QUANTUM WALKS ON NECKLACES AND MIXING

Mixing of Quantum Walks on Generalized Hypercubes

We study continuous-time quantum walks on graphs which generalize the hypercube. The only known family of graphs whose quantum walk instantaneously mixes to uniform is the Hamming graphs with small arities. We show that quantum uniform mixing on the hypercube is robust under the addition of perfect matchings but not much else. Our specific results include: • The graph obtained by augmenting the...

Mixing in Continuous Quantum Walks on Graphs

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that a continuous quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs th...

Mixing and decoherence in continuous-time quantum walks on cycles

We prove analytical results showing that decoherence can be useful for mixing time in a continuous-time quantum walk on finite cycles. This complements the numerical observations by Kendon and Tregenna (Physical Review A 67 (2003), 042315) of a similar phenomenon for discrete-time quantum walks. Our analytical treatment of continuous-time quantum walks includes a continuous monitoring of all ve...

Non-uniform Mixing in Continuous Quantum Walks

In this note, we study the mixing properties of continuous-time quantum random walks on graphs. We prove that the only graphs in the family of balanced complete multipartite graphs that have a uniform mixing property are K2, K3, K4, and K2,2. This is unlike the classical case where the uniform mixing property is satisfied by all such graphs. Our proof exploits the circulant structure of these g...

Average mixing of continuous quantum walks