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arXiv:2202.09080 · Physical Review A

Implementation of a discrete-time quantum walk with a circulant matrix on a graph by optical polarizing elements

Yusuke Mizutani, Tomoyuki Horikiri, Leo Matsuoka, Yusuke Higuchi, Etsuo Segawa

Published Aug 5, 2022• 3 citations
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Abstract

In this paper, we introduce a quantum walk whose local scattering at each vertex is denoted by a unitary circulant matrix, namely, the circulant quantum walk. We also introduce another quantum walk induced by the circulant quantum walk, namely, the optical quantum walk, whose underlying graph is a 2-regular directed graph and obtained by blowing up the original graph in some way. We propose a design of an optical circuit which implements the stationary state of the optical quantum walk. We show that if the induced optical quantum walk does not have $+1$ eigenvalue, then the stationary state of the optical quantum walk gives that of the original circulant quantum walk. From this result, we give a useful condition for the setting of the circulant quantum walks which can be implemented by this optical circuit.

DOI
10.1103/physreva.106.022402

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In this paper, we introduce a quantum walk whose local scattering at each vertex is denoted by a unitary circulant matrix, namely, the circulant quantum walk. We also introduce another quantum walk induced by the circulant quantum walk, namely, the optical quantum walk, whose underlying graph is a 2-regular directed graph and obtained by blowing up the original graph in some way. We propose a design of an optical circuit which implements the stationary state of the optical quantum walk. We show that if the induced optical quantum walk does not have +1+1+1 eigenvalue, then the stationary state of the optical quantum walk gives that of the original circulant quantum walk. From this result, we give a useful condition for the setting of the circulant quantum walks which can be implemented by this optical circuit.
The random walk formalism is used across a wide range of applications, from modelling share prices to predicting population genetics. Likewise, quantum walks have shown much potential as a framework for developing new quantum algorithms. Here we present explicit efficient quantum circuits for implementing continuous-time quantum walks on the circulant class of graphs. These circuits allow us to sample from the output probability distributions of quantum walks on circulant graphs efficiently. We also show that solving the same sampling problem for arbitrary circulant quantum circuits is intractable for a classical computer, assuming conjectures from computational complexity theory. This is a new link between continuous-time quantum walks and computational complexity theory and it indicates a family of tasks that could ultimately demonstrate quantum supremacy over classical computers. As a proof of principle, we experimentally implement the proposed quantum circuit on an example circulant graph using a two-qubit photonics quantum processor.
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This thesis focuses on applying analytical and numerical techniques to the problem of designing quantum circuits for quantum walks. We develop efficient quantum circuit implementations of continuous-time quantum walks using diagonalization of the Hamiltonian. Next, for the discrete-time Szegedy quantum walk model, we provide a theoretical framework for the development of efficient quantum circuits, and apply it to a wide variety of graphs. We also investigate the optimisation of numerical techniques to the general problem of implementing a unitary matrix. Lastly, we discuss design principles for the development of efficient quantum circuit implementations of quantum walks of different types.
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