# Measurement-Induced Boolean Dynamics and Controllability for Quantum Networks

## TL;DR

This paper investigates how sequential quantum measurements influence the dynamics and controllability of quantum networks, revealing probabilistic Boolean models and their Markovian or non-Markovian nature, with implications for quantum control.

## Contribution

It introduces a novel analysis of measurement-induced quantum dynamics, deriving explicit Boolean recursive models and exploring their impact on network controllability.

## Key findings

- Global measurements induce Markov chain dynamics.
- Local measurements lead to non-Markovian behavior.
- Measurement observables and Hamiltonian determine state transitions.

## Abstract

In this paper, we study dynamical quantum networks which evolve according to Schr\"odinger equations but subject to sequential local or global quantum measurements. A network of qubits forms a composite quantum system whose state undergoes unitary evolution in between periodic measurements, leading to hybrid quantum dynamics with random jumps at discrete time instances along a continuous orbit. The measurements either act on the entire network of qubits, or only a subset of qubits. First of all, we reveal that this type of hybrid quantum dynamics induces probabilistic Boolean recursions representing the measurement outcomes. With global measurements, it is shown that such resulting Boolean recursions define Markov chains whose state-transitions are fully determined by the network Hamiltonian and the measurement observables. Particularly, we establish an explicit and algebraic representation of the underlying recursive random mapping driving such induced Markov chains. Next, with local measurements, the resulting probabilistic Boolean dynamics is shown to be no longer Markovian. The state transition probability at any given time becomes dependent on the entire history of the sample path, for which we establish a recursive way of computing such non-Markovian probability transitions. Finally, we adopt the classical bilinear control model for the continuous Schr\"odinger evolution, and show how the measurements affect the controllability of the quantum networks.