Action Principles for Quantum Automata and Lorentz Invariance of Discrete Time Quantum Walks

TL;DR

This paper introduces a discrete action principle for quantum automata, specifically for Discrete Time Quantum Walks, ensuring energy-momentum conservation and Lorentz invariance, enabling self-consistent models of spinor-gauge interactions.

Contribution

It develops a covariant, energy-momentum preserving action framework for DTQWs, extending their applicability to interacting spinor-gauge field models.

Findings

Discrete action principle for quantum automata proposed

Covariant formulation preserves energy and momentum

Discrete stress-energy tensor derived for DTQWs

Abstract

A discrete action principle for general quantum automata is proposed. This action principle is particularized to Discrete Time Quantum Walks (DTQWs) and then extended into an energy and momentum preserving, manifestly covariant formulation. Space-time coordinates are introduced as new variables of the action and their equations of motion enforce energy and momentum conservation. This guarantees that the proposed action can be used to build future, DTQW-based self-consistent models of spinors interacting with gauge fields. A discrete stress-energy tensor for the DTQW is also obtained by functional differentiation of the action with respect to the gradients of the coordinates viewed as functions of the discrete grid points. The manifest covariance of the formulation highlights the special role played by the grid reference frame in the DTQW dynamics. The main discussion is complemented by three appendices.