Approximate bang-bang control assisted rapid switching feedback stabilization for stochastic qubit systems

Abstract

For stochastic qubit systems under measurement feedback, an approximate bang-bang control assisted rapid switching control scheme with fewer switching times between the two subsets of state space is proposed in this paper. An eigenstate for single-qubit systems and any GHZ entangled state for multi-qubit systems are prepared successfully. In our control scheme, the state space is partitioned into two parts: a subset S1 containing the target state and its complementary set S2. On each subset, a Lyapunov function is selected and the corresponding control law is designed, where the control laws in S1 are of approximate bang-bang form and therefore can speed up the control process and the control laws in S2 ensure the strictly monotonic descent of the corresponding Lyapunov function and therefore can reduce the switching times. In particular, for multi-qubit systems with degenerate measured observables, we use two control channels to distinguish the target state from other isospectral eigenstates. By properly constructing the control Hamiltonians and using stochastic Lyapunov stability theory, we prove that the switching control laws in this paper render the system trajectory to switch at most twice on the boundary of S1 and S2 and to converge to the target state contained in S1 rapidly. Numerical simulations verify the effectiveness and rapidity of the proposed rapid control scheme.