Emergent Quantum Walk Dynamics from Classical Interacting Particles

TL;DR

This interdisciplinary approach connects the Classical models to the broad range of applications where DTQWs are successfully employed and provides a minimal lattice-based microscopic understanding of the emergence of quantum-like dynamics in active matter systems.

Abstract

The dynamics of a discrete-time quantum walk (DTQW) can be realized within a purely classical interacting particle system composed of some boxes and a large but finite number of balls, and can, in principle, be implemented in a tabletop experimental setting. The distribution of the balls evolves under stochastic, occupation-dependent update rules at each lattice site, producing quantum-walk dynamics without invoking a wavefunction. The update parameters are fixed by the parameters of coin and shift operations of the DTQW. This framework naturally yields a generalized active spin model and provides a minimal lattice-based microscopic understanding of the emergence of quantum-like dynamics in active matter systems. This interdisciplinary approach connects the Classical models to the broad range of applications where DTQWs are successfully employed.

Figures (4)

• Figure 1: Schematic illustration of the system of two-box. A collection of $N$ balls is distributed between two boxes labeled $0$ and $1$, each carrying an associated phase tag $\eta_0$ and $\eta_1$. The transformation process $\mathbb{T}_c$ updates both the occupation numbers and the phase tags, mapping the initial configuration to a new one according to prescribed rules.
• Figure 2: Schematic illustration of the box–ball model for a discrete-time quantum walk. Each lattice site $x\in\mathbb{Z}$ contains two boxes labeled by $c=0,1$, with associated phase tags $\eta_{x,c}$. At time $t=0$, the $N$ balls (one of which is marked) are distributed according to the preparation process $\mathbb{P}$. The transformation $\mathbb{T}_c$ updates both the occupation numbers $N_{0,c}$ and the phase tags, followed by the conditional shift that moves the balls and associated phase tags to left or right depending on $c$. The configuration at $t=1$ illustrates the resulting redistribution after one full step.
• Figure 3: Probability distributions arising from the $\mathbb{M}$ process in the box model of the Hadamard coined quantum walk, with initial state $\frac{1}{\sqrt{2}}\lvert0\rangle - \frac{i}{\sqrt{2}}\lvert1\rangle$ and at time $t = 100$, are shown for different total numbers of particles $N$. As $N$ increases, the distribution approaches that of the ideal Hadamard quantum walk, indicated by the dotted line.
• Figure 4: Snapshot of the trajectories of the marked particle in the box model ($N=10^9$) of the 1D Hadamard quantum walk, initialized in the state $\frac{1}{\sqrt{2}}\lvert0\rangle-\frac{i}{\sqrt{2}}\lvert1\rangle$, shown at time $t=200$