Multiparticle quantum walk with a gaslike interaction

We analyze the dynamics of multiparticle discrete-time quantum walk on the two-dimensional lattice, with an interaction inspired on a classical model for gas collision, called the HPP model. In this classical model, the direction of motion changes only when the particles collide head-on, preserving momentum and energy. In our quantum model, the dynamics is driven by the usual quantum-walk evolution operator if the particles are on different nodes, and is driven by the HPP rules if the particles are in the same node, linearly extended for superpositions. Using this form of evolution operator, we numerically analyze three physical quantities for the two-walker case: The probability distribution of the position of one walker, the standard deviation of the position of one walker, and the entanglement between the walkers as a function of the number of steps. The numerical analysis implies that the entanglement between the walkers as a function of the number of steps initially increases and quickly tends to a constant value, which depends on the initial condition. We compare the results obtained using the HPP interaction with the equivalent ones using the phase interaction, which is based on an evolution operator that inverts the sign of the coin operator if the walkers are in the same position.

Figure 1

Part of a two-dimensional lattice displaying 16 nodes and four cells per node. A node has label $(x,y)$ such that $x+y$ is even. A square necessarily contains two cells. A full cell represents one particle and a empty cell represents no particle.

Figure 2

Two head-on collisions, where the particles are represented by the arrows (two cells are full and two cells are empty). The right-hand lattice shows the direction the particles take after the collision. In all other cases, the particles keep the same direction.

Figure 3

If the position of a walker is node (0, 0), there are four nodes it can go after the action of the shift operator. The nodes are (1, 1), $(-1,1), (-1,-1)$, and (1,$-1$) associated with the coin states (0, 0), (1, 0), (1, 1), and (0, 1), respectively. The walker never steps on nodes $(x,y)$ such that $x+y$ is odd.

Figure 4

Probability distribution of a Grover walk after 29 time steps (dimensionless plot) with initial state of Eq. (8).

Figure 5

Probability distribution of the position of the first walker after 18 time steps (dimensionless plot) with the HPP interaction (the initial state is described in each panel).

Figure 6

Probability distribution of the position of the first walker after 18 time steps (dimensionless plot) with the phase interaction (the initial state is described in each panel).

Figure 7

Standard deviation of the position of the first walker as a function of the number of time steps (Time) for the initial conditions described at the upper-left corner (dimensionless plot). Continuous lines refer to $U_{\text{HPP}}$ and dashed lines to $U_{\text{phase}}$.

Figure 8

Von Neumann entropy of the reduced density matrix of the first walker as a function of the number of time steps (Time) for the initial conditions described at the lower-right corner (dimensionless plot). Continuous lines refer to $U_{\text{HPP}}$ and dashed lines to $U_{\text{phase}}$.