STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Quantum Walks on the Hypercube
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
Exponential algorithmic speedup by a quantum walk
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
On Distributions Computable by Random Walks on Graphs
SIAM Journal on Discrete Mathematics
Fastest Mixing Markov Chain on a Graph
SIAM Review
Decoherence in quantum walks – a review
Mathematical Structures in Computer Science
On mixing in continuous-time quantum walks on some circulant graphs
Quantum Information & Computation
Mixing and decoherence in continuous-time quantum walks on cycles
Quantum Information & Computation
Mixing of quantum walk on circulant bunkbeds
Quantum Information & Computation
Decoherence in quantum walks – a review
Mathematical Structures in Computer Science
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We study the set of probability distributions visited by a continuous-time quantum walk on graphs. An edge-weighted graph G is universal mixing if the instantaneous or average probability distribution of the quantum walk on G ranges over all probability distributions on the vertices as the weights are varied over non-negative reals. The graph is uniform mixing if it visits the uniform distribution. Our results include the following: • All weighted complete multipartite graphs are instantaneous universal mixing. This is in contrast to the fact that no unweighted complete multipartite graphs are uniform mixing (except for the four-cycle K2,2). • For all n ≥ 1, the weighted claw K1,n is a minimally connected instantaneous universal mixing graph. In fact, as a corollary, the unweighted K1,n is instantaneous uniform mixing. This adds a new family of uniform mixing graphs to a list that so far contains only the hypercubes. • Any weighted graph is average almost-uniform mixing unless its spectral type is sublinear in the size of the graph. This provides a nearly tight characterization for average uniform mixing on circulant graphs. • No weighted graphs are average universal mixing. This shows that weights do not help to achieve average universal mixing, unlike the instantaneous case. Our proofs exploit the spectra of the underlying weighted graphs and path collapsing arguments.