History states of one-dimensional quantum walks

TL;DR

This paper applies the history state formalism to one-dimensional quantum walks, revealing insights into system-time entanglement, its independence from initial conditions, and providing methods to evaluate and generate such states.

Contribution

It introduces a formalism to describe quantum walks via history states, analyzes entanglement properties, and offers a quantum circuit for state generation, advancing understanding of quantum walk dynamics.

Findings

System-time entanglement is independent of initial spin orientation for certain coin operators.

Entanglement of the global operator relates to its entangling power and can be analytically computed.

A quantum circuit for generating the history state of quantum walks is provided.

Abstract

We analyze the application of the history state formalism to quantum walks. The formalism allows one to describe the whole walk through a pure quantum history state, which can be derived from a timeless eigenvalue equation. It naturally leads to the notion of system-time entanglement of the walk, which can be considered as a measure of the number of orthogonal states visited in the walk. We then focus on one-dimensional discrete quantum walks, where it is shown that such entanglement is independent of the initial spin orientation for real Hadamard-type coin operators and real initial states (in the standard basis) with definite site parity. Moreover, in the case of an initially localized particle it can be identified with the entanglement of the unitary global operator that generates the whole history state, which is related to its entangling power and can be analytically evaluated. Besides, it is shown that the evolution of the spin subsystem can also be described through a spin history state with an extended clock. A connection between its average entanglement (over all initial states) and that of the operator generating this state is also derived. A quantum circuit for generating the quantum walk history state is provided as well.