# Analytical expression for variance of homogeneous-position quantum walk with decoherent position

## TL;DR

This paper derives an analytical formula for the variance of a homogeneous-position decoherent quantum walk, showing how noise affects its spread and validating the formula's accuracy through comparison with previous studies.

## Contribution

It provides a new analytical expression for the variance of decoherent quantum walks with general position noise, including effects of tunneling to multiple neighbors.

## Key findings

- Quadratic term of variance remains unaffected by decoherence.
- Linear term of variance increases with decoherence.
- Analytical expression is accurate even after dozens of steps.

## Abstract

We have derived an analytical expression for variance of homogeneous-position decoherent quantum walk (HPDQW) with general form of noise on its position, and have shown that, while the quadratic ($t^2$) term of variance never changes by position decoherency, the linear term ($t$) does and always increases the variance. We study the walker with ability to tunnel out to $d$ nearest neighbors as an example and compare our result with former studies. We also show that, although our expression have been derived for asymptotic case, the rapid decay of time-dependent terms cause the expressions to be correct with a good accuracy even after dozens of steps.