Analysis and Applications of Quantum Walks
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ANALYSIS AND APPLICATIONS OF QUANTUM WALKS L. E. Fedichkin and F. P. Meshchaninov
UDC 519.217, 530.145
Abstract. We consider continuous unitary quantum walks of a single particle and of closed systems and discuss more realistic approaches to modeling quantum walks. Keywords and phrases: quantum walks, database search, quantum computing. AMS Subject Classification: 60K99, 81Q99
CONTENTS 1. Unitary Quantum Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2. Quantum Walks of Open Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
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Unitary Quantum Walks
Quantum walks are direct analogs of classical random walks; both types of random walks can be continuous or discrete in time (below, we call them briefly continuous and discrete walks). Quantum walks are often considered as tools for developing algorithms that work much faster than their classical analogs. The most famous algorithms on quantum walks are the quantum Monte Carlo method based on the scheme of a Markov chain, the Grover algorithm, and Ambainis search algorithm in the database. A model of artificial intelligence, which makes decisions based on quantum walks along a graph consisting of memory cells, was also constructed (see [7]). In the simplest models and algorithms based on them, walk graphs are considered as closed systems. This means that at any moment of time, the state of the system can be described by a certain vector |ψ. If we consider time-continuous random walks, then the evolution of the system is described by the Schr¨ odinger equation, and in the case of discrete walks, each step of a walk is accomplished by applying the corresponding operator, which, as a rule, is independent of the number of the step. More realistic models take into account external noise phenomenologically (see, e.g., [2, 12]) by introducing the probability and the operator of error occurring due to noises. The most realistic approach to simulation of quantum walks with account of the influence of individual noise sources was performed in [10, 11, 16, 17]. 1. Continuous unitary quantum walks. In this section, we consider continuous, unitary quantum walks of a particle. Quantum and classical walks on complete graphs G = (V, E) with N nodes are described by similar systems of differential equations. The adjacency matrix is introduced as follows: 1, (j, k) ∈ E, Ajk = 0, (j, k) ∈ / E. Continuous random walks are described by a Markov process with a fixed probability of transition of a particle to an adjacent node per unit time γ. The evolution of the probability distribution over the Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 151, Quantum Probability, 2018.
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graph is described by the system of first-order differential equations dpj (t) =γ Ljk pk (t), dt
j = 1, N ,
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