Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On the hitting times of quantum versus random walks
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Quantum walk based search algorithms
TAMC'08 Proceedings of the 5th international conference on Theory and applications of models of computation
Tree search and quantum computation
Quantum Information Processing
QI'11 Proceedings of the 5th international conference on Quantum interaction
SIAM Journal on Computing
Intricacies of quantum computational paths
Quantum Information Processing
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Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider {\em quantum\/} walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the ``Hadamard walk''''. In particular, we show that after~$t$ time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval~$[-t/\sqrt{2},\;t/\sqrt{2}]$. This implies that the same walk defined on the circle mixes in {\em linear\/} time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.