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International Journal of Theoretical Physics

A Modified Quantum Search Algorithm

Hossein Mehri‐Dehnavi, Hasan Dashtianeh, Hossein Yahyavi Kuchaksaraei, Masumeh Mahmoodi Khoshdareh, Hossein Movahhedian, Robabeh Rahimi

Published Sep 7, 2018• 5 citations
Publisher
DOI
10.1007/s10773-018-3880-6

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We propose in this paper a rough counting algorithm and modified version of Grover's search algorithm. The proposed counting algorithm uses Deutsch-Jozsa's algorithm to roughly estimate the number of items satisfying some search conditions by reconstructing the output distribution. The modified Grover's search combines the counting into one part of the algorithm and uses the counting result to determine the number of Grover's iterations taken to get such items. We provide an analysis of expected probability of the proposed algorithm taking into account the problem size, the number of satisfying items and the query complexity to show that the proposed modified Grover's search is optimal.
Quantum walks are the quantum mechanical counterpart to classical random walks. Similarly to random walks, in quantum walks a walker moves on a graph structure from one node to another governed by some stochastic process. Unlike random walks, quantum walks exhibit quantum superposition. This allows the walker to be in multiple nodes simultaneously. Due to quantum superposition quantum walks have been applied to cryptography, optimization, graph theory and spatial search problems. The goal of a spatial search problem is to find a target node from a graph structure by moving from one node to another along the edges of the graph. On certain graph structures quantum walks are able to find the target node of the search faster than classical random walks. However, determining the speedup provided by the quantum walk can be difficult to do analytically, especially when there are multiple target nodes. Therefore, it is of importance to investigate numerical approaches for approximating the dynamics of quantum walks. This Bachelor's thesis concerns a heuristic method for analyzing the dynamics of discrete-time coined quantum walk (DTCQW) search algorithms used for finding a specific node or a set of nodes on a two-dimensional lattice graph. Instead of simulating the walk fully, the heuristic method uses the eigenvectors and eigenvalues of a unitary evolution operator that governs the walk dynamics to approximate the probability of the DTCQW search algorithm finding a node that is a target of the search. This work goes over two slightly different DTCQW search algorithms. One of these is a instance of the DTCQW search algorithm that has been investigated in previous literature (referred to as the standard DTCQW search algorithm). The other one is a slightly modified version of the standard DTCQW search algorithm (referred to as the modified DTCQW search algorithm) In this work, the method was first applied to the standard DTCQW search algorithm to compare the results to previously known analytical results. The goal of this was to showcase that by using the method one can arrive at similar results as with analytical approaches. In addition, the method was used to investigate the modified DTCQW search algorithm. This was done to give numerical showing that the modified DTCQW search algorithm is able to find certain target node formations with a higher probability than the standard DTCQW search algorithm. The results of this work show that the method can approximate the analytical results using only a couple of eigenvector-eigenvalue pairs of the unitary evolution operator. Therefore, it can be considered as a cost-efficient alternative for simulating the complete quantum walk. Using the method it can be seen that the modified search algorithm finds target nodes that are adjacent to each other in the graph structure with a high probability. Additionally, the results show that the modified search algorithm only has a small chance of finding target nodes that are not adjacent to other target nodes. This gives numerical showing that the modified search algorithm behaves differently from the standard search algorithm and could therefore be used for different applications.
Grover's search algorithm has attracted great attention due to its quadratic speedup over classical algorithms in unsorted database search problems. However, Grover's algorithm is inefficient in multi-target search problems, except in the case of 1/4 of the data in the database satisfying the search conditions. Long presented a modified version of Grover's search algorithm by introducing a phase-matching condition that can search for the target state with a zero theoretical failure rate. In this work, we present an optimized exact multi-target search algorithm based on the modified Grover's algorithm by transforming the canonical diffusion operator to a more efficient diffusion operator, which can solve the multi-target search problem with a 100% success rate while requiring fewer gate counts and shallower circuit depth. After that, the optimized multi-target algorithm for four different items, including 2-qubit with 2 targets, 5-qubit with 2 targets, 5-qubit with 4 targets, and 6-qubit with 3 targets, are implemented on MindSpore framework. The experimental results show that, compared with Grover's algorithm and the modified Grover's algorithm, the proposed algorithm can reduce the quantum gate count by at least 21.1% and the depth of the quantum circuit by at least 11.4% and maintain a 100% success probability. Our code are available at https://github.com/mindsporelab/models/tree/master/research/arxiv_papers/GROVER-OP. Thanks for the support provided by MindSpore Community.