Simplifying Continuous-Time Quantum Walks on Dynamic Graphs

TL;DR

This paper explores methods to simplify continuous-time quantum walks on dynamic graphs by identifying specific scenarios that enable reductions, facilitating more efficient quantum circuit implementations.

Contribution

The paper introduces six scenarios for simplifying dynamic graphs in quantum walks, enhancing the efficiency of quantum circuit design.

Findings

Simplification via commuting graphs

Parallel implementation of single-qubit gates

Utilization of hypercube uniform mixing

Abstract

A continuous-time quantum walk on a dynamic graph evolves by Schr\"odinger's equation with a sequence of Hamiltonians encoding the edges of the graph. This process is universal for quantum computing, but in general, the dynamic graph that implements a quantum circuit can be quite complicated. In this paper, we give six scenarios under which a dynamic graph can be simplified, and they exploit commuting graphs, identical graphs, perfect state transfer, complementary graphs, isolated vertices, and uniform mixing on the hypercube. As examples, we simplify dynamic graphs, in some instances allowing single-qubit gates to be implemented in parallel.