# A limit distribution for a quantum walk driven by a five-diagonal unitary matrix

## TL;DR

This paper derives an explicit long-time limit distribution for a quantum walk driven by a five-diagonal unitary matrix, revealing how phase terms influence the walk's spatial distribution over time.

## Contribution

It introduces a new quantum walk model with a five-diagonal unitary matrix and provides an explicit form of its long-time limit distribution, highlighting phase effects.

## Key findings

- Explicit limit distribution derived
- Phase term affects quantum walk behavior
- Walk delocalizes over time

## Abstract

In this paper, we work on a quantum walk whose system is manipulated by a five-diagonal unitary matrix, and present long-time limit distributions. The quantum walk launches off a location and delocalizes in distribution as its system is getting updated. The five-diagonal matrix contains a phase term and the quantum walk becomes a standard coined walk when the phase term is fixed at special values. Or, the phase term gives an effect on the quantum walk. As a result, we will see an explicit form of a long-time limit distribution for a quantum walk driven by the matrix, and thanks to the exact form, we understand how the quantum walker approximately distributes in space after the long-time evolution has been executed on the walk.