Feedback Exponential Stabilization of GHZ States of Multiqubit Systems

In this article, we consider stochastic master equations describing the evolution of a multiqubit system interacting with electromagnetic fields undergoing continuous-time measurements. By considering multiple $z$-type (Pauli $z$ matrix on different qubits) and $x$-type (Pauli $x$ matrix on all qubits) measurements and one control Hamiltonian, we provide general conditions on the feedback controller and the control Hamiltonian ensuring almost sure exponential convergence to a predetermined Greenberger–Horne-Zeilinger (GHZ) state, which is assumed to be a common eigenstate of the measurement operators. We provide explicit expressions of feedback controllers satisfying such conditions. We also consider the case of only $z$-type measurements and multiple control Hamiltonians. We show that local stability in probability holds true, however due to the absence of random displacements generated by $x$-type measurements, the reachability of a neighborhood of a predetermined GHZ state is not clear. In this case, we provide a heuristic discussion on some conditions which may ensure asymptotic convergence toward the target state. Finally, we demonstrate the effectiveness of our methodology for a three-qubit system through numerical simulations.