Abstract
We mainly study exceptional configuration for coined quantum walk search. For searching on a two-dimensional grid by AKR algorithm, we find some new classes of exceptional configurations that cannot be found by the AKR algorithm effectively and the known diagonal configuration can be regarded as its special case. Meanwhile, we give two modified quantum walk models that can improve the success probability in the exceptional configurations by numerical simulation. Furthermore, we introduce the concept of generalized exceptional configuration and consider search by quantum walk on a cycle with Grover coin. We find that the most common coin combination model (G, −), where G is a Grover diffusion transformation, is a generalized exceptional configuration when just searching one marked vertex on the cycle. In the end, we find generalized exceptional configuration has a different evolution of quantum coherence from exceptional configuration. These extend largely the range of exceptional configuration of quantum walk search in some sense.
Highlights
Quantum walks, as a quantum analogue of classical random walks, have been a useful model in designing quantum algorithms for a variety of problems[1,2,3,4,5,6]
We mainly consider the exceptional configurations on two-dimensional grid and generalized exceptional configurations on one-dimensional cycle under the framework of coined quantum walk
We find a wider class of exceptional configurations on N × N grid for the AKR algorithm
Summary
As a quantum analogue of classical random walks, have been a useful model in designing quantum algorithms for a variety of problems[1,2,3,4,5,6]. We will introduce a more general conception “generalized exceptional configuration” for search problems (the initial state is always the equal superposition state), which means that, for an arrangement of marked vertices, the probability of success is always the same as their probability in the initial state no matter how many steps we take. We introduce the concept of “generalized exceptional configuration” by extending the range of exceptional configuration of quantum walk search and we find the success probability of the most natural (G, −)-Type coined quantum walk search algorithm will not grow over time when searching one marked vertex on the cycle.
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