Sliding Mode Control of TwoLevel Quantum Systems
Abstract
This paper proposes a robust control method based on sliding mode design for twolevel quantum systems with bounded uncertainties. An eigenstate of the twolevel quantum system is identified as a sliding mode. The objective is to design a control law to steer the system’s state into the sliding mode domain and then maintain it in that domain when bounded uncertainties exist in the system Hamiltonian. We propose a controller design method using the Lyapunov methodology and periodic projective measurements. In particular, we give conditions for designing such a control law, which can guarantee the desired robustness in the presence of the uncertainties. The sliding mode control method has potential applications to quantum information processing with uncertainties.
quantum control, sliding mode control, bounded uncertainty, periodic projective measurement, Lyapunov methodology.
I Introduction
The manipulation and control of quantum systems is becoming an important task in many fields [1][3], such as atomic physics [4], molecular chemistry [5] and quantum information [6]. It is desirable to develop quantum control theory in a systematic way in order to adapt it to the development of quantum technology [7]. Several useful tools from classical control theory have been introduced to the control analysis and design of quantum systems. For example, optimal control theory has been used to assist in control design for closed and dissipative quantum systems [8][14]. A learning control method has been presented for guiding the control of chemical reactions [5], [15]. Quantum feedback control approaches including measurementbased feedback and coherent feedback have been used to improve performance for several classes of tasks such as preparing quantum states, quantum error correction, controlling quantum entanglement [16][30]. Robust control tools have been introduced to enhance the robustness of quantum feedback networks and linear quantum stochastic systems [31], [32].
Although some progress has been made, more research effort is necessary in controlling quantum phenomena. In particular, the robustness of quantum control systems has been recognized as a key issue in developing practical quantum technology [33][35]. In this paper, we focus on the robustness problem for quantum control systems. In [32], James and coworkers have formulated and solved a quantum robust control problem using the method for linear quantum stochastic systems. Here, we develop a variable structure control approach with sliding modes to enhance the robustness of quantum systems. The variable structure control strategy is a widely used design method in classical control theory and industrial applications where one can change the controller structure according to a specified switching logic in order to obtain desired closedloop properties [36], [37]. In [38] and [39], Dong and Petersen have formulated and solved a variable structure control problem for the control of quantum systems. However, the results in [38] only involve openloop control design using an idea of changing controller structures and do not consider the robustness which can be obtained through sliding mode control. Ref. [38] and Ref. [40] have briefly discussed the possible application of sliding mode control to quantum systems. In [41], two approaches based on sliding mode design have been proposed for the control of quantum systems and potential applications of sliding mode control to quantum information processing have been presented. Following these results, this paper formally presents a sliding mode control method for twolevel quantum systems to deal with bounded uncertainties in the system Hamiltonian [42]. In particular, we propose two approaches of designing the measurement period for different situations which are dependent on the bound on the uncertainties and the allowed probability of failure.
Variable structure control design with sliding modes generally includes two main steps: selecting a sliding surface (sliding mode) and controlling the system to and maintaining it in this sliding surface. Being in the sliding surface guarantees that the quantum system has the desired dynamics. We will select an eigenstate of the free Hamiltonian of the controlled quantum system as a sliding mode. In the second step, direct feedback control is not directly applicable since we generally cannot acquire state information without destroying the quantum system’s state. Hence, we propose a new method to accomplish this task, which is based on the Lyapunov methodology and periodic projective measurements. The Lyapunov methodology is a powerful tool for designing control laws in classical control theory and has also been applied to quantum control problems [43][48]. Most existing results on Lyapunov control of quantum systems focus on designing a control law to ensure that the controlled quantum system’s state asymptotically converges to the target state. The existing Lyapunov design methods in quantum control rely on perfect knowledge of the initial quantum states and system Hamiltonian. In our approach, once the Lyapunov control steers the quantum system into a sliding mode domain, we make a projective measurement on the system. Hence, the Lyapunov design method can tolerate small drifts (uncertainties) when carrying out our control tasks, which will be demonstrated by simulation in Section II.C. Periodic projective measurements are employed to maintain the system’s state in the sliding mode domain when uncertainties exist in the system Hamiltonian. If the measurement period is small enough and the initial state is an eigenstate, the frequent measurements make the system collapse back to the initial state. This is related to the quantum Zeno effect (for details, see [49], [50] and [51]). In contrast to the quantum Zeno effect, our objective is to design a measurement period which is as large as possible. The framework of the proposed method involves unitary control (Lyapunov control) and projective measurement. In this sense, it is similar to the discretetime quantum feedback stabilization problem in [52] and [53]. However, these papers do not consider possible uncertainties in the system Hamiltonian and use generalized measurements rather than periodic projective measurements. The main feature of the proposed method is that the control law can guarantee control performance when bounded uncertainties exist in the system Hamiltonian.
This paper is organized as follows. Section II introduces a quantum control model, defines the sliding mode and formulates the control problem. In Section III, we present a sliding mode control method based on the Lyapunov methodology and periodic projective measurements for twolevel quantum systems with bounded uncertainties. Using the known information about uncertainties (e.g., the uncertainty bound and type of uncertainties), we propose two approaches (i.e., Eqs. (13) and (14)) for designing the measurement period to guarantee the control performance. An illustrative example is presented to demonstrate the proposed method. The detailed proofs of the main theorems are presented in Section IV. Conclusions are given in Section V.
Ii Sliding modes and problem formulation
In this section, we first introduce a twolevel quantum control model. Then a sliding mode is defined using an eigenstate. Finally the control problem considered in this paper is formulated.
Iia Quantum Control Model
In this paper, we focus on twolevel purestate quantum systems. The quantum state can be represented by a twodimensional unit vector in a Hilbert space . Since the global phase of a quantum state has no observable physical effect, we do not consider the effect of global phase. If we denote the Pauli matrices as follows:
(1) 
we may select the free Hamiltonian of the twolevel quantum system as . Its two eigenstates are denoted as and . To control a quantum system, we introduce the following control Hamiltonian , where and is a set of timeindependent Hamiltonians. For simplicity, the control Hamiltonian for twolevel systems can be written as , where
(2) 
The controlled dynamical equation can be described as (we have assumed by using atomic units in this paper)
(3) 
This control problem is converted into the following problem: given an initial state and a target state, find a set of controls in (3) to drive the controlled system from the initial state to the target state.
In practical applications, we often use the density operator (or density matrix) to describe the quantum state of a quantum system. For a pure state , the corresponding density operator is . For a twolevel quantum system, the state can be represented in terms of the Bloch vector :
(4) 
The evolution equation of can be written as
(5) 
where and is the total system Hamiltonian.
After we represent the state with the Bloch vector, the pure states of a twolevel quantum system correspond to the surface of the Bloch sphere, where , , . An arbitrary pure state for a twolevel quantum system can be represented as
(6) 
IiB Sliding Modes
Sliding modes play an important role in variable structure control [36]. Usually, the sliding mode is constructed so that the system has desired dynamics in the sliding surface. For a quantum control problem, a sliding mode may be represented as a functional of the state and the Hamiltonian ; i.e., . For example, an eigenstate of the free Hamiltonian (i.e., where is one eigenvalue of ) can be selected as a sliding mode. We can define . If the initial state is in the sliding mode; i.e., , we can easily prove that the quantum system will maintain its state in this surface under only the action of the free Hamiltonian . In fact, , and we have
(7) 
That is, an eigenstate of can be identified as a sliding mode. For twolevel quantum systems, we may select either or as a sliding mode. Without loss of generality, we identify the eigenstate of a twolevel quantum system as the sliding mode in this paper.
IiC Problem Formulation
In Section II.B, we have identified an eigenstate as a sliding mode. This means that if a quantum system is driven into the sliding mode, its state will be maintained in the sliding surface under the action of the free Hamiltonian. However, in practical applications, it is inevitable that there exist noises and uncertainties. In this paper, we suppose that the uncertainties can be approximately described as perturbations in the Hamiltonian. That is, the uncertainties can be denoted as . The unitary errors in [33] belong to this class of uncertainties and uncertainties in onequbit (one quantum bit) gate also correspond to this class of uncertainties [41]. For a spin system in solidstate nuclear magnetic resonance (NMR), external noisy magnetic fields and unwanted coupling with other spins may lead to uncertainties in this class. Further, we assume the uncertainties are bounded; i.e.,
(8) 
When , . That is, there exist no uncertainties, which is trivial for our problem. Hence, in the following we assume . An important advantage of classical sliding mode control is its robustness. Our main motivation for introducing sliding mode control to quantum systems is to deal with uncertainties. We further suppose that the corresponding system without uncertainties is completely controllable and arbitrary unitary control operations can be generated. This assumption can be guaranteed for a twolevel quantum system if we can realize arbitrary rotations along the axis and axis () (e.g., see [54] for details).
The control problem under consideration is stated as follows: design a control law to drive and then maintain the quantum system’s state in a sliding mode domain even when bounded uncertainties exist in the system Hamiltonian. Here a sliding mode domain may be defined as , where is a given constant. Here we assume since the case only occurs in the sliding mode surface and the case is always true. Hence, the two cases with and are trivial for our problem. The definition of the sliding mode domain implies that the system has a probability of at most (which we call the probability of failure) to collapse out of when making a measurement. This behavior is quite different from that which occurs in traditional sliding mode control. Hence, we expect that our control laws will guarantee that the system’s state remains in except that a measurement operation may take it away from with a small probability (not greater than ). The control problem considered in this paper includes three main subtasks: (i) for any initial state (assumed to be known), design a control law to drive the system’s state into a defined sliding mode domain ; (ii) design a control law to maintain the system’s state in ; (iii) design a control law to drive the system’s state back to if a measurement operation takes it away from . For simplicity, we suppose that there exist no uncertainties during the control processes (i) and (iii).
Iii Sliding mode control based on Lyapunov methods and projective measurements
Iiia General Method
The first task is to design a control law to drive the controlled system to the chosen sliding mode domain . Lyapunovbased methods are widely used to accomplish this task in traditional sliding mode control. If the gradient of a Lyapunov function is negative in the neighborhood of the sliding surface, then the controlled system’s state will be attracted to and maintained in . The Lyapunov methodology has also been used to design control laws for quantum systems [43][48]. However, these existing results do not consider the issue of robustness against uncertainties. Since the measurement of a quantum system will inevitably destroy the measured state, most existing results on Lyapunovbased control for quantum systems in fact use a feedback design to construct an openloop control. That is, Lyapunovbased control can be used to first design a feedback law which is then used to find the openloop control by simulating the closedloop system. Then the control can be applied to the quantum system in an openloop way. Hence, the traditional sliding mode control methods using Lyapunov control cannot be directly applied to our problem.
Although quantum measurement often has deleterious effects in quantum control tasks, recent results have shown that it can be combined with unitary transformations to complete some quantum manipulation tasks and enhance the capability of quantum control [40], [55][59]. For example, Vilela Mendes and Man’ko [40] showed nonunitarily controllable systems can be made controllable by using “measurement plus evolution”. Quantum measurement can be used as a control tool as well as a method of information acquisition. It is worth mentioning that the effect of measurement on a quantum system as a control tool can be achieved through the interaction between the system and measurement apparatus. In this paper, we will combine the Lyapunov methodology and projective measurements (with the measurement operator ) to accomplish the sliding mode control task for twolevel quantum systems. The projective measurement with on a twolevel system makes the system’s state collapse into (corresponding to eigenvalue of ) or (corresponding to eigenvalue of ).
The steps in the control algorithm are as follows (see Fig. 1):

Select an eigenstate of as a sliding mode , and define the sliding mode domain as .

For a known initial state , construct a Lyapunov function to find the control law that can drive into the sliding mode .

For a specified probability of failure and , construct the control period so that the control law can drive the system’s state into in a time period .

For an initial condition which is another eigenstate , design a Lyapunov function and construct the period by using a similar method to that in 3).

According to and , design the period for periodic projective measurements.

Use the designed control law to drive the system’s state into in and make a projective measurement at . Then repeat the following operations: make periodic projective measurements with the period to maintain the system’s state in ; if the measurement result corresponds to , we use the corresponding control law to drive the state back into .
From the above control algorithm, we see that the design of Lyapunov functions and the selection of the period for the projective measurements are the two most important tasks. To design a control law for quantum systems, several Lyapunov functions have been constructed, such as state distancebased and average valuebased approaches [43][48]. Here we select a function based on the HilbertSchmidt distance between a state and the sliding mode state as a Lyapunov function [46], [48]; i.e.,
It is clear that . The firstorder time derivative of is
(9) 
where () and denotes the argument of a complex number . To ensure , we choose the control laws as in [48]:
(10) 
where may be used to adjust the control amplitude and satisfies . Define when .
When one employs a Lyapunov methodology to design a control law, LaSalle’s invariance principle is a useful tool to analyze its convergence. That is, if is an autonomous dynamical system with phase space and is a Lyapunov function on satisfying for all and , any bounded solution converges to the invariant set as (for details, see [60]). For twolevel quantum systems, LaSalle’s invariance principle can guarantee that the quantum state converges to the sliding mode under the control law in (10) (for details, see [48]). The convergence is asymptotic. Hence, we make a projective measurement with the measurement operator when we apply the Lyapunov control to the system for (corresponding to the initial condition ) or (corresponding to the initial condition ), which will drive the system into with a probability not less than .
Another important task is to design the measurement period . We can estimate a bound according to the bound on the uncertainties and the allowed probability of failure . Then, we construct a period to guarantee control performance according to the estimated bound. An extreme case is . That is, after the quantum system’s state is driven into the sliding mode, we make frequent measurements. This corresponds to the quantum Zeno effect [50], which is the inhibition of transitions between quantum states by frequent measurement of the state (see, e.g., [49] and [50]). Frequent measurements (i.e., ) can guarantee that the state is maintained in the sliding mode in spite of the presence of uncertainties. However, it is usually a difficult task to make such frequent measurements. We may conclude that the smaller is, the bigger the cost of accomplishing the periodic measurements becomes. Hence, in contrast to the quantum Zeno effect, we wish to design a measurement period as large as possible. In the following subsection, we will propose two approaches of designing for different situations.
IiiB The Design of the Measurement Period
We select the sliding mode as . If there exist no uncertainties and we have driven the system’s state to the sliding mode at time , it will be maintained in the sliding mode using only the free Hamiltonian ; i.e., . That is, if the quantum system’s state is driven into the sliding mode, it will evolve in the sliding surface. However, in practical applications, some uncertainties are unavoidable, which may drive the system’s state away from the sliding mode. We wish to design a control law to ensure the desired robustness in the presence of uncertainties. Assume that the state at time is . If we make measurements on this system, the probability that it will collapse into (the probability of failure) is
(11) 
where . We have assumed that the possible uncertainties can be described by , where unknown , and satisfy . We now give detailed discussions to design the measurement period for possible uncertainties.
First we consider a special case (). This case corresponds to phaseflip type bounded uncertainties. For any (where ), if , we have
(12) 
This type of uncertainty does not drive the system’s state away from the sliding mode. Hence we ignore this type of uncertainty in our method.
Now we consider the unknown uncertainties (where ) and have the following theorem.
Theorem 1
For a twolevel quantum system with the initial state at the time , the system evolves to under the action of (where and ). If , where
(13) 
the system’s state will remain in (where ). When one makes a projective measurement with the measurement operator at the time , the probability of failure is not greater than .
Using Theorem 1, we may try to maintain the system’s state in (i.e., the subtask (ii)) by implementing periodic projective measurements with the measurement period . If we have more knowledge about the uncertainties, it is possible to improve the measurement period . Now assume that the uncertainty is () and . We have the following theorem.
Theorem 2
For a twolevel quantum system with the initial state at the time , the system evolves to under the action of (where , and ). If and , where
(14) 
the system’s state will remain in (where ). When one makes a projective measurement with the measurement operator at the time , the probability of failure is not greater than .
Remark 1
The proofs of Theorem 1 and Theorem 2 will be presented in Section IV. The two theorems mean the following fact. For a twolevel quantum system with unknown uncertainties (where ), if its initial state is in the sliding mode , we can ensure that the probability of failure is not greater than a given constant () through implementing periodic projective measurements with the measurement period using (13). Further, if we know that and satisfy the relationship and there exists only one type of uncertainty (i.e., or , where ), we can design a measurement period using (14) which is larger than . The proof of Theorem 2 also shows that is an optimal measurement period. This measurement period will guarantee the required robustness. It is easy to prove the relationship for arbitrary . The detailed proof will be presented in the Appendix. Based on the above analysis, the selection rule for is summarized in Table I. Moreover, from (13) and (14), it is clear that for a constant , and when . That is, for a given bound on the uncertainties, if we expect to guarantee the probability of failure , it requires us to implement frequent measurements such that the measurement period . Another special case is , which leads to and . That is, to deal with very large uncertainties, we need to make frequent measurements () to guarantee the desired robustness. From (13), we also know that for a given , monotonically decreases with increasing . This means that we need to employ a smaller measurement period to deal with uncertainties with a larger bound .
Type of uncertainties  (  

Allowed probability of failure 


The measurement period 

IiiC An Illustrative Example
Now we present an illustrative example to demonstrate the proposed method. Assume . Consider two cases: (a) ; (b) . For simplicity, we assume . Hence, . We first design the control and using (10). Here, we consider control only using . Using (10), we select and . Let the time stepsize be given by . We can obtain the probability curve of shown in Fig. 2, the control value shown in Fig. 3 and . For , we can design the measurement period using (13). For , we can design the measurement period using (13). Since when , if the uncertainties take the form of (, we can improve the measurement period to using (14). It is clear that in these two cases. For some practical quantum systems such as spin systems in NMR, we can use strong control actions (e.g., ) to drive the system from into within a short time period [8]. These facts make the assumption of no uncertainties in the control process reasonable. Moreover, the fact that the measurement period is much greater than the control time required to go to from indicates the possibility of realizing such a periodic measurement on a practical quantum system.
Remark 2
In the process of designing the control law for driving the system’s state from to , we employ an approach based on the Lyapunov methodology. An advantage of such an approach is that it is relatively easy to find a control law by simulation. It is worth noting that most existing applications of the Lyapunov methodology to quantum systems do not involve measurement. Here, we combine the Lyapunovbased control and projective measurements for controlling quantum systems, which in some applications make our method more useful than the Lyapunovbased control for quantum systems proposed in previous papers. In [41], an approach based on timeoptimal control design has also been proposed to complete this task. The advantage of such an approach is that we take the shortest time to complete the control task. However, it is generally difficult to find a complete timeoptimal solution for highdimensional quantum systems. For the above simple task, it has been proven that the timeoptimal control employs a bangbang control strategy [14]. Using the method in [14], we should take in and then use in . In this case, the total time required is ().
Remark 3
In the process of designing the Lyapunov control for driving the system’s state from to , we ignore possible uncertainties. By simulation, we find that small uncertainties can also be tolerated in this process. For example, if and the uncertainty is the noise with a uniform distribution on the interval , the probability curves of are shown in Fig. 4 when we apply the control obtained from Fig. 3 to the quantum system. The probabilities of for the cases with uncertainties are very close to the probability of for the case without uncertainties. By more simulation, we find that the final probability of is for ( where or ), the final probability of is for () and the final probability of is for (). If we use a smaller probability of failure (e.g., ) as the terminal condition of the Lyapunov control or employ a bigger for the same , these simulations suggest that it is possible to ensure that the Lyapunov control will drive the system’s state into the sliding mode domain even when there exist small uncertainties.
Iv Proof of Theorems
This section will present the detailed proofs of Theorem 1 and Theorem 2. The proof of Theorem 1 involves the following steps: (I) Compare the probabilities of failure for and ( and are constant); (II) Compare the probabilities of failure for and (); (III) Use the previous results to compare the probabilities of failure for and (); (IV) Use to estimate the measurement period. The basic steps for the proof of Theorem 2 include: (I) Formulate the problem of finding the “worst” case as an optimal control problem of ; (II) Obtain the optimal control solution for nonsingular cases; (III) Exclude the possibility of singular cases; (IV) Use the “worst” case to estimate the measurement period. Considering that the arguments in the proof of Theorem 2 are useful for the proof of Theorem 1, we will first present the proof of Theorem 2 and then prove Theorem 1.
Iva Proof of Theorem 2
{proof}We now consider as a control input and select the performance measure as
(17) 
From (11), we know that the “worst” case (i.e., the case maximizing the probability of failure) corresponds to minimizing . Also, we introduce the Lagrange multiplier vector and obtain the corresponding Hamiltonian function as follows:
(18) 
where . That is
(19) 
According to Pontryagin’s minimum principle [61], a necessary condition for to minimize is
(20) 
The necessary condition provides a relationship to determine the optimal control . If there exists a time interval of finite duration during which the necessary condition (20) provides no information about the relationship between , , , we call the interval a singular interval [61]. If we do not consider singular cases (i.e., ), the optimal control should be chosen as follows:
(21) 
That is, the optimal control strategy for is bangbang control; i.e., . Now we consider which leads to the state equation
(22) 
where . The corresponding solution is
(23) 
where . From (23), we know that is a monotonically decreasing function in when . Hence, we only consider the case where .
Now consider the optimal control problem with a fixed final time and a free final state . According to Pontryagin’s minimum principle, . From this, it is straightforward to verify that . Now let us consider another necessary condition which leads to the following relationships:
(24) 
where . The corresponding solution is
(25) 
We obtain
(26) 
It is easy to show that the quantity occurring in (21) does not change its sign when and .
Now we further exclude the possibility that there exists a singular case. Suppose that there exists a singular interval (where ) such that when
(27) 
We also have the following relationship
(28) 
where we have used (16) and the following costate equation
(29) 
If , we have . By the principle of optimality [61], we may consider the case . Using (27), (28) and , we have and . Using the relationship of , we obtain or . If , the initial state and the final state are the same state . However, if we use the control , from (23) we have . Hence, this contradicts the fact that we are considering the optimal case . If , there exists such that . By the principle of optimality [61], we may consider the case . From the two equations (27) and (28), we know that which contradicts . Hence, no singular condition can exist if .
If , using (21) we must select when . From (26), we know that there exist no satisfying . Hence, there exist no singular cases for our problem.
From the above analysis, is the optimal control when . Hence . From (11), it is clear that the probabilities of failure satisfy . That is, the probability of failure is not greater than for . When , is monotonically decreasing and is monotonically increasing. When , using (23) we have . That is, the probability of failure . Hence, we can design the measurement period using the case of when .
Using (11) and (23), for we obtain the probability of failure
(30) 
Hence, we can design the maximum measurement period as follows
(31) 
For (where ), we can obtain the same conclusion as that in the case (where ).
IvB Proof of Theorem 1
Lemma 3
For a twolevel quantum system with the initial state (i.e., ), the system evolves to and under the action of