Controllability of Quantum Walks on Graphs

TL;DR

This paper studies the controllability of discrete-time quantum walks on graphs with vertex-dependent coin operations, providing criteria for complete controllability based on graph properties and Lie algebra analysis.

Contribution

It characterizes the set of reachable states using Lie algebra techniques and establishes conditions under which quantum walks are fully controllable.

Findings

Quantum walks are fully controllable if graph degree exceeds half the number of nodes.

Controllability depends only on the graph structure, not on specific walk details.

Explicit algorithms for control and state transfer steps are provided.

Abstract

In this paper, we consider discrete time quantum walks on graphs with coin focusing on the decentralized model, where the coin operation is allowed to change with the vertex of the graph. When the coin operations can be modified at every time step, these systems can be looked at as control systems and techniques of geometric control theory can be applied. In particular, the set of states that one can achieve can be described by studying controllability. Extending previous results, we give a characterization of the set of reachable states in terms of an appropriate Lie algebra. We then prove general results and criteria relating controllability to the algebraic and topological properties of the walk. As a consequence of these results, we prove that if the degree of the underlying graph is larger than $\frac{N}{2}$, where $N$ is the number of nodes, the quantum walk is always completely controllable, i.e., it is possible to having it to evolve according to an arbitrary unitary evolution. Another result is that controllability for decentralized models only depends on the graph and not on the particular quantum walk defined on it. We also provide explicit algorithms for control and quantify the number of steps needed for an arbitrary state transfer. The results of the paper are of interest in quantum information theory where quantum walks are used and analyzed in the development of quantum algorithms.