Quantum walk on the Bloch sphere

TL;DR

This paper proposes a method to implement discrete-time quantum walks on the Bloch sphere using spin clusters and analyzes their behavior, confirming quadratic variance enhancement and addressing nonorthogonality issues.

Contribution

It introduces a novel scheme for quantum walks on the Bloch sphere using spin clusters and interactions, expanding quantum walk implementations.

Findings

Quadratic variance enhancement confirmed

Walker states are generally nonorthogonal

Increasing spins in the cluster mitigates nonorthogonality

Abstract

A scheme for implementing the discrete-time quantum walk on the Bloch sphere is proposed, which is closely related to the SU(2) group. A spin cluster serves as the walker, whereas its location on the Bloch sphere is described by the spin coherent state. An additional spin that interacts with the spin cluster plays the role of a coin, whose state determines the rotation of the spin cluster. The Wigner function is calculated to visualize the movement of the walker on the Bloch sphere, with which the probability distribution and the standard deviation are also achieved. The quadratic enhancement of variance for the quantum walk on the Bloch sphere is confirmed. Compared to the ideal quantum walk on a circle, the walker's states on the Bloch sphere are generally nonorthogonal, whose drawbacks can be eliminated by increasing the number of spins in the spin cluster.