# One Dimensional Quantum Walks with Two-step Memory

## TL;DR

This paper introduces a new model of one-dimensional quantum walks with two-step memory, providing a general amplitude formula and analyzing its probability distribution through simulations, revealing distinct and similar features compared to existing models.

## Contribution

The paper develops a general formula for amplitudes of two-step-memory quantum walks with Hadamard coin and compares its behavior to one-step memory and memoryless walks.

## Key findings

- Probability distribution differs from one-step memory quantum walk.
- Distribution shows similarities to memoryless Hadamard quantum walk.
- Simulation confirms theoretical differences and similarities.

## Abstract

In this paper we investigate one dimensional quantum walks with two-step memory, which can be viewed as an extension of quantum walks with one-step memory. We develop a general formula for the amplitudes of the two-step-memory walk with Hadamard coin by using path integral approach, and numerically simulate its process. The simulation shows that the probability distribution of this new walk is different from that of the Hadamard quantum walk with one-step memory, while it presents some similarities with that of the normal Hadamard quantum walk without memory.