Optimization for the propagation of a multiparticle quantum walk in a one-dimensional lattice

The quantum walk is a quantum counterpart of the classical random walk that exhibits nonclassical behaviors and outperforms the classical random walk in various aspects. It has been known that a single particle can be propagated by a discrete-time quantum walk with a quadratic time scaling in the variance of position distribution, beating the linear time scaling in a classical random walk. In this paper, we consider the discrete-time quantum walk for multiple particles in a one-dimensional lattice, and investigate the optimization of the joint coin state to enhance the spatial propagation of the particles in the lattice. We study the asymptotic evolution of position distribution for multiple particles in the long-time limit, and analytically optimize the joint coin state to derive the maximum variance of the position distribution between the particles after the evolution of the quantum walk. An interesting result is that an optimized coin state always possesses specific exchange symmetry which can be characterized by a graph consisting of two disconnected complete subgraphs and the exchange symmetry can significantly influence the position correlations between the particles, showing the critical role of coin symmetry in the propagation of multiple particles by the quantum walk. We further study the entanglement of the optimized coin states to show the relation of the coin correlations to the particle position distribution.