Mixing in Continuous Quantum Walks on Graphs
نویسندگان
چکیده
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 that have the instantaneous uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs lack this mixing property as well.
منابع مشابه
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...
متن کاملMixing 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...
متن کاملUniform Mixing and Association Schemes
We consider continuous-time quantum walks on distance-regular graphs. Using results about the existence of complex Hadamard matrices in association schemes, we determine which of these graphs have quantum walks that admit uniform mixing. First we apply a result due to Chan to show that the only strongly regular graphs that admit instantaneous uniform mixing are the Paley graph of order nine and...
متن کاملYqis 2015
Quantum walks on graphs [? ] are an extension of random walks to the quantum domain. Towards mixing, they can seemingly provide a quadratic speedup compared to reversible classical random walks. On the other hand, lifted randomwalks, as proposed by Diaconis et al. [? ], are a non-quantum extension proven to deliver the same speedup [? ]). However, the construction of lifted walks makes explicit...
متن کاملMixing 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...
متن کامل