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Quantum Ensemble Classification: A Sampling-Based Learning Control Approach Chunlin Chen, Member, IEEE, Daoyi Dong, Senior Member, IEEE, Bo Qi, Ian R. Petersen, Fellow, IEEE, and Herschel Rabitz

Abstract— Quantum ensemble classification (QEC) has significant applications in discrimination of atoms (or molecules), separation of isotopes, and quantum information extraction. However, quantum mechanics forbids deterministic discrimination among nonorthogonal states. The classification of inhomogeneous quantum ensembles is very challenging, since there exist variations in the parameters characterizing the members within different classes. In this paper, we recast QEC as a supervised quantum learning problem. A systematic classification methodology is presented by using a sampling-based learning control (SLC) approach for quantum discrimination. The classification task is accomplished via simultaneously steering members belonging to different classes to their corresponding target states (e.g., mutually orthogonal states). First, a new discrimination method is proposed for two similar quantum systems. Then, an SLC method is presented for QEC. Numerical results demonstrate the effectiveness of the proposed approach for the binary classification of two-level quantum ensembles and the multiclass classification of multilevel quantum ensembles. Index Terms— Inhomogeneous ensembles, quantum discrimination, quantum ensemble classification (QEC), sampling-based learning control (SLC).

I. I NTRODUCTION

O

PTIMAL discrimination [1], [2] and classification [3] of quantum states or quantum systems are a central topic in the quantum information technology [4]. This interdisciplinary research area involves pattern recognition in machine learning, Manuscript received March 11, 2014; revised September 23, 2015; accepted March 5, 2016. This work was supported in part by the National Natural Science Foundation of China under Grant 61004049, Grant 61273327, and Grant 61374092, in part by the Australian Research Council under Grant DP130101658 and Grant FL110100020, and in part by the U.S. National Science Foundation under Grant CHE-0718610. (Corresponding author: Daoyi Dong.) C. Chen is with the Department of Control and Systems Engineering, School of Management and Engineering, Nanjing University, Nanjing 210093, China, and also with the Department of Chemistry, Princeton University, Princeton, NJ 08544 USA (e-mail: [email protected]). D. Dong and I. R. Petersen are with the School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia (e-mail: [email protected]; [email protected]). B. Qi is with the Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China (e-mail: [email protected]). H. Rabitz is with the Department of Chemistry, Princeton University, Princeton, NJ 08544 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2016.2540719

quantum control in quantum technology [5]–[8], and very common laboratory problems of isolating similar species in chemical physics. In the existing research, the discrimination of two similar quantum systems (e.g., similar molecules) has been extensively investigated in the chemical and physical communities [9]–[20]. For example, multiobjective optimization and closed-loop learning control techniques have been employed for the dynamic discrimination of similar molecular systems [2], [9]. Results on mechanistic investigation and controllability analysis of quantum dynamic discrimination have been presented [11], [14]. The resource demands for perfect discrimination between the quantum operations have been theoretically investigated [17]. Many practical quantum systems exist in the form of quantum ensembles. A quantum ensemble consists of a large number of (e.g., 1023 ) single quantum systems (e.g., identical spin systems or molecules). Each single quantum system in a quantum ensemble is referred to as a member of the ensemble. Quantum ensembles have the wide applications in emerging quantum technology, including quantum computation [21], long-distance quantum communication [22], and magnetic resonance imaging [23]. In practical applications, the members of a quantum ensemble could show variations in the parameters that characterize the system dynamics [24], [25]. Such an ensemble is called an inhomogeneous quantum ensemble [26]. For example, the spins of an ensemble in nuclear magnetic resonance (NMR) experiments may have a large dispersion in the strength of the applied radio frequency field (inhomogeneity) and their natural frequencies (Larmor dispersion) [25]. In complex systems, there are intrinsic inhomogeneities of even chemically identical molecules due to different conformations and diverse environments [27]. The classification of inhomogeneous quantum ensembles is a significant issue and has great potential applications in the discrimination of atoms (or molecules), the separation of isotopes, and quantum information extraction. However, quantum mechanics forbids deterministic discrimination among the nonorthogonal states [1]. A useful idea is to first drive the members of a quantum ensemble from an initial state to different orthogonal states corresponding to different classes (e.g., eigenstates) before classifying them. It often becomes easy to discriminate (or separate) different orthogonal states. Usually, it is impractical to employ different

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control inputs for individual members of a quantum ensemble in physical experiments. Hence, it is important to develop new approaches for designing external control fields that can simultaneously steer the ensemble of inhomogeneous systems from an initial state to different target states when variations exist in their internal parameters. Some quantum control techniques, such as the multidimensional pseudospectral method [23], [28], the Lyapunov control methodology [29], the gradient-based optimal method [30]–[32], and the samplingbased learning control (SLC) approach [26], may provide inspiration for the solution to this problem. In this paper, we recast the quantum ensemble classification (QEC) task as a supervised quantum learning problem and present a systematic classification methodology by using an SLC method [26], [33]. Although the SLC approach has been presented in [26] for the control of inhomogeneous quantum ensembles, it needs to be adapted for the more challenging classification problem of quantum ensembles. The key ideas of the SLC approach are to construct a generalized system, develop an efficient algorithm to learn an optimal control, and verify the performance of the learned optimal control. In this paper, a gradient-based algorithm is employed in the SLC method to learn an optimal control strategy for the quantum classification problem. Other learning algorithms, such as stochastic search algorithms, can also be used in the SLC method. Lots of existing numerical results for the optimal control of molecular systems [34], the selective pulse design of spin systems [30]–[32], and the learning control of quantum ensembles [26] have shown that well-chosen gradient-based algorithms have the advantage of low computational cost and fast convergence (compared with the stochastic search algorithms), and the generalized system constructed using samples includes rich information on a quantum ensemble. Hence, an SLC method using a gradient-based algorithm is especially suitable for the challenging classification task of quantum ensembles. In this proposed method, we first learn an optimal control strategy to steer the members in a quantum ensemble belonging to different classes into their corresponding target states and, then, employ a physical readout process (e.g., projective measurement, fluorescence images of molecules [27], and Stern–Gerlach experiments for spin systems [4], [35]) to classify these classes. For example, the states of single molecules could be read out using a visualization technique, where the highly photostable chromophore dinaphtoquaterrylenebis (dicarboximide) (DNQDI) is embedded in thin polymer films in concentrations sufficiently low to allow individual DNQDI molecules to be spatially resolved in an epifluorescence confocal microscope (for details, see [27]). It is feasible to read out the intensity of the single-molecule fluorescence after they are excited with laser pulses. For a spin ensemble, when some members are driven to spin up and the others are driven to spin down (orthogonal to spin up), it is feasible to physically separate the two classes of members using Stern–Gerlach experiments [4]. We first develop a new approach for the discrimination of two similar quantum systems and the binary classification of quantum ensembles. Then, we apply the proposed approach to the multiclass

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classification of multilevel quantum ensembles. The main contributions of this paper can be summarized as follows. 1) The classification problem of inhomogeneous quantum ensembles is formally formulated as a supervised quantum learning problem. To the best of our knowledge, this paper is the first systematic investigation of the classification problem for inhomogeneous quantum ensembles. 2) The SLC method is adapted for the binary classification of inhomogeneous quantum ensembles. 3) The proposed SLC-based classification approach is extended to multiclass classification problems of quantum ensembles. 4) A collection of numerical results is presented, which shows the effectiveness of the proposed method for quantum discrimination and QEC. This paper is organized as follows. Section II formulates the learning problem for QEC. A control design method is presented in Section III for quantum discrimination of similar quantum systems. In Section IV, an SLC method is proposed for the binary classification of quantum ensembles, and the numerical results are demonstrated for an ensemble of two-level systems. The proposed approach is applied to the multiclass classification of multilevel quantum ensembles in Section V. Finally, the conclusion is drawn in Section VI. II. P ROBLEM F ORMULATION We focus on finite-dimensional closed quantum systems. The state evolution of a quantum system is described by the following Schrödinger equation (setting the reduced Plank constant h¯ = 1): ⎧ ⎨ d |ψ(t) = −i H (t)|ψ(t) (1) dt ⎩t ∈ [0, T ], |ψ(0) = |ψ 0

where |ψ(t) (quantum state) is a unit complex vector on the underlying √ Hilbert space, H (t) is the system Hamiltonian, and i = −1. The dynamics of the system is governed by a time-dependent Hamiltonian of the form H (t) = H0 + Hc (t) = H0 +

M

u m (t)Hm

(2)

m=1

is the free Hamiltonian of the system and where H0 M Hc (t) = m=1 u m (t)Hm is the time-dependent control Hamiltonian that represents the interaction of the system with the external fields u m (t) (real-valued and square-integrable functions). Hm is the Hermitian operators through which the controls couple to the system. The solution of (1) is given by |ψ(t) = U (t)|ψ0 , where the propagator U (t) satisfies the following equation (I is an identity matrix): ⎧ ⎨d U (t) = −i H (t)U (t) (3) dt ⎩t ∈ [0, T ], U (0) = I. In this paper, we consider the classification problem for a quantum ensemble of similar members with different Hamiltonians, which is referred to as QEC. QEC can be taken as a generalization of quantum dynamic discrimination

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and has similar characteristics to selective excitation problems in NMR [36]. Suppose that for an inhomogeneous quantum ensemble, we are given an unknown member belonging to a certain class; how well can we predict the class that the unknown member belongs to? In the classical machine learning, this problem can be solved using the typical supervised learning algorithms with a training set. However, this problem is much more difficult for quantum systems, because we cannot achieve deterministic discrimination for given quantum systems unless they lie in mutually orthogonal states. We have to drive the members from different classes to appropriate orthogonal states (e.g., eigenstates) before we can discriminate them with high accuracy. The SLC approach presented for the control of inhomogeneous quantum ensembles can be combined with supervised learning for QEC. We define the sampling set in the training step for the QEC problem as follows. Definition 1: A sampling set for the training step consists of N quantum systems (each of them labeled with an associated class) that are chosen from the quantum ensemble and the set is denoted by D N = {(H 1(t), y1 ), (H 2(t), y2 ), . . . , (H N (t), y N )}

(4)

where H n (t) (n = 1, 2, . . . , N) describes the nth quantum system in the sampling set and yn is the associated class that this quantum system belongs to. For ease of presentation, we first consider an inhomogeneous ensemble consisting of two classes of members (i.e., classes A and B) and propose an SLC approach for this binary QEC problem using a two-level quantum ensemble example. We further extend the proposed approach to the classification problem with multiclasses and multilevel quantum ensembles. For the binary QEC problem, the Hamiltonian of each member has the following form: ⎧ M ⎪ ⎪ ⎪ HεA,ε (t) = g0A (ε0 )H0 + guA (εu ) u m (t)Hm ⎪ ⎨ 0 u m=1 (5) M ⎪ ⎪ ⎪ B B B ⎪ u m (t)Hm ⎩ Hε0,εu (t) = g0 (ε0 )H0 + gu (εu ) m=1

where g0A (·) and guA (·) are known functions, while the inhomogeneity parameters ε0 and εu in the Hamiltonian HεA0,εu (t) for class A are characterized by the distribution functions d0A (ε0 ) and duA (εu ), respectively. We assume that the parameters ε0 and εu are time independent. A similar explanation is applicable to the Hamiltonian HεB0 ,εu (t) for class B. We assume that these parameter variations in ε0 and εu originate from different inhomogeneities (i.e., ε0 and εu are independent of each other). It is straightforward to extend the proposed method to the case where εu is dependent on ε0 if an approximate dependence relationship is known. Fig. 1 shows an example to describe the inhomogeneity of the quantum ensemble consisting of two classes of members with parameters in each class having Gaussian distribution. The function d0A (ε0 ) (d0B (ε0 )) characterizes the distribution of inhomogeneity in the free Hamiltonian for class A (B), and the function duA (εu ) (duB (εu )) characterizes the distribution

Fig. 1. Example of an inhomogeneous quantum ensemble consisting of two classes (A and B) with the inhomogeneity parameters having Gaussian distribution.

of inhomogeneity in the control Hamiltonian for class A (B). Fig. 1 shows a 2-D Gaussian distribution case regarding the parameters ε0 and εu . For a binary QEC task, the objective is to design a control strategy u(t) = {u m (t), m = 1, 2, . . . , M} to simultaneously stabilize the members in class A (with different values of ε0 and εu ) from an initial state |ψ0 to the same target state |ψtarget A and, at the same time, to stabilize the members in class B (with different values of ε0 and εu ) from |ψ0 to another target state |ψtarget B . In many practical situations, it is natural to assume that the inhomogeneity parameters in different classes have a specific distribution (e.g., Gaussian distribution [30]). These parameter distributions between different classes may overlap. Usually, there is not an effective method to distinguish between the members in the overlapping region between classes. Hence, there is a theoretical upper bound of classification precision (less than 100%). A binary QEC problem can be described by the following definition. Definition 2 (Binary QEC): A binary QEC task is to construct a binary quantum classifier to maximize the classification accuracy, where this binary quantum classifier consists of three steps. 1) Training Step: Learn an optimal control strategy u(t) with the sampling set D N = {(H 1(t), y1 ), (H 2(t), y2 ), . . . , (H N (t), y N )} where yn ∈ {A, B} ( A and B are symbolic constants) and H n (t) (n = 1, 2, . . . , N) is the time-dependent Hamiltonian describing the nth member in the sampling set. 2) Coherent Control Step: Apply the learned optimal control strategy u(t) to all the members of the quantum ensemble. 3) Classification Step: Predict the class y j of an unknown quantum system in the quantum ensemble using a corresponding physical readout process, where j = 1, 2, . . . , Ne and Ne is the number of members in the quantum ensemble. For example, a schematic of the classification process for a spin-1/2 quantum ensemble is shown in Fig. 2. As shown in Fig. 2, an ensemble of inhomogeneous spin-1/2 systems is prepared with an initial state. After learning using a training set from the quantum ensemble, we can find an optimal control



Fig. 2.

Schematic of the binary classification for a spin-1/2 quantum ensemble.

strategy to simultaneously drive all the members of class A to the target state (spin up) and all the members of class B to another target state (spin down). Then, we can use a Stern–Gerlach experiment to physically separate the two classes. From Definition 2, it is clear that the key task of the classification problem is to learn an optimal control strategy in the training step for the binary quantum classifier. The training performance is described by a performance function J (u) for each learned control strategy u = {u m (t), m = 1, 2, . . . , M}. The binary QEC problem can then be formulated as a maximization problem as follows:

max J (u) := max w A AV JεA0 ,εu (u) + w B AV JεB0 ,εu (u) u

u

s.t. t ∈ [0, T ] A

B

ψ ε0 ,εu (0) = ψε0 ,εu (0) = |ψ0 ⎧

d A ⎪ ⎪ ψε0 ,εu (t) = −i HεA0,εu (t) ψεA0 ,εu (t) ⎪ ⎪ ⎪ ⎨ dt M A (t) = g A (ε )H + g A (ε ) H u m (t)Hm 0 ε0 ,εu u u 0 0 ⎪ ⎪ ⎪ ⎪ m=1 ⎪

2 ⎩ A Jε0 ,εu (u) := ψεA0 ,εu (T ) ψtarget A ⎧

d B ⎪ ⎪ ψε0 ,εu (t) = −i HεB0,εu (t) ψεB0 ,εu (t) ⎪ ⎪ dt ⎪ ⎨ M B (t) = g B (ε )H + g B (ε ) H u m (t)Hm 0 0 u ε0 ,εu u 0 ⎪ ⎪ ⎪ ⎪ m=1 ⎪

2 ⎩ B Jε0 ,εu (u) := ψεB0 ,εu (T ) ψtarget B

such as multiobjective optimization [2] and closed-loop learning control [9]. The quantum discrimination problem can be taken as a special case of the binary QEC problem with the number of members in an ensemble Ne = 2. In this sense, a control design for the quantum discrimination is the foundation of QEC. The gradient-based optimal methods have been widely applied to a selective pulse design in the NMR [30]–[32]. In this section, we develop a gradient-based learning control method for the quantum discrimination of two similar quantum systems, and then, we extend the method to the control design for binary QEC in Section IV. A. Learning Control Design for Quantum Discrimination Suppose that two similar quantum systems to be discriminated (separated), a and b, have the following Hamiltonians: ⎧ M ⎪ ⎪ ⎪ H a (t) = g0 ε0a H0 + gu εua u m (t)Hm ⎪ ⎨ m=1 (7) M ⎪ b b ⎪ ⎪ b ⎪ u m (t)Hm ⎩ H (t) = g0 ε0 H0 + gu εu m=1

where ε0a , εua , ε0b , g0 (·) and gu (·).

(6)

where w A , w B ∈ [0, 1] are the weights assigned to classes A and B, respectively, satisfying w A +w B = 1. JεA0 ,εu (u) is a measure of classification accuracy for each member in class A regarding the target state |ψtarget A and AV[ JεA0 ,εu (u)] denotes the average value of JεA0 ,εu (u) over class A regarding all the members in this class. A similar expression holds for class B. It is clear that JεA0 ,εu and JεB0 ,εu implicitly depend on the control strategy u(t) through the Schrödinger equation. The performance J (u) represents the weighted accuracy of classification with an upper bound of 1. The larger J (u) is, the better the classification performance is. III. D ISCRIMINATION OF T WO S IMILAR Q UANTUM S YSTEMS Optimal dynamic discrimination between two similar quantum systems has been investigated using different techniques,

εub

and are predefined constants for functions a and b are prepared in the same initial state |ψ0 . The objective is to find an optimal control strategy u(t) (t ∈ [0, T ]) to drive the state of system a to the target state a b and the state of system b to the target state |ψtarget |ψtarget a b at the same time. Usually, we let ψtarget |ψtarget = 0, so that we can completely discriminate system a from system b. The control performance J (u) is redefined for the discrimination problem as J (u) := wa J a (u) + wb J b (u)

(8)

where wa , wb ∈ [0, 1] are the weights assigned to the associated systems, respectively, and a 2 J a (u) := ψ a (T ) ψtarget

b 2 . J b (u) := ψ b (T ) ψtarget (9) Here, we set wa = wb = 0.5 for the discrimination problem. In order to find an optimal control strategy u ∗ = {u ∗m (t), (t ∈ [0, T ]), m = 1, 2, . . . , M} for the discrimination problem, it is a good choice to follow the direction of the gradient of J (u) as an ascent direction. For ease of notation, we present the method for M = 1. We introduce a timelike


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variable s to characterize different control strategies u (s) (t). Then, a gradient in the control space can be defined as du (s) (10) = ∇ J (u (s)) ds where ∇ J (u) denotes the gradient of J (u) with respect to the control u. If u (s) is the solution of (10) starting from an arbitrary initial condition u (0) and u (s) ≥ u (0) , then the value of J (u) is increasing along u (s). This can be observed from the fact (d/ds)J (u (s)) = (du (s)/ds) ≥ 0. In other words, starting from an initial guess u 0 , we solve the following initial value problem: ⎧ (s) ⎪ ⎨ du = ∇ J (u (s) ) = w ∇ J a (u (s) ) + w ∇ J b (u (s) ) a b ds (11) ⎪ ⎩u (0) = u 0 in order to find a control strategy which maximizes J (u). This initial value problem can then be solved numerically by a forward Euler method over the s-domain, which is easy to implement and has low computational cost, that is u(s + s, t) = u(s, t) + s∇ J (u (s)).

(12)

As for practical applications, we present its iterative approximation version to find the optimal control u ∗ (t), where we use k as an index of iterations instead of the variable s and denote the control at iteration step k as u k (t). Equation (12) can be rewritten as u k+1 (t) = u k (t) + ηk ∇ J (u k )

∇ J (u k ) = wa ∇ J a (u k ) + wb ∇ J b (u k ).

(14)

In addition, we have the gradient of J a (u k ) with respect to the control u as follows (a detailed derivation can be found in [26]): a a a ψtarget G 1 (t) ψ0 (15) ∇ J a (u k ) = 2 ψ a (T ) ψtarget where (·) denotes the imaginary part of a complex number, G a1 (t) = Ua (T )Ua† (t)gu (εua )H1Ua (t), and the propagator Ua (t) satisfies d Ua (t) = −i H a (t)Ua (t), U (0) = I. dt A similar expression can also be derived for ∇ J b (u k ). When we generalize the gradient-based method to the case with M > 1, for each control u m (t) (m = 1, 2, . . . , M) of the control strategy u(t), we have a a a ψtarget G m (t) ψ0 ∇ J u km = 2wa ψ a (T ) ψtarget b b b b G (t) ψ0 ψ (16) + 2wb ψ (T ) ψ where

target

Remark 1: The numerical solution of a control design using Algorithm 1 is always difficult with a time varying continuous control strategy u(t). In the practical implementation, we usually divide the time duration [0, T ] equally into a number of time slices t and assume that the controls are constant within each time slice. Instead of t ∈ [0, T ], the time index is tq = q T /Q, where Q = T / t and q = 1, 2, . . . , Q.

(13)

where ηk is the updating step (learning rate) for the kth iteration and

target

Algorithm 1 Gradient-Based Iterative Learning for Quantum Discrimination 1: Set the index of iterations k = 0 2: Choose a set of arbitrary controls u k=0 (t) = {u 0m (t), m = 1, 2, . . . , M}, t ∈ [0, T ] 3: repeat (for each iterative process) 4: Compute the propagator Uak (t) and Ubk (t) for systems a and b, respectively, with the control strategy u k (t) 5: repeat (for each control u m (t) (m = 1, 2, . . . , M) of the control vector u k (t)) k (t) := ∇ J (u k ) and compute ∇ J (u k ) using 6: δm m m equation (16) k k 7: u k+1 m (t) = u m (t) + ηk δm (t) 8: until m = M 9: k =k +1 10: until the learning process ends 11: The optimal control strategy u ∗ (t) = {u ∗m (t)} = {u km (t)}, m = 1, 2, . . . , M

m

G am (t) = Ua (T )Ua† (t)gu εua Hm Ua (t) G bm (t) = Ub (T )Ub† (t)gu εub Hm Ub (t).

A gradient-based iterative learning algorithm for the discrimination of quantum systems is shown in Algorithm 1.

B. Numerical Examples To demonstrate this learning control method for the discrimination of two similar quantum systems, we consider two-level (e.g., spin-1/2) systems. We denote the Pauli matrices σ = (σx , σ y , σz ) as follows: 0 1 0 −i 1 0 , σy = , σz = . (17) σx = 1 0 i 0 0 −1 For a two-level quantum system, we may assume the free Hamiltonian H0 = (1/2)σz . Its two eigenstates are denoted by |0 (e.g., spin up) and |1 (e.g., spin down). To control a two-level quantum system, we use the control Hamiltonian of Hu = (1/2)u 1 (t)σx + (1/2)u 2 (t)σ y . Hence 1 1 1 σz + u 1 (t)σx + u 2 (t)σ y . (18) 2 2 2 For two similar two-level systems, the Hamiltonian of each system can be described as H (t) = H0 + Hu (t) =

H (t) = g0 (ε0 )H0 + gu (εu )Hu (t) 1 1 = g0 (ε0 )σz + gu (εu )(u 1 (t)σx + u 2 (t)σ y ). (19) 2 2 We assume g0 (ε0 ) = ε0 and gu (εu ) = εu . The state of the two quantum systems can be represented in the eigenbasis of H0 by |ψ(t) = c0 (t)|0+c1 (t)|1. Denote C(t) = (c0 (t), c1 (t))T , where c0 (t) and c1 (t) are complex numbers, and x T represents the transpose of x. We have εu f (u) c˙0 (t) 0.5ε0i c0 (t) = (20) −0.5ε0i c˙1 (t) −εu f ∗ (u) c1 (t)



Fig. 4. Evolution of the states of system a [(ε0a , εua ) = (0.9, 0.9)] and system b [(ε0b , εub ) = (1.1, 1.1)] regarding their populations (|c0a (t)|2 and |c0b (t)|2 ) at the state |0, respectively.

Fig. 3. Learning performance of discrimination between system a [(ε0a , εua ) = (0.9, 0.9)] and system b [(ε0b , εub ) = (1.1, 1.1)]. (a) Evolution of performance functions J a (u) and J b (u). (b) Learned optimal control strategy u(t) where a.u. have been used.

where f (u) = u 2 (t) − 0.5i u 1 (t), (ε0 , εu ) = (ε0a , εua ) for system a and (ε0 , εu ) = (ε0b , εub ) for system b. Define the performance function as 1 1 a J (u) + J b (u) 2 2 a 2 1 b

2 1 + C (T ) C b = C a (T ) Ctarget target . 2 2

J (u) =

(21)

The task is to find a control u(t) to maximize the performance function in (21). For a given small threshold ε > 0, if |J (u k+1 ) − J (u k )| < ε for uninterrupted n e steps, a suitable control law for the problem has then been found. In this paper, we set ε = 10−4 , n e = 100, and use atomic units (a.u.) by setting h = 1 in all numerical experiments. Now, we employ Algorithm 1 to find the optimal control strategy u ∗ (t) = {u ∗m (t), m = 1, 2} and, then, apply the optimal control strategy for discriminating system a from system b. The parameter settings are listed as follows. The initial state C0 = (1, 0)T , i.e., |ψ0 = |0, and the target state a a for system a Ctarget = (1, 0)T , i.e., |ψtarget = |0 and the b b target state for system b Ctarget = (0, 1)T , i.e., |ψtarget = |1. The ending time T = 5 (in a.u.) and the total time duration [0, T ] is equally discretized into Q = 500 time slices with each time slice t = (tq −tq−1 )|q=1,2,...,Q = T /Q = 0.01; the learning rate ηk = 0.2, and the control strategy is initialized as u k=0 (t) = {u 01 (t) = sin t, u 02 (t) = sin t}. In the first example, two similar systems a and b are characterized with parameters (ε0a , εua ) = (0.9, 0.9) and (ε0b , εub ) = (1.1, 1.1), respectively. The numerical results are shown in Figs. 3–5. As shown in Fig. 3(a), the learning process converges very quickly, and the performance function J (u)

Fig. 5. Demonstration of the state transition trajectories of system a [(ε0a , εua ) = (0.9, 0.9)] and system b [(ε0b , εub ) = (1.1, 1.1)] on the Bloch sphere. (a) Initial states for systems a and b; (b) Target state for system a; (c) Target state for system b.

converges to 0.999 after about 2000 steps of iterative learning with an optimized control strategy u(t) = {u 1 (t), u 2 (t)} in Fig. 3(b). The learning process takes 3 min using a laptop with CPU at 2.5 GHz, Windows 8, MATLAB 7.10.0. Then, we apply the learned optimal control strategy to systems a and b. The evolution of their states can be clearly demonstrated regarding their populations at the state |0, as shown in Fig. 4. At time t = T = 5, |c0a (T )|2 = 1.0000, and |c0b (T )|2 = 0.0000, which indicates that, after the coherent control step, we can discriminate system a from system b using a projective measurement, and the success probability is almost 100%. A 3-D visualized demonstration is further shown in Fig. 5 for the state transition trajectories of systems a and b on the Bloch sphere. For a two-level system, its state can also be represented using a Bloch vector r = (x, y, z), where x = tr{|ψψ|σx }, y = tr{|ψψ|σ y }, z = tr{|ψψ|σz }, and tr(·) is the trace operator. As shown in Fig. 5, using the same learned control strategy, the state trajectories of systems a and b are successfully driven from the same initial state |ψ0 = |0 [i.e., r0 = (0, 0, 1)] to different target states a a b = |0 [i.e., rtarget = (0, 0, 1)] and |ψtarget = |1 of |ψtarget b [i.e., rtarget = (0, 0, −1)], respectively. In the second example, two similar systems a and b are characterized with parameters (ε0a , εua ) = (0.95, 0.95) and (ε0b , εub ) = (1.05, 1.05). The numerical results are shown


7

amplitudes of controls in Fig. 6(b) are larger than those in Fig. 3(b)]. This phenomenon is comprehensively tested through further numerical experiments with varied parameters. All of the results show that the gradient-based iterative learning method is successful for the discrimination of similar quantum systems and also support the previous findings that optimal dynamic discrimination is feasible for many similar quantum systems in physics and chemistry communities [2], [9]–[15]. IV. Q UANTUM E NSEMBLE C LASSIFICATION VIA SLC Binary classification is to classify the members of a given set of objects into two classes on the basis of whether they have certain properties or not. As introduced in Definition 2, for a binary QEC problem, we have to learn from a sampling set as defined in Definition 1 and find out an optimal control strategy for all the members in the quantum ensemble. In this section, we combine an SLC approach into the quantum discrimination method introduced above to solve the QEC problem [i.e., the maximization problem formulated as (6)]. Fig. 6. Learning performance of discrimination between system a [(ε0a , εua ) = (0.95, 0.95)] and system b [(ε0b , εub ) = (1.05, 1.05)]. (a) Evolution of performance functions J a (u) and J b (u). (b) Learned optimal control strategy u(t) in a.u.

Fig. 7. Evolution of the states of system a [(ε0a , εua ) = (0.95, 0.95)] and system b [(ε0b , εub ) = (1.05, 1.05)] regarding the population at the state |0.

in Figs. 6 and 7. Similar to the first example, we can also successfully learn an optimal control strategy [Fig. 6(b)] to drive systems a and b to different target eigenstates from the same initial state. The evolution of their states is shown in Fig. 7 regarding their populations at the state |0. By comparing the second example with the first one, it is clear that the difference lies in the similarity between Hamiltonians. For the second example, the increase in the similarity of Hamiltonians makes it more difficult to discriminate system a from system b. More learning steps (about 15 000 steps) are needed for the second example than the first one (around 2000 steps). A larger control strength is also needed for the second example than that in the first one [i.e., the

A. SLC for Quantum Ensemble Classification The first key issue for QEC is how to obtain the sampling set for the training step. In general, there are two ways to construct a sampling set for QEC: 1) the data are provided initially and the sampling set can be constructed directly, but we do not know the distribution of parameters that characterize the members belonging to different classes and 2) no initial sampling data are provided, but we know the distribution of parameters and we can choose samples using the distribution information. The first way is very common in classical machine learning problems, while the second way is more suitable for the classification of quantum systems. In the quantum domain, it is difficult to obtain a specific description for a single system in a quantum ensemble, while we can characterize an ensemble of similar systems with a distribution of parameters (e.g., Gaussian distribution, Boltzmann distribution, and uniform distribution). According to the distribution of parameters for a quantum ensemble, we can choose sample members to construct the sampling set for the classification task. To find an optimal control strategy using the sampling set, we adopt a modified SLC approach, whose key idea originated in [26] and [33] as a general framework for the optimal control design of inhomogeneous quantum ensembles and the robust control design of quantum systems with uncertainties. In the SLC approach, a generalized system is constructed by sampling members from the inhomogeneous ensemble. In this paper, we adopt the key idea from SLC and solve the supervised quantum learning problem of QEC via constructing a generalized system using the sampling set. Suppose we have obtained a sampling set D N = {(H n (t), yn )} (n = 1, 2, . . . , N) for the binary QEC problem, where yn ∈ {A, B} and H n (t) is the time-dependent Hamiltonian that describes the nth member of the quantum ensemble. Now, we split D N into two subsets according to the value of yn (n = p or q) and rewrite the sampling set for



training as follows: DN = DNA ∪ DNB , N = N A + NB D N A = HεAp ,ε p (t), y p = A , p = 1, 2, . . . , N A u 0 D N B = HεBq ,εq (t), yq = B , q = N A +1, N A + 2, . . . , N 0

u

p

p

M

(22)

where HεAp ,ε p (t) = g0A (ε0 )H0 + guA (εu ) m=1 u m (t)Hm and 0 u q q M HεBq ,εq (t) = g0B (ε0 )H0 + guB (εu ) m=1 u m (t)Hm . u

0

Using the sampling set (22), we can construct a generalized system as follows: A

⎛ ⎞ ψ 1 (t) ε0 ,εu1 ⎜ ⎟ .. ⎜ ⎟ . ⎜

⎟ ⎜ A ⎟ ψ (t) ⎟ d ⎜ ⎜ ε0N A ,εuN A

⎟ ⎜ ⎟ dt ⎜ ψ BN A +1 N A +1 (t) ⎟ ⎜ ε0 ,εu ⎟ ⎜ ⎟ .. ⎜ ⎟ . ⎝

⎠ B ψ N (t) ε ,ε N

⎛0 u ⎞ HεA1,ε1 (t) ψεA1 ,ε1 (t) 0 u 0 u ⎜ ⎟ .. ⎜ ⎟ ⎜ ⎟ .

⎜ ⎟ ⎜ ⎟ H AN A N A (t) ψ AN A N A (t) ⎜ ⎟ ε0 ,εu ε0 ,εu ⎟. = −i ⎜ (23)

⎜HB ⎟ B ⎜ N A +1 N A +1 (t) ψ N A +1 N A +1 (t) ⎟ ε0 ,εu ⎜ ε0 , εu ⎟ ⎜ ⎟ .. ⎜ ⎟ . ⎝ ⎠

B B H N (t) ψ N (t) ε0 ,εuN

ε0 ,εuN

The performance function for this generalized system is defined by JN (u) := w A J A + w B J B

(24)

where J

A

NA NA A

1 1 A ψ p p (T ) ψtarget A 2 = Jε p ,ε p (u) = ε0 ,εu 0 u NA NA p=1

JB = =

1 NB 1 NB

p=1

N q=N A +1

JεBq ,εq (u) u

0

N B ψ q q=N A +1

q

ε0 ,εu

2 (T ) ψtarget B .

(25)

The task of the training step is to find a control strategy that maximizes the performance function defined in (24). From (16), (24), and (25), we have ∇ JN u km =

NA

2wa

ψεAp ,ε p (T ) ψtarget A ψtarget A G Ap,m (t)|ψ0 0 u NA p=1

+

2wb NB

N

B

ψεBq ,εq (T ) ψtarget B ψtarget B G q,m (t) ψ0

q=N A +1

0

u

(26)

Algorithm 2 SLC for Binary QEC 1: Set the index of iterations k = 0 2: Choose a set of arbitrary controls u k=0 (t) = {u 0m (t), m = 1, 2, . . . , M}, t ∈ [0, T ] 3: repeat (for each iterative process) 4: repeat (for each member in training subset D N A , p = 1, 2, . . . , N A ) 5: Compute the propagator Uεkp ,ε p (t) with the control 6: 7: 8:

9: 10: 11: 12: 13: 14: 15: 16:

0

u

0

u

strategy u k (t) until p = N A repeat (for each member in training subset D N B , q = N A + 1, N A + 2, . . . , N) Compute the propagator Uεkq ,εq (t) with the control

strategy u k (t) until q = N repeat (for each control u m (t) (m = 1, 2, . . . , M) of the control vector u k (t)) k (t) := ∇ J (u k ) and compute ∇ J (u k ) using δm N m N m equation (26) k k u k+1 m (t) = u m (t) + ηk δm (t) until m = M k =k+1 until the learning process ends The optimal control strategy u ∗ (t) = {u ∗m (t)} = {u km (t)}, m = 1, 2, . . . , M

where

p G Ap,m (t) = Uε p ,εup (T )Uε†p ,ε p (t)guA εu Hm Uε p ,εup (t) 0 0 0 u q B G q,m (t) = Uεq ,εuq (T )Uε†q ,εq (t)guB εu Hm Uεq ,εuq (t). 0

0

u

0

Then, we design the SLC algorithm (Algorithm 2) for binary QEC using the gradient-based method to approximate an optimal control strategy u ∗ = {u ∗m (t)}. Remark 2: Both Algorithms 1 and 2 are developed using a gradient-based method. Although the gradient-based algorithm has been widely investigated in NMR and molecular chemistry, existing results mainly focus on the quantum optimal control problems. In this paper, we adapt the gradient-based method for a QEC problem aiming at providing excellent performance in classification precision as well as robustness. The algorithm convergence is closely related to quantum control problems. According to the theory of quantum control landscapes [37], [38], gradient-based algorithms are effective for many practical quantum control problems [39], [40]. For the classification problem considered in this paper, the numerical results show that the gradient-based method works well. For those complex quantum control or classification problems (e.g., control of multilevel open quantum systems), stochastic learning techniques (e.g., genetic algorithms) may be required to achieve convergence. Moreover, the initial control guess may have an impact on the convergence and convergence speed. However, a large number of numerical results show that it is easy to find an initial guess to achieve convergence. Different initial guesses that can lead to convergence also achieve almost the same classification accuracy


9

although corresponding optimal control fields are usually different. Remark 3: As for the specific techniques of choosing samples (N members of the ensemble), we generally choose them according to the functions g0A (·), guA (·), g0B (·), and guB (·), and the distribution of the inhomogeneity parameters ε0 and εu . It is clear that the basic motivation of the proposed SLC approach is to design a control law using only a few sampling members instead of the whole ensemble (consisting of a large number of members). Therefore, it is necessary to choose the samples that are representative for the quantum ensemble. In general, if we know the distribution of the parameter dispersion, it is practical and convenient to choose artificial members for the construction of the generalized system. In numerical examples, we will demonstrate the detailed method for choosing samples, and more related topics have also been discussed in [26] and [33]. In addition, overlapped distributions of the inhomogeneity parameters for class A and class B may lead to the problem of overlapped classification, which is a challenging task even for the classical classification problems [41]. In Section IV-B, we demonstrate the classification performance for both the cases with and without class overlapping. B. Numerical Examples We consider two-level quantum systems. For two similar classes of members in an inhomogeneous quantum ensemble, the Hamiltonians can be described as 1 1 HεA0,εu (t) = g0A (ε0 )σz + guA (εu )(u 1 (t)σx + u 2 (t)σ y ) 2 2 1 1 HεB0,εu (t) = g0B (ε0 )σz + guB (εu )(u 1 (t)σx + u 2 (t)σ y ). 2 2 (27) Assume g0A (ε0 ) = ε0 with distribution d0A (ε0 ), guA (εu ) = εu with distribution duA (εu ), g0B (ε0 ) = ε0 with distribution d0B (ε0 ), and guB (εu ) = εu with distribution duB (εu ). Suppose the distributions of ε0 and εu for class A are d0A (ε0 ) = (ε0 − μ0A /σ0A ) and duA (εu ) = (εu − μuA /σuA ), √ x respectively, where (x) = −∞ (1/ 2π)exp(−(1/2)ν 2 )dν is the distribution function of the standard normal distribution. We may choose some equally spaced samples in the ε0 − εu space. For example, we may choose the intervals of [μ0A − 3σ0A , μ0A + 3σ0A ] and [μuA − 3σuA , μuA + 3σuA ], and divide them into NεA0 +1 and NεAu +1 subintervals, respectively, where NεA0 and NεAu are usually positive odd numbers. Then, the number of samples for class A is N A = NεA0 NεAu , where p p ε0 and εu can be chosen from the combination of p0

pu

(ε0 and εu ) as follows: ⎧ A 0 0 ⎪ 0 = 1, 2, . . . , N A 0 ⎪ε p ∈ ε p = μ A −3σ A + (2 p −1)3σ , p ⎨ ε0 0 0 0 0 NεA 0 u A u ⎪ (2 p −1)3σu p p u = 1, 2, . . . , N A . ⎪ ⎩εu ∈ εu = μuA −3σuA + , p εu NA εu

(28) In practical applications, the numbers of NεA0 and NεAu can be chosen by experience or be tried through numerical computation. As long as the generalized system can model the quantum

TABLE I PARAMETERS C HARACTERIZING THE I NHOMOGENEITY D ISTRIBUTION

ensemble and is effective to find the optimal control strategy, we prefer smaller numbers NεA0 and NεAu to speed up the training process and simplify the generalized system. A similar expression to (28) defines the samples for class B. We use the performance function as defined in (24) with w A = w B = 0.5. Now, we use Algorithm 2 to find the optimal control strategy. The parameter settings are listed as follows. w A = w B = 0.5, the initial state for each member of the quantum ensemble C0 = (1, 0)T , i.e., |ψ0 = |0, and the target state for members belonging to class A Ctarget A = (1, 0)T , i.e., |ψtarget A = |0. The target state for elements belonging to class B Ctarget B = (0, 1)T , i.e., |ψtarget B = |1. The ending time T = 8 (in a.u.) and the total time duration [0, T ] is equally discretized into Q = 800 time slices with each time slice t = (tq − tq−1 )|q=1,2,...,Q = T /Q = 0.01. NεA0 = NεAu = NεB0 = NεBu = 5. The learning rate ηk = 0.2. The control strategy is initialized as u k=0 (t) = {u 01 (t) = sin t, u 02 (t) = sin t}. In the training step, we use J (u) as the performance function that represents the measure of weighted accuracy for QEC. After we apply the optimized control u ∗ to the inhomogeneous quantum ensemble, we use fidelity to characterize how well every member is classified. The fidelity between the final state |ψεA0 ,εu (T ) of a member belonging to class A and the target state |ψtarget A is defined as follows [4]:

F ψεA0 ,εu (T ) , ψtarget A = ψεA0 ,εu (T ) ψtarget A . (29) A similar representation can be defined for the final state |ψεB0 ,εu (T ) of a member belonging to class B and the target state |ψtarget B . It is clear that the accuracy of QEC can be calculated with 1 (AV[ J A ] + AV[ J B ]) 2

1 = AV F 2 ψεA0 ,εu (T ) , |ψtarget A 2

+ AV F 2 ψεB0 ,εu (T ) , |ψtarget B .

ζ = J (u) =

(30)

We demonstrate and analyze several groups of numerical examples for cases 1–3 with class overlapping, whose parameters characterizing the inhomogeneity are listed, as shown in Table I. It can be calculated that the theoretical upper bound on classification precision for case 2 is 99.73% and the upper bounds on classification precision for cases 1 and 3 are very close to 100%. The learning control performance for case 1 is shown in Figs. 8 and 9. As shown in Fig. 8, the learning algorithm



Fig. 8. Learning performance of binary QEC for case 1. (a) Evolution of performance function J A and J B . (b) Learned optimal control for QEC in a.u.

Fig. 10. Learning performance of binary QEC in case 2. (a) Evolution of performance function J A and J B . (b) Learned optimal control for QEC in a.u.

Fig. 9. Control performance of binary QEC for case 1. (a) Demonstration of the state transition of all members on the Bloch sphere using the same learned control. (b) Trajectories of state transition for members in class A from |ψ0 = |0 to |ψtarget A = |0. (c) Trajectories of state transition for members in class B from |ψ0 = |0 to |ψtarget B = |1. (d) Control performance regarding fidelity.

Fig. 11. Control performance of binary QEC for case 2. (a) Demonstration of the state transition of all members on the Bloch sphere using the same learned control. (b) Trajectories of state transition for members in class A from |ψ0 = |0 to |ψtarget A = |0. (c) Trajectories of state transition for members in class B from |ψ0 = |0 to |ψtarget B = |1. (d) Control performance regarding fidelity.

converges quickly after about 8000 iterations and finds an optimized control for the coherent control step of binary QEC. Applying the learned control to 300 randomly selected testing samples (150 for class A and 150 for class B), the control performance is shown in Fig. 9. The mean value of fidelity for the testing of class A is 0.9976 and for class B is 0.9985. With an additional 104 testing samples for both classes A and B, the classification accuracy in case 1 is estimated as ζ = 99.62%. Compared with case 1, we study the effect of larger dispersion on the Hamiltonian in case 2 with larger deviation. The learning control performance for case 2 is shown

in Figs. 10 and 11. As shown in Fig. 10(a), many more learning steps are needed to find a satisfactory control, and the learned optimal control is shown in Fig. 10(b). Due to a larger dispersion, the control performance is a little worse than that in case 1. The mean value of fidelity for the testing of class A is 0.9821 and for class B is 0.9905. With an additional 104 testing samples for both classes A and B, the classification accuracy in case 2 is estimated as ζ = 97.35%. In case 3, we further study the effect of a larger difference of Hamiltonian between classes A and B. The difference between


11

Fig. 13. Control performance of binary QEC in case 3. (a) Demonstration of the state transition of all members on the Bloch sphere using the same learned control. (b) Trajectories of state transition for members in class A from |ψ0 = |0 to |ψtarget A = |0. (c) Trajectories of state transition for members in class B from |ψ0 = |0 to |ψtarget B = |1. (d) Control performance regarding fidelity. Fig. 12. Learning performance of binary QEC in case 3. (a) Evolution of performance function J A and J B . (b) Learned optimal control in a.u. for QEC.

the means of the distribution in case 1 |μ0A − μ0B | = 0.3 and |μuA − μuB | = 0.3, while, in case 3, |μ0A − μ0B | = 0.4 and |μuA − μuB | = 0.4. The learning control performance for case 3 is shown in Figs. 12 and 13. As shown in Fig. 12(a), much fewer learning steps are needed to find a satisfactory control, and the learned optimal control is shown in Fig. 12(b). Due to larger difference between classes A and B, the control performance is better than that in case 1. The mean value of fidelity for testing of class A is 0.9992 and for class B is 0.9996. With an additional 104 testing samples for both classes A and B, the classification accuracy in case 3 is estimated as ζ = 99.88%. From the numerical results in the above three cases, we have the following conclusions. First, the SLC approach is effective for the binary QEC problem and can achieve a high level of classification accuracy. Second, the classification performance is deteriorated with larger dispersion on the Hamiltonian and smaller difference of Hamiltonians between classes A and B. More numerical experiments are carried out, and similar findings are obtained. The tradeoff relationship between performance and dispersion (or relevant parameters) has been widely investigated in many contexts, such as NMR [42]. For ease of demonstration, we may use the same deviation σ for all the distributions in a certain case. Define the dispersion on the Hamiltonian as Disp = 3σ and define the difference of the Hamiltonian between classes A and B as Diff = 1/2(|μ0A − μ0B | + |μuA − μuB |). The collective results are shown in Fig. 14 with a 3-D Pareto front. For a detailed discussion about Pareto front [43], please refer to [2].

Fig. 14.

3-D Pareto front of binary QEC.

Remark 4: An interesting inverse problem is to find a set of control fields that can be applied to a quantum ensemble for identifying the different distributions of inhomogeneous parameters in different classes. This problem can be formulated as a parameter identification problem that is worth investigating in the future research. In this paper, the classification problem under consideration involves class overlapping, which is more challenging than that without class overlapping. The proposed approach can be easily applied to the QEC problem without class overlapping and can obtain even better performance. For example, in case 1, its counterpart without class overlapping can be characterized with truncated normal distribution. Let the probability density function of a truncated normal distribution be x−μ 1 σφ σ p(x, μ, σ, l, r ) = l−μ r−μ − σ σ where φ(x) is the probability density function of the standard normal distribution. The probability density functions for the



TABLE II C OMPARISON B ETWEEN THE C LASSIFICATION A CCURACY ζ W ITH C LASS OVERLAPPING AND THE C LASSIFICATION A CCURACY ζ W ITHOUT C LASS OVERLAPPING

We consider an inhomogeneous quantum ensemble of three-level -type atomic systems [26]. The evolving state |ψ(t) of the -type system can be expanded in terms of the eigenstates as follows: |ψ(t) = c1 (t)|1 + c2 (t)|2 + c3 (t)|3

truncated normal distributions are set as follows: p0A = p ε0 , μ0A , σ0A , −∞, μ0 puA = p εu , μuA , σuA , −∞, μu p0B = p ε0 , μ0B , σ0B , μ0 , +∞ puB = p εu , μuB , σuB , μu , +∞ where μ0 = (μ0A + μ0B /2) and μu = (μuA + μuB /2). Using the same approach and parameter settings in case 1, we can achieve the classification accuracy ζ = 99.66%. Similar comparison is investigated in cases 2 and 3. These results are shown in Table II. V. M ULTICLASS C LASSIFICATION OF M ULTILEVEL Q UANTUM E NSEMBLES In machine learning, multiclass classification involves classifying instances into more than two classes. Some classification algorithms naturally permit the use of more than two classes [44]. A useful strategy is the one-versus-all strategy, where a single classifier is trained per class to distinguish that class from all other classes [45]. The SLC-based QEC approach proposed for QEC in Section IV can be extended to the multiclass classification of multilevel quantum ensembles using the one-versus-all strategy. For example, suppose an inhomogeneous quantum ensemble consists of three classes of members (i.e., classes A, B, and C). First, by applying the binary QEC approach introduced in Section IV, we can classify them into two classes (one for members belonging to class A and the other for all the members belonging to classes B and C). By a proper physical operation, we can separate the members in class A from classes B and C. Then, we use the binary QEC approach again to classify the members belonging to class B from the members belonging to class C. According to the numerical results demonstrated in Section IV, good classification performance is also expected for these multiclass classification problems. However, the one-versus-all strategy may cause additional cost, since the process involves multiple times of binary classification and multiple learning control procedures. In this section, we use a different strategy from the one-versus-all strategy to extend the proposed SLC-based classification method to the multiclass classification of multilevel quantum ensembles. In this strategy, only one quantum coherent control procedure is needed to implement the multiclass QEC.

(31)

where |1, |2, and |3 are the basis states of the lower, middle, and upper atomic states, respectively, corresponding to the free Hamiltonian ⎛ ⎞ 1.5 0 0 H0 = ⎝ 0 1 0⎠. (32) 0 0 0 Denote C(t) = (c1 (t), c2 (t), c3 (t))T . To control such a three-level system, we use the control Hamiltonian of Hu = u 1 (t)H1 + u 2 (t)H2 , where ⎛ ⎛ ⎞ ⎞ 0 0 0 0 0 1 H1 = ⎝ 0 0 1 ⎠, H2 = ⎝ 0 0 0 ⎠. (33) 0 1 0 1 0 0 Similarly, we describe the inhomogeneous three-level quantum ensemble with the Hamiltonian of each member as Hε0 ,εu (t) = ε0 H0 + εu ((u 1 (t)H1 + u 2 (t)H2 ).

(34)

Suppose that the inhomogeneous quantum ensemble consists of three classes of members labeled with classes A, B, and C, respectively. For this multiclass QEC problem, we first use the same control field to drive the members belonging to classes A, B, and C from an initial state |ψ0 to three different target eigenstates (|1, |2, and |3), respectively, so that we can classify them with an additional physical operation (e.g., projective measurement). We can modify Algorithm 2 into its multiclass version and, then, apply it to the three-level inhomogeneous quantum ensemble for finding an optimal control strategy u ∗ (t) = {u ∗m (t), m = 1, 2} to maximize the performance function 1 J (u) = (AV[ J A ] + AV[ J B ] + AV[ J C ]) 3

1 = AV F 2 ψεA0 ,εu (T ) , |1 3

+ AV F 2 ψεB0 ,εu (T ) , |2

+ AV F 2 ψ C (T ) , |3 . (35) ε0 ,εu

The parameter are √ settings √ √ listed as follows. The initial state C0 = (1/ 3, 1/ 3, 1/ 3). The three target eigenstates for classes A, B, and C are Ctarget A = (1, 0, 0) (i.e., |1), Ctarget B = (0, 1, 0) (i.e., |2), and Ctarget A = (0, 0, 1) (i.e., |3), respectively. The ending time T = 10 (in a.u.) and the total time duration [0, T ] is equally discretized into Q = 1000 time slices. The learning rate is ηk = 0.2. The control is initialized as u 0 (t) = {u 01 (t) = sint, u 02 (t) = sint}. The parameters ε0 and εu characterize the inhomogeniety of the quantum ensemble, and they have different normal distributions that are described with the distribution functions d0 (ε0 ) = (ε0 − μ0 /σ0 ) and du (εu ) = (εu − μu /σu ), where for class A (μ0A = 1, 3σ0A = 0.05) and (μuA = 0.8, 3σuA = 0.05), for class B (μ0B = 0.8, 3σ0B = 0.05),


13

the classification accuracy ζ = 98.80%, which verifies the effectiveness of the proposed SLC-based approach for QEC. VI. C ONCLUSION

Fig. 15. Learning control performance for multiclass QEC. (a) Evolution of performance function J (u) with its three component values J A (u), J B (u), and J C (u). (b) Learned optimal control for the multiclass QEC problem in a.u. (c) Control performance regarding fidelity.

and (μuB = 1, 3σuB = 0.05), and for class C (μC 0 = 1.2, C = 0.05). To construct 3σ0C = 0.05) and (μC = 1.2, 3σ u u the generalized system for learning the optimal control, the sampling method as described in (28) is adopted with setting NεA0 = NεAu = NεB0 = NεBu = NεC0 = NεCu = 3. The learning performance is shown in Fig. 15. The evolution of the performance function J (u) is shown in Fig. 15(a) with its three component values J A (u), J B (u), and J C (u). The results demonstrate that the SLC-based classification method is effective for the multiclass QEC of multilevel quantum ensembles and has good scalability. The learned optimized control for the coherent control step of QEC is shown in Fig. 15(b). As shown in Fig. 15(c), 300 randomly selected samples for each class of members are tested, and all of them are controlled to their corresponding target eigenstates, respectively, with high fidelity. The mean value of fidelity for testing of class A is 0.9897, for class B is 0.9953, and for class C is 0.9976. With an additional 104 testing samples for each class, we have

In this paper, we present a systematic classification approach for inhomogeneous quantum ensembles by combining an SLC approach with quantum discrimination. The classification process is accomplished via simultaneously steering members belonging to different classes to different corresponding target states (e.g., eigenstates). A new discrimination method is first presented for quantum systems with similar Hamiltonians. Then, an SLC method is proposed for QEC. Numerical experiments are carried out to test the performance of the proposed approach for the binary classification of two-level quantum ensembles and the multiclass classification of three-level -type quantum ensembles. In these numerical examples, we obtain the samples in the training step by assuming that each inhomogeneity parameter has a uniform distribution. However, it is straightforward to employ different distribution assumptions to prepare samples for learning. Numerical results showed that even if the inhomogeneity parameters have other distributions, sampling members according to a relevant uniform distribution can achieve better performance than sampling them according to other distributions (e.g., a truncated Gaussian distribution) [26]. In addition, it is straightforward to extend the SLC method with the use of stochastic searching algorithms (such as genetic algorithm and differential evolution algorithm) instead of gradient-type algorithms. Moreover, we have used a.u. for the control amplitudes and control time, and did not consider specific parameter values for particular quantum systems. The proposed method can be applied to real systems, such as inhomogeneous spin ensembles, inhomogeneous atomic ensembles, and ensembles of similar molecules. When the control amplitudes have realistic limitations for these real systems, the control time T may need to be adjusted to achieve a high level of classification accuracy. Although these limitations may cause slight variations in the performance, the proposed method is expected to work well as long as the classification control problem is still controllable with these limitations. All the numerical results in this paper demonstrate the effectiveness of the proposed approach for different classification tasks of quantum ensembles. ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments that have greatly improved the original manuscript. R EFERENCES [1] M. Mohseni, A. M. Steinberg, and J. A. Bergou, “Optical realization of optimal unambiguous discrimination for pure and mixed quantum states,” Phys. Rev. Lett., vol. 93, no. 20, p. 200403, 2004. [2] V. Beltrani, P. Ghosh, and H. Rabitz, “Exploring the capabilities of quantum optimal dynamic discrimination,” J. Chem. Phys., vol. 130, no. 16, p. 164112, 2009. [3] M. Gut.a˘ and W. Kotłowski, “Quantum learning: Asymptotically optimal classification of qubit states,” New J. Phys., vol. 12, p. 123032, Dec. 2010.


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Chunlin Chen (M’06) received the B.E. degree in automatic control and the Ph.D. degree in control science and engineering from the University of Science and Technology of China, Hefei, China, in 2001 and 2006, respectively. He was with the Department of Chemistry, Princeton University, Princeton, NJ, USA, from 2012 to 2013. He held visiting positions with the University of New South Wales, Sydney, NSW, Australia, and the City University of Hong Kong, Hong Kong. He is currently an Associate Professor with the Department of Control and System Engineering, School of Management and Engineering, Nanjing University, Nanjing, China. His current research interests include quantum control, intelligent control, and machine learning.


Daoyi Dong (SM’11) received the B.E. degree in automatic control and the Ph.D. degree in engineering from the University of Science and Technology of China, Hefei, China, in 2001 and 2006, respectively. He is currently a Senior Lecturer with the University of New South Wales, Canberra, ACT, Australia. He was with the Institute of Systems Science, Chinese Academy of Sciences, Beijing, China, and the Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou, China. He held visiting positions with Princeton University, Princeton, NJ, USA, RIKEN, Wako-shi, Japan, and The University of Hong Kong, Hong Kong. His current research interests include quantum control, multiagent systems, and intelligent control. Dr. Dong is a recipient of an International Collaboration Award and an Australian Post-Doctoral Fellowship from the Australian Research Council. He is a co-recipient of the Guan Zhao-Zhi Award at The 34th Chinese Control Conference and the Best Theory Paper Award at The 11th World Congress on Intelligent Control and Automation. He serves as an Associate Editor of the IEEE T RANSACTIONS ON N EURAL N ETWORKS AND L EARNING S YSTEMS . Bo Qi was born in China in 1982. He received the B.S. degree in mathematics from Shandong University, Jinan, China, in 2004, and the Ph.D. degree in control theory from the Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences (CAS), Beijing, China, in 2009. He was a Visiting Researcher with the Centre for Quantum Dynamics, Griffith University, Brisbane, QLD, Australia, from 2011 to 2012. He is currently an Associate Professor with AMSS, CAS. His current research interests include quantum control, quantum filtering, and quantum information theory.

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Ian R. Petersen (F’00) was born in VIC, Australia. He received the Ph.D. degree in electrical engineering from the University of Rochester, Rochester, NY, USA, in 1984. He was a Post-Doctoral Fellow with Australian National University, Canberra, ACT, Australia. In 1985, he joined the Australian Defence Force Academy, University of New South Wales, Canberra, where he is currently a Scientia Professor and an Australian Research Council Laureate Fellow with the School of Engineering and Information Technology. His current research interests include robust control theory, quantum control theory, and stochastic control theory. Prof. Petersen is a fellow of the Australian Academy of Science. He has served as an Associate Editor of the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL, Systems and Control Letters, Automatica, and the SIAM Journal on Control and Optimization. He is an Editor of Automatica.

Herschel Rabitz received the Ph.D. degree in chemical physics from Harvard University, Cambridge, MA, USA, in 1970. He is currently a Professor with the Department of Chemistry, Princeton University, Princeton, NJ, USA. His current research interests include the interface of chemistry, physics, and engineering, with principal areas of focus, including molecular dynamics, biophysical chemistry, chemical kinetics, and optical interactions with matter. An overriding theme throughout his research is the emphasis on molecular-scale systems analysis.

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