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Register a new userQuantum Walk in Periodic Potential on a Line and a Model of Interacting Opinions
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Authors
C.-L. Ho
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Two subjects are discussed in this work: localisation and recurrence in a model of quantum walk in a periodic potential, and a model of opinion dynamics with multiple choices of opinions.
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We present numerical study of a model of quantum walk in periodic potential on the line. We take the simple view that different potentials affect differently the way the coin state of the walker is changed. For simplicity and definiteness, we assume the walkers coin state is unaffected at sites without potential, and is rotated in an unbiased way according to Hadamard matrix at sites with potential. This is the simplest and most natural model of a quantum walk in a periodic potential with two coins. Six generic cases of such quantum walks were studied numerically. It is found that of the six cases, four cases display significant localization effect, where the walker is confined in the neighborhood of the origin for sufficiently long times. Associated with such localization effect is the recurrence of the probability of the walker returning to the neighborhood of the origin.
We present a model of interacting multiple choices of opinions. At each step of the process, a listener is persuaded by his/her neighbour, the lobbyist, to modify his/her opinion on two different choices of event. Whether or not the listener will be convinced by the lobbyist depends on the difference between his/her opinion with that of the lobbyist, and with that of the revealed social opinion (the social pressure). If the listener is convinced, he/she will modify his/her opinion and update his/her revealed preference, and proceed to persuade his/her next neighbour. If the listener is not convinced by the lobbyist, he/she will retain his/her revealed preference, and try to persuade the lobbyist to change his/her opinion. In this case, the direction of opinion propagation is reversed. A consensus is reached when all the revealed preference is the same. Our numerical results show that consensus can always be attained in this model. However, the time needed to achieve consensus, or the so-called convergence time, is longer if the listener is more concerned with the public opinion, or is less likely to be influenced by the lobbyist.
We investigate a system of two atoms in an optical lattice, performing a quantum walk by state-dependent shift operations and a coin operation acting on the internal states. The atoms interact, e.g., by cold collisions, whenever they are in the same potential well of the lattice. Under such conditions they typically develop a bound state, so that the two atoms effectively perform a quantum walk together, rarely moving further from each other than a few lattice sites. The theoretical analysis is based on a theory of quantum walks with a point defect, applied to the difference variable. We also discuss the feasibility of an experimental realization in existing quantum walk experiments.
We present an investigation of many-particle quantum walks in systems of non-interacting distinguishable particles. Along with a redistribution of the many-particle density profile we show that the collective evolution of the many-particle system resembles the single-particle quantum walk evolution when the number of steps is greater than the number of particles in the system. For non-uniform initial states we show that the quantum walks can be effectively used to separate the basis states of the particle in position space and grouping like state together. We also discuss a two-particle quantum walk on a two- dimensional lattice and demonstrate an evolution leading to the localization of both particles at the center of the lattice. Finally we discuss the outcome of a quantum walk of two indistinguishable particles interacting at some point during the evolution.
This paper explores two circuit approaches for quantum walks: the first consists of generalised controlled
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