Zero forcing, linear and quantum controllability for systems evolving on networks

Abstract

We study the dynamics of systems on networks from a linear algebraic perspective. The control theoretic concept of controllability describes the set of states that can be reached for these systems. Under appropriate conditions, there is a connection between the quantum (Lie theoretic) property of controllability and the linear systems (Kalman) controllability condition. We investigate how the graph theoretic concept of a zero forcing set impacts the controllability property. In particular, we prove that if a set of vertices is a zero forcing set, the associated dynamical system is controllable. The results open up the possibility of further exploiting the analogy between networks, linear control systems theory, and quantum systems Lie algebraic theory. This study is motivated by several quantum systems currently under study, including continuous quantum walks modeling transport phenomena. Additionally, it proposes zero forcing as a new notion in the analysis of complex networks.

12 pages, 1 figure

References in corpus (6)

• Continuous-Time Quantum Walks: Models for Coherent Transport on Complex Networks
• Full control by locally induced relaxation
• Local controllability of quantum networks
• Quantum System Identification
• Coherent open-loop optimal control of light-harvesting dynamics
• Controllable Subsets in Graphs

Cited by in corpus (19)

• Zero forcing sets and controllability of dynamical systems defined on graphs
• From pulses to circuits and back again: A quantum optimal control perspective on variational quantum algorithms
• On the Universality of the Quantum Approximate Optimization Algorithm
• Driven quantum dynamics: will it blend?
• Logic circuits from zero forcing
• Lyapunov control-inspired strategies for quantum combinatorial optimization
• The zero forcing polynomial of a graph
• Efficiently Controllable Graphs
• Fractional Zero Forcing via Three-color Forcing Games
• Leaky Forcing: A New Variation of Zero Forcing
• Zero forcing number, constrained matchings and strong structural controllability
• Quantum Speed Limits from Symmetries in Quantum Control
• Robustness of random-control quantum-state tomography
• Failed zero forcing and critical sets on directed graphs
• Zero forcing number of graphs with a power law degree distribution
• Failed power domination on graphs
• Leaky Positive Semidefinite Forcing on Graphs
• On the Relationships between Zero Forcing Numbers and Certain Graph Coverings
• Minimum leader selection for Structural Controllability of Undirected Graphs with Leader-follower Framework