Role of symmetry in quantum search via continuous-time quantum walk

TL;DR

This paper explores how graph symmetries influence the invariant subspaces in continuous-time quantum walk-based search, highlighting the importance of asymmetries in the Hamiltonian for effective quantum search.

Contribution

It introduces a group representation theory-based method to identify invariant subspaces and emphasizes the role of asymmetries in quantum search efficiency.

Findings

Invariant subspaces correspond to trivial representations of symmetry groups.

Symmetries determine the dimensionality reduction in quantum walk analysis.

Asymmetric parts of the Hamiltonian are crucial for successful quantum search.

Abstract

For quantum search via the continuous-time quantum walk, the evolution of the whole system is usually limited in a small subspace. In this paper, we discuss how the symmetries of the graphs are related to the existence of such an invariant subspace, which also suggests a dimensionality reduction method based on group representation theory. We observe that in the one-dimensional subspace spanned by each desired basis state which assembles the identically evolving original basis states, we always get a trivial representation of the symmetry group. So we could find the desired basis by exploiting the projection operator of the trivial representation. Besides being technical guidance in this type of problem, this discussion also suggests that all the symmetries are used up in the invariant subspace and the asymmetric part of the Hamiltonian is very important for the purpose of quantum search.