and J. Watrous. Sharp quantum vs. classical query complexity separations. Algorithmica, 34:449--461, 2002.]]Google ScholarCross Ref
- E. Bernstein and U. Vazirani. Quantum complexity theory. SIAM J. on Computing, 26:1411--1473, 1997.]] Google ScholarDigital Library
- A. M. Childs, E. Farhi, and S. Gutmann. An example of the difference between quantum and classical random walks. Quantum Information Processing, 1:35--43, 2002.]] Google ScholarDigital Library
- W. van Dam and S. Hallgren. Efficient quantum algorithms for shifted quadratic character problems. quant-ph/0011067.]]Google Scholar
- W. van Dam and G. Seroussi. Efficient quantum algorithms for estimating Gauss sums. Unpublished.]]Google Scholar
- D. Deutsch. Quantum theory, the Church-Turing principle, and the universal quantum computer. Proc. Roy. Soc. London A, 400:96--117, 1985.]]Google ScholarCross Ref
- D. Deutsch and R. Jozsa. Rapid solution of problems by quantum computation. Proc. Roy. Soc. London A, 439:553--558, 1992.]]Google ScholarCross Ref
- E. Farhi and S. Gutmann. Quantum computation and decision trees. Phys. Rev. A, 58:915--928, 1998.]]Google ScholarCross Ref
- J. Goldstone. Personal communication, Sept. 2002.]]Google Scholar
- M. Grigni, L. Schulman, M. Vazirani, and U. Vazirani. Quantum mechanical algorithms for the nonabelian hidden subgroup problem. In Proc. 33rd ACM Symp. on Theory of Computing, pages 68--74, 2001.]] Google ScholarDigital Library
- S. Hallgren. Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem. In Proc. 34th ACM Symp. on Theory of Computing, pages 653--658, 2002.]] Google ScholarDigital Library
- S. Hallgren, A. Russell, and A. Ta-Shma. Normal subgroup reconstruction and quantum computation using group representations. In Proc. 32nd ACM Symp. on Theory of Computing, pages 627--635, 2000.]] Google ScholarDigital Library
- G. Ivanos, F. Magniez, and M. Santha. Efficient quantum algorithms for some instances of the non-abelian hidden subgroup problem. quant-ph/0102014.]]Google Scholar
- J. Kempe. Quantum random walks hit exponentially faster. quant-ph/0205083.]]Google Scholar
- A. Kitaev. Quantum measurements and the abelian stabilizer problem. quant-ph/9511026.]]Google Scholar
- S. Lloyd. Universal quantum simulators, Science 273:1073--1078, 1996.]]Google ScholarCross Ref
- D. A. Meyer. From quantum cellular automata to quantum lattice gasses. J. Stat. Phys., 85:551--574, 1996.]]Google ScholarCross Ref
- C. Moore and A. Russell. Quantum walks on the hypercube. In Proc. of RANDOM 02, Vol. 2483 of LNCS, pages 164--178, 2002.]] Google ScholarDigital Library
- M. Mosca and A. Ekert. The hidden subgroup problem and eigenvalue estimation on a quantum computer. In Proc. 1st NASA International Conf. on Quantum Computing and Quantum Communication, Vol. 1509 of LNCS, pages 174--188, 1999.]] Google ScholarDigital Library
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information Cambridge University Press, Cambridge, 2000.]] Google ScholarDigital Library
- P. W. Shor. Algorithms for quantum computation: discrete logarithms and factoring. SIAM J. on Computing, 26:1484--1509, 1997.]] Google ScholarDigital Library
- D. Simon. On the power of quantum computation. SIAM J. on Computing, 26:1474--1483, 1997.]] Google ScholarDigital Library
- V. G. Vizing. On an estimate of the chromatic class of a p-graph. Metody Diskret. Analiz., 3:25--30, 1964.]]Google Scholar
- J. Watrous. Quantum algorithms for solvable groups. In Proc. 33rd ACM Symposium on Theory of Computing, pages 60--67, 2001.]] Google ScholarDigital Library
- J. Watrous. Quantum simulations of classical random walks and undirected graph connectivity. J. Computer and System Sciences, 62:376--391, 2001.]] Google ScholarDigital Library
- A. Wigderson. The complexity of graph connectivity. In Proc. 17th Mathematical Foundations of Computer Science Conf., Vol. 629 of LNCS, pages 112--132, 1992.]] Google ScholarDigital Library
Index Terms
- Exponential algorithmic speedup by a quantum walk
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