Quantum walk based state transfer algorithms on the complete M-partite graph

TL;DR

This paper explores quantum walk algorithms for search and state transfer on complete M-partite graphs, demonstrating high-fidelity transfer between partitions and proposing an active switch to improve same-partition transfer fidelity.

Contribution

It introduces quantum walk-based search and state transfer algorithms on M-partite graphs, including a novel active switch method for improved same-partition transfer fidelity.

Findings

Search algorithm finds marked vertices with unit probability in large graphs.

High-fidelity state transfer between different partitions is achievable.

Active switching enhances transfer fidelity within the same partition.

Abstract

We investigate coined quantum walk search and state transfer algorithms, focusing on the complete $M$-partite graph with $N$ vertices in each partition. First, it is shown that by adding a loop to each vertex the search algorithm finds the marked vertex with unit probability in the limit of a large graph. Next, we employ the evolution operator of the search with two marked vertices to perform a state transfer between the sender and the receiver. We show that when the sender and the receiver are in different partitions the algorithm succeeds with fidelity approaching unity for a large graph. However, when the sender and the receiver are in the same partition the fidelity does not reach exactly one. To amend this problem we propose a state transfer algorithm with an active switch, whose fidelity can be estimated based on the single vertex search alone.