Abstract
We investigate the behavior of coherence in scattering quantum walk search on complete graph under the condition that the total number of vertices of the graph is significantly larger than the marked number of vertices we are searching, N ≫ v. We find that the consumption of coherence represents the increase of the success probability for the searching, also it is related to the efficiency of the algorithm in oracle queries. If no coherence is consumed or an incoherent state is utilized, the algorithm will behave as the classical blind search, implying that coherence is responsible for the speed-up in this quantum algorithm over its classical counterpart. The effect of noises, in particular of photon loss and random phase shifts, on the performance of algorithm is studied. Two types of noise are considered because they arise in the optical network used for experimental realization of scattering quantum walk. It is found that photon loss will reduce the coherence and random phase shifts will hinder the interference between the edge states, both leading to lower success probability compared with the noise-free case. We then conclude that coherence plays an essential role and is responsible for the speed-up in this quantum algorithm.
Highlights
Random walk is an important prototype for efficient classical algorithms[1,2]
We calculate the coherence in the scattering quantum walk search algorithm on complete graph and associate it with the probability of success under the condition N v > 1
If the coherence consumption is smaller than a given value, the quantum algorithm we investigated will be less efficient than the classical search with memory
Summary
The quantum walk search algorithm we studied here is the scattering quantum walk search[18]. The scattering quantum walk is defined on a graph (V, E) with V being the set of the total vertices and E being the set of edges connecting vertices. Where state |m, l〉 is an edge state going from vertex m to l The evolution of this quantum walk is defined by the local unitary operator for each vertex. If we denote the set of marked vertices as and the set of all vertices as , the oracle can be defined as a function of vertex. In this quantum algorithm, a controlled unitary operator works as an oracle, i.e. where the first one is a state of vertex and the second one is a qubit. When the result of the oracle is stored in the qubit, another controlled will be applied
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