Relationship Between Quantum Walk and Relativistic Quantum Mechanics

TL;DR

This paper explores the deep mathematical and physical connections between quantum walks and relativistic quantum mechanics, revealing how the structure of quantum walks mirrors fundamental relativistic equations and concepts.

Contribution

It demonstrates the analogy between quantum walk equations and relativistic equations like Klein-Gordon and Dirac, and shows how the coin introduces reversibility and relativistic structure.

Findings

Quantum walk equations resemble Klein-Gordon and Dirac equations.

The coin acts as an analog of spinor degrees of freedom.

Decoherence introduces entropy, making the walk irreversible.

Abstract

Quantum walk models have been used as an algorithmic tool for quantum computation and to describe various physical processes. This paper revisits the relationship between relativistic quantum mechanics and the quantum walks. We show the similarities of the mathematical structure of the decoupled and coupled form of the discrete-time quantum walk to that of the Klein-Gordon and Dirac equations, respectively. In the latter case, the coin emerges as an analog of the spinor degree of freedom. Discrete-time quantum walk as a coupled form of the continuous-time quantum walk is also shown by transforming the decoupled form of the discrete-time quantum walk to the Schrodinger form. By showing the coin to be a means to make the walk reversible, and that the Dirac-like structure is a consequence of the coin use, our work suggests that the relativistic causal structure is a consequence of conservation of information. However, decoherence (modelled by projective measurements on position space) generates entropy that increases with time, making the walk irreversible and thereby producing an arrow of time. Lieb-Robinson bound is used to highlight the causal structure of the quantum walk to put in perspective the relativistic structure of quantum walk, maximum speed of the walk propagation and the earlier findings related to the finite spread of the walk probability distribution. We also present a two-dimensional quantum walk model on a two state system to which the study can be extended.