Quantum control theory and applications: A survey <s> Theorem 8 The system <s> Abstract This paper formulates and solves a variable structure control problem for the state control of quantum systems. Following the concept of wavefunction controllability, we first present a specific quantum system which is wavefunction uncontrollable under the action of any one of two controllers with different fixed structures. However, we prove that the system is wavefunction controllable if the control input is allowed to switch between the two controllers. An extension and the potential applications of the proposed control strategy are briefly discussed and the general steps of variable structure control design for this class of quantum systems are also presented. The variable structure control scheme provides an alternative quantum engineering approach for the manipulation of quantum systems. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Theorem 8 The system <s> This paper proposes a new robust control method for quantum systems with uncertainties involving sliding mode control (SMC). Sliding mode control is a widely used approach in classical control theory and industrial applications. We show that SMC is also a useful method for robust control of quantum systems. In this paper, we define two specific classes of sliding modes (i.e., eigenstates and state subspaces) and propose two novel methods combining unitary control and periodic projective measurements for the design of quantum sliding mode control systems. Two examples including a two-level system and a three-level system are presented to demonstrate the proposed SMC method. One of main features of the proposed method is that the designed control laws can guarantee desired control performance in the presence of uncertainties in the system Hamiltonian. This sliding mode control approach provides a useful control theoretic tool for robust quantum information processing with uncertainties. <s> BIB002 </s> Quantum control theory and applications: A survey <s> Theorem 8 The system <s> This article focuses on wavefunction controllability of a class of quantum systems with switching control. First, we present a specific quantum system which is not wavefunction controllable under the action of any one of two controllers with different fixed structures. However, we prove that the system is wavefunction controllable if the control input is allowed to switch between the two controllers. Then we present a sufficient condition for wavefunction controllability of a class of quantum systems with switching control and briefly discuss the problem of control design for such quantum control systems. Finally, two illustrative examples are presented to demonstrate a possible control design process for the proposed method. <s> BIB003
is pure state controllable if the controller B u can be arbitrarily switched between controllers B u m and B u n (m = n, 1 ≤ m, n ≤ M ), and the following assumptions hold: (I) The combined graph G for the system (31) is connected. (II) The combined graph G remains connected after elimination of edge pairs (|φ j , |φ l ), (|φ a , |φ b ) such that ν jl = ν ab (degenerate transitions). The proof of this theorem can be found in BIB003 . The variable structure control strategy provides a useful method for open-loop control design for a class of quantum systems and is also applicable to qubit preparation for quantum information processing (for details, see BIB001 , BIB003 ). Here, for open-loop control of quantum systems, we only use the concept of variable structure control and consider control inputs which are allowed to switch between a number of different control structures. Moreover, note that sliding mode control is an important aspect in variable structure control of classical systems. Also, the sliding mode method can be used to deal with uncertainties in the feedback control of quantum systems and has potential applications to quantum state preparation and quantum error correction BIB002 .