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arXiv:1408.1616 · Journal of Computational and Theoretical Nanoscience

Complexity Analysis of Quantum Walk Based Search Algorithms

B. L. Douglas, Jingbo B. Wang

Published Jun 20, 2013• 7 citations
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Abstract

We present several families of graphs that allow both efficient quantum walk implementations and efficient quantum walk based search algorithms. For these graphs, we construct quantum circuits that explicitly implement the full quantum walk search algorithm, without reference to a `black box' oracle. These circuits provide a practically implementable method to explore quantum walk based search algorithms with the aim of eventual real-world applications. We also provide a numerical analysis of a quantum walk based search along a twisted toroid family of graphs, which requires O($\sqrt{n}$ log($n$)) elementary 2-qubit quantum gate operations to find a marked node.

DOI
10.1166/jctn.2013.3095

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Quantum walks, being the quantum analog of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the ``glued trees'' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed and discuss potential applications to quantum search for marked vertices along these graphs.
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Search algorithms based on quantum walks have emerged as a promising approach to solve computational problems across various domains, including combinatorial optimization, and cryptography. Stating a generic search problem in terms of a (quantum) search over a graph makes the efficiency of the algorithmic method depend on the structure of the graph itself. In this work, we propose a complete implementation of a quantum walk search on Johnson graphs, speeding up the solution of the subset-sum problem. We provide a detailed design of each sub-circuit, quantifying their cost in terms of gate number, depth, and width. We compare our solution against a Grover quantum search, showing a reduction of the T-count and T-depth for practically solvable problems. The proposed design provides a building block for the construction of efficient quantum search algorithms that can be modelled on Johnson graphs, filling the gap with the existing theoretical complexity analyses.
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Quantum walks, being the quantum analogue of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the `glued trees' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed, and discuss potential applications to quantum search for marked vertices along these graphs.