Average position in quantum walks with a U(2) coin

TL;DR

This paper analyzes discrete-time quantum walks with a general U(2) coin, deriving a formula for the average position and exploring symmetry properties based on initial states.

Contribution

It provides a new analytical expression for the average position in quantum walks with a U(2) coin and examines symmetry properties related to initial states.

Findings

Average position formula: <x> = max(<x> sin(α+γ))

Symmetry properties of quantum walks with U(2) coins

Initial state impacts on walk behavior

Abstract

We investigated discrete-time quantum walks with an arbitary unitary coin. Here we discover that the average position $<x> =\max(<x> \sin(\alpha+\gamma)$, while the initial state is $1/\sqrt{2}(\mid0L>+i\mid0R>)$. We prove the result and get some symmetry properties of quantum walks with a U(2) coin with $\mid0L>$ and $\mid0R>$ as the initial state.