# Coined Quantum Walks on Weighted Graphs

## TL;DR

This paper introduces a new type of quantum walk on weighted graphs, generalizing existing models, and demonstrates its applications to quantum search algorithms with improved success probabilities.

## Contribution

It defines a discrete-time coined quantum walk on weighted graphs and extends lackadaisical quantum walks to real-valued self-loop weights, with applications to quantum search and dispersion analysis.

## Key findings

- Weighted quantum walk equivalent to a deformed Grover walk with faster dispersion
- Improved success probability in weighted quantum search for certain self-loop weights
- Generalization of lackadaisical quantum walks to real-valued self-loops

## Abstract

We define a discrete-time, coined quantum walk on weighted graphs that is inspired by Szegedy's quantum walk. Using this, we prove that many lackadaisical quantum walks, where each vertex has $l$ integer self-loops, can be generalized to a quantum walk where each vertex has a single self-loop of real-valued weight $l$. We apply this real-valued lackadaisical quantum walk to two problems. First, we analyze it on the line or one-dimensional lattice, showing that it is exactly equivalent to a continuous deformation of the three-state Grover walk with faster ballistic dispersion. Second, we generalize Grover's algorithm, or search on the complete graph, to have a weighted self-loop at each vertex, yielding an improved success probability when $l < 3 + 2\sqrt{2} \approx 5.828$.