Rapid Lyapunov control of finite-dimensional quantum systems

Rapid Lyapunov control of finite-dimensional quantum systems

Abstract

Rapid state control of quantum systems is significant in reducing the influence of relaxation or decoherence caused by the environment and enhancing the capability in dealing with uncertainties in the model and control process. Bang-bang Lyapunov control can speed up the control process, but cannot guarantee convergence to a target state. This paper proposes two classes of new Lyapunov control methods that can achieve rapidly convergent control for quantum states. One class is switching Lyapunov control where the control law is designed by switching between bang-bang Lyapunov control and standard Lyapunov control. The other class is approximate bang-bang Lyapunov control where we propose two special control functions which are continuously differentiable and yet have a bang-bang type property. Related stability results are given and a construction method for the degrees of freedom in the Lyapunov function is presented to guarantee rapid convergence to a target eigenstate being isolated in the invariant set. Several numerical examples demonstrate that the proposed methods can achieve improved performance for rapid state control of quantum systems.

q

A1,A2]Sen Kuang, A2]Daoyi Dong, A2]Ian R. Petersen

uantum systems; switching control; approximate bang-bang control; rapid Lyapunov control

1 Introduction

Quantum control has the potential to play important roles in the development of quantum information technology and quantum chemistry, and has received wide attention from different fields such as quantum information, chemical physics and quantum optics [1, 2, 3, 4, 5, 6]. Transfer control between quantum states is one of the basic tasks in quantum control. Different control strategies such as optimal control [7, 8, 9, 10], adiabatic control [11, 12], Lyapunov control methods [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], and LQG control [24, 25, 26], and sliding mode control [27, 28] have been presented for controller design in quantum systems. Among these control strategies, Lyapunov control methods have been extensively studied for quantum systems due to their simplicity and intuitive nature in the design of control fields [29, 30, 31, 32, 33]. In Lyapunov control, a Lyapunov function is constructed using information on states or operators related to the quantum system [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 29, 30, 31, 32, 33] and the associated control law is designed based on the Lyapunov function (feedback design). Then the control law can be implemented in an open-loop way. From the viewpoint of control theory, one hopes that any system trajectory converges to a desired target state. Unfortunately, the LaSalle invariance principle used in Lyapunov control methods cannot guarantee convergence of any system trajectory to a target state. Some methods such as using implicit Lyapunov functions or switching control methods have been developed to achieve approximate or asymptotic convergence for some specific quantum control tasks (see, e.g., [14, 16, 19, 21]).

For quantum systems, rapid state control is of importance because a realistic quantum system cannot be perfectly separated from its environment, which will cause a relaxation or decoherence effect. In the context of quantum information processing, rapid control is a basic requirement for performance improvement in quantum computing. In practical applications, robustness has been recognized as a key requirement for the development of quantum technology [34, 35, 36, 37, 38, 39, 40, 41]. Rapid control may make the control law more robust to uncertainties in the model or in the control process. Time optimal control methods have been proposed to achieve rapid control for quantum systems [42, 43, 44]. However, it is very difficult to obtain the optimal control law for general quantum systems [42, 43, 44]. In [22], an optimal Lyapunov design method has been proposed to design a control law for rapid state transfer in quantum systems. Under power-type and strength-type constraints on the control fields, two kinds of Lyapunov control laws were proposed. In particular, the strength-type constraint led to a bang-bang Lyapunov control. In [23], the convergence problem for bang-bang Lyapunov control law is further discussed for two-level quantum systems.

The bang-bang Lyapunov control method in [22, 23] can be used to achieve rapid state control for some practical quantum systems with a high level of fidelity. However, since the control function of bang-bang Lyapunov control is not continuously differentiable, the LaSalle invariance principle cannot be directly used to guarantee convergence. We show that a high-frequency oscillation with an infinitesimal period may occur in bang-bang Lyapunov control, which prevents effective transfer to the target state. Such control fields can also not be realized in the laboratory. In order to achieve rapidly convergent control in state transfer, we propose two classes of new Lyapunov control methods in this paper: switching Lyapunov control and approximate bang-bang Lyapunov control. We first derive a sufficient condition for a two-level system that shows a high-frequency oscillation with an infinitesimal period in bang-bang Lyapunov control. Then we design a switching strategy, i.e., switching between bang-bang Lyapunov control and standard Lyapunov control. For approximate bang-bang Lyapunov control, we design two special control functions that incorporate bang-bang and smoothness properties. These proposed Lyapunov design methods can achieve rapidly convergent control in quantum systems, which is demonstrated by several numerical examples involving a two-level system, a three-level system and a two-qubit superconducting system.

This paper is organized as follows. Section 2 presents the system model, and analyzes the robustness of open-loop quantum control. Section 3 discusses Lyapunov functions with various degrees of freedom, presents several stability results, and develops a construction method for designing the degrees of freedom in the Lyapunov function. A switching strategy between bang-bang and standard Lyapunov control schemes is proposed and the switching condition is investigated in Section 4. In Section 5, we propose two approximate bang-bang Lyapunov control methods. Three numerical examples are presented to demonstrate the performance of the proposed rapid Lyapunov control strategies in Section 6. Conclusions are presented in Section 7.

Notation

  • : the imaginary unit, i.e., ;

  • : the commutator of the matrices and , i.e., ;

  • : the repeated commutator with depth , i.e., ;

  • : the induced -norm of the matrix , or the -norm of the vector ;

  • : the transpose of the matrix ;

  • : the conjugate transpose of the matrix ;

  • : the conjugate transpose of the state vector ;

  • : the complex conjugate of the complex number ;

  • : the modulus of the complex number ;

  • : the sets of all real numbers;

  • : the sets of all complex numbers;

  • : the trace of the matrix ;

  • : the spectrum of the matrix , i.e., the set of all eigenvalues of ;

  • : the real part of the complex number ;

  • : the imaginary part of the complex number .

2 System model and robustness of control

2.1 Models of finite-dimensional quantum systems

Assume that the quantum system under consideration is an -dimensional and controllable closed system [45], described by the following Liouville-von Neumann equation:

(1)

where is a density matrix describing the state of the system; is the internal Hamiltonian, and is the control Hamiltonian that describes the interaction between the external control fields and the system ( and are time-independent Hermitian matrices); and are external real-valued control fields. In the energy representation, has a diagonal form, i.e., . We call the transition frequency between the energy levels and . Denote as the eigenvector of corresponding to the eigenvalue , i.e., where the -th element is 1 and other elements are 0. All form an orthogonal basis of the dimensional complex Hilbert space .

Since the Hamiltonian is a Hermitian matrix, the evolution of the system (1) is unitary. For a given initial state , the quantum state at time , , can always be written as

(2)

where is a unitary matrix with . Equation (2) indicates that the system state at any time always has the same spectrum with the initial state .

Using (1) and the property of the unitary operator , we can undertake a differentiation calculation on both sides of (2) to obtain the Schrödinger equation for :

(3)

with .

We assume that the control objective is to steer the system to an eigenstate of , . Due to the isospectral evolution property of closed quantum systems and the fact that pure states () and mixed states () have different spectra, we assume that the initial state is a pure state since the target eigenstate is a pure state. Moreover, quantum pure states have wide applications in quantum information processing.

Also, the following conditions are assumed on the system:

(4)
(5)

Condition (4) means that the transition frequencies between the target eigenstate and other eigenstates are distinguishable, and that is non-degenerate, i.e., its all eigenvalues are mutually different. Condition (5) implies that there exists a direct coupling between the target eigenstate and any other eigenstate.

2.2 Robustness of open-loop quantum control

For a closed quantum system, we may use the model (1) to design an open-loop control law and then apply the control law to the practical system. Since any physical system is unavoidably affected by some uncertainties, the robustness of the control law should be considered. Possible perturbations to the quantum system (1) include perturbations of the internal Hamiltonian , and perturbations in the control Hamiltonian , inaccuracy in the control law, and inaccuracy in the initial states. Here, we mainly discuss perturbations in the Hamiltonian. The aim is to show that rapid control can make the control law more robust to uncertainties in the model or in the control process. This is another motivation (besides reducing the relaxation and decoherence effect) to develop rapid Lyapunov control in the following.

We denote the perturbations in the internal and control Hamiltonians as and , respectively, where is a real diagonal matrix and are Hermitian matrices. Thus, the internal and control Hamiltonians with perturbations can be written as and , respectively. We call the model (1) the nominal system and the system with and the perturbed system. Define . Then, the dynamics of the perturbed system can be described as

(6)

where the Hermitian matrix is the uncertainty in the Hamiltonian .

We assume that the uncertainty satisfies and that the introduction of does not break Conditions (4) and (5), i.e., , where and is the diagonal elements of ; and . Now, we examine the effect of the uncertainty on the quantum system. {thm} We assume . For any initial pure state , the states of the nominal system (1) and the perturbed system (6) satisfy . If at a finite time , then for an arbitrarily given , when , the distance between and the target state satisfies . {pf} Similar to (3), we write the perturbed system (6) in terms of its time evolution operators as follows:

(7)

Since both and are unitary matrices, we let

(8)

Differentiating both sides of (8) with respect to and considering (3) and (7), we have

(9)

Define . We have . The Dyson series solution of (9) is the following time-ordered integral

(10)

where . Considering , we have

(11)

For any initial state , we have

(12)

Considering , we have

(13)

For the perturbed system (6), when ,

(14)

Hence, when , we have .  Theorem 2.2 can be regarded as a generalization to quantum systems of the continuous dependence on parameters of solutions to differential equations. Theorem 2.2 shows that, for given and , if the nominal system (1) can approach the target state within a shorter time period, the perturbed system (6) can tolerate larger perturbations when guaranteeing given performance. That is to say, a rapidly convergent control for the nominal system (1) may lead to improved robustness. This paper develops rapidly convergent Lyapunov control methods for the system (1).

3 Lyapunov quantum control and stability

3.1 Lyapunov control design

Consider the following Lyapunov function:

(15)

where is a positive semi-definite Hermitian operator that needs to be constructed for completing a given control task.

The time derivative of the Lyapunov function (15) is calculated as

(16)

We design the control laws by guaranteeing in (16). Considering the fact that in (16) is independent of the control field while is an unknown Hermitian matrix to be constructed, we let

(17)

Since the diagonal matrix is non-degenerate, (17) implies that is also a diagonal matrix. We denote . Using (17), (16) can be written as

(18)

where . For notational simplicity, we also denote as hereinafter.

Thus, by guaranteeing in (18), we design a control law with the following general form:

(19)

where the control function satisfies: 1) is continuously differentiable with respect to ; 2) ; and 3) . In particular, we call the following control law the standard Lyapunov control in this paper:

(20)

where the control gain is used to adjust the amplitude of the control field .

3.2 General stability results

The control law (19) means that the whole “closed-loop” system is a nonlinear autonomous system. We use the LaSalle invariance principle to analyze the stability of the system. The LaSalle principle ensures that the system (1) with the control fields (19) necessarily converges to the largest invariant set contained in .

Assume and let . The invariance property guarantees that , which holds when , i.e.

(21)

Substituting the solution of into (21) and using (17), one has

(22)

Since the time function sequence is linearly independent, and and are diagonal, we have

(23)

Since and are Hermitian matrices, (3.2) reduces to

(24)

For even and odd , (3.2) has the following forms, respectively:

(25)
(26)

We denote , and define

(27)
(28)
(29)
(30)

Then (25) and (26) read

(31)
(32)

Since the system (1) evolves unitarily, the positive limit set of any evolution trajectory has the same spectrum as its initial state. Thus, the invariant set that the system with the control law (19) will converge to can be characterized in the following theorem. {thm} Given an arbitrary initial pure or mixed state , and under the action of the control fields (19), the system (1) converges to the invariant set , where represents the spectrum of “”, and and are defined by (3.2)-(30). From Theorem 3.2, the invariant set that the system converges to is dependent on the transition frequencies of the system, the diagonal values of , the initial state and the connectivity of . When these factors satisfy some particular conditions, it is possible to obtain a simpler form for the invariant set (see [31]).

3.3 Construction of Hermitian operator

In this subsection, we study the construction method of to achieve convergence to the target eigenstate . Thus, we only consider the case of initial pure states. For the system (1), all possible initial pure states can be divided into two classes: initial states satisfying either or . In this subsection, we give a method for constructing to achieve convergence to the target state for these two classes of initial states, respectively.

When the initial state satisfies , we have the following result. {thm} Consider the system (1) satisfying Conditions (4), (5) and with the control fields (19). Assume that the target eigenstate and the initial pure state satisfy . If the diagonal elements of satisfy , then is isolated in the invariant set and the system state starting from necessarily converges to . {pf} Using Conditions (4) and (5), we can simplify the invariant set in Theorem 3.2. For convenience of expression, we assume that the target eigenstate is the -th eigenstate of , i.e., . If , we have and . Thus, (31) and (32) are equivalent to
and , respectively. Using Condition (4), one can obtain . Using Condition (5), we can obtain the relationship . Hence, all states in the invariant set are of the form , where “” represents an arbitrary eigenvalue of the initial state .

Since is a pure state, has one eigenvalue 1 and eigenvalues 0. Hence, the states in the invariant set have the form of or , where and are Hermitian matrices. For , all eigenvalues of are 0, which leads to , i.e., . For , has one eigenvalue 1 with multiplicity 1 and one eigenvalue 0 with multiplicity . In other words, has one eigenvalue 1 and eigenvalues 0. It is clear that the target eigenstate is isolated in .

For any initial pure state which satisfies , one has or . When , the conclusion naturally holds. When , we have . Hence, when and , the system (1) necessarily converges to the target state .  When the initial state satisfies , we have . That is to say, under the construction relation of in Theorem 3.3, the control law (19) cannot enable any state transfer. In this case, there exists a such that . Thus, we may use the following switching control to achieve convergence to the target state:

(33)

where , and is a small time duration.

When is small, the state is not in the invariant set . If we take as a new initial state, then Theorem 3.3 guarantees that the control law (19) can achieve convergence to the target state. Thus, we have the following conclusion. {thm} Consider the system (1) satisfying Conditions (4), (5) and with the switching control (33). Assume that the target eigenstate and the initial pure state satisfy . If the switching time satisfies and the diagonal elements of satisfy , then the system state starting from necessarily converges to . For general continuously differentiable control function (19), the construction relation in Theorems 3.3 and 3.3 ensures convergence to the target eigenstate. Based on the construction relation of , we propose two new methods including switching Lyapunov control and approximate bang-bang Lyapunov control to achieve rapidly convergent Lyapunov control.

4 Switching between Lyapunov control schemes

To speed up the control process, Ref. [22] proposed two design methods for quantum systems with power-type constraints and strength-type constraints such that in (18) takes the minimum value at each moment. For the case with strength-type constraints, the “optimal” control law is the following bang-bang Lyapunov control:

(34)

where is the maximum admissible strength of each control field, i.e., .

The bang-bang Lyapunov control in (34) makes in (18) satisfy , and can speed up completing some quantum control tasks. Especially, the state may move rapidly towards the target state at the early stages of the control [22]. However, convergence cannot be guaranteed since the control function is not continuously differentiable. Here, we first show that the bang-bang Lyapunov control may lead to a high-frequency oscillation phenomena [22, 23], which prevents effective state transfer towards the target state. Then, we propose two classes of switching Lyapunov control strategies to achieve rapidly convergent control, i.e., switching between the bang-bang Lyapunov control and the standard Lyapunov control, and switching between bang-bang Lyapunov control schemes with variable control strengths.

4.1 High-frequency oscillation in bang-bang Lyapunov control

In this subsection, we present a sufficient condition for two-level quantum systems that high-frequency oscillation phenomena occur in the bang-bang Lyapunov control [22], which can be used to determine switching conditions for the design of switching Lyapunov control. We first give the following definition.

{defn}

The control law (34) is said to have a high-frequency oscillation with an infinitesimal period at time if ,

for all . Since the control in (34) can take on only one of three constant values (0 and ) at any time, the high-frequency oscillation in Definition 4.1 means that the control always jumps between these values after an arbitrarily small time duration in the interval . Such a control field cannot be realized in practice.

Now consider the system model (1) in the case of two energy levels, and denote its internal Hamiltonian as and its control Hamiltonian as :

(35)

where .

We define the first eigenstate as the excited state, as the ground state, and

(36)

Let the excited state be the target state. According to Theorem 3.3 or Theorem 3.3, can be chosen as . Thus, . From (34), holds at any zero point of the bang-bang Lyapunov control. For convenience of analysis, in this paper we denote such moments as to differentiate them from the initial moment 0. Thus, we have

(37)

Equation (4.1) equals that . For the two-level system, we have the following result. {thm} Consider the two-level system

(38)

with the Hamiltonians (35) and the bang-bang Lyapunov control (34) (where ). Assume that the initial state of the system is an arbitrary pure state. We denote the state at any zero point of the control field (i.e., ) as . Then, a sufficient condition for the bang-bang Lyapunov control (34) to have a high-frequency oscillation with an infinitesimal period is

(39)
{pf}

Assume that from the state , a constant control acts on the system and lasts to time . Write the state at time as . We have

(40)

Denote , , and , then in (40) can be calculated as

(41)