1. Introduction
In quantum phenomena, as in the classical systems, the existence of uncertainties and noises are unavoidable. For example, in superconducting qubits, the coupling energy of a Josephson junction may have fluctuations [
1,
2]. Noises and fluctuations may exist in magnetic fields and electric fields in cavity quantum electrodynamics (QED) [
3,
4]. The spins of an ensemble in nuclear magnetic resonance (NMR) experiments may not be exactly known with respect to the strength of the applied radio frequency field [
5].
The classification of inhomogeneous quantum ensembles is a significant issue which has many applications in the discrimination of atoms (or molecules), the separation of isotopic molecules, and quantum information extraction. Thus, treating the quantum systems with uncertainties is an important and applicable subject which needs to be considered.
A quantum ensemble consists of a large number of single quantum systems. In the practical world, some of the quantum systems exist in the form of quantum ensembles. Each single quantum system in a quantum ensemble is referred to as a member of the ensemble [
6]. Quantum ensembles have wide applications in emerging quantum technology, including long-distance quantum communication [
7], quantum computation [
8], and magnetic resonance imaging [
9].
Control of inhomogeneous quantum ensembles is an important issue in practical applications. Control of inhomogeneous quantum systems for discrimination between two or more similar systems, for instance, is an attractive field of study [
10]. In practical applications, the members of quantum ensembles could have variations in some parameters of dynamic systems. These situations are referred to as inhomogeneous quantum ensembles [
6].
There are many approaches which can be used for solving quantum control problems with uncertainties. For instance, an optimal control for NMR pulse sequences is designed by applying gradient algorithms [
11]. Additionally, a sequential convex programming method is proposed for designing robust quantum manipulations [
12]. Dong and his collogues have designed a development of the variable structure control approach with sliding modes to improve the robustness of quantum systems in which a sliding mode control method is presented for two-level quantum systems to treat bounded uncertainties in the system Hamiltonian [
13]. In addition to these works, a Lyapunov control method is presented to attain a universal quantum control [
14]. For the first time a sampling-based learning control (SLC) of inhomogeneous quantum ensembles is presented for overcoming the compensation for parameter dispersion [
6]. As an important application, the sampling-based learning controller is used for designing of a superconducting quantum control of systems [
15]. Construction of universal quantum gates by using a sampling-based learning control are presented in order to find robust optimal control fields in the presence of different fluctuations and uncertainties [
16]. Furthermore, an extended sampling-based learning control for designing a robust quantum unitary transformation in quantum information processing is presented and implemented [
17]. In other applications, to prevent a control field failing in laser-assisted collisions, a sampling-based robust control is used [
18].
In [
19], a systematic sampling-based learning control method with gradient-based learning algorithms for steering the components of inhomogeneous quantum ensembles with uncertainties to the same ideal state is investigated by Dong and coworkers. There are some challenges in gradient algorithms. For instance, they may fall into a local optimum depending on the initial choices of problem variables or, in complex situations, function derivatives may not be easily found.
Genetic-type algorithms (GAs) have being used in optimization problem-solving. For this purpose, by applying cross-over and mutation operators on current solutions, new solutions are generated and, statistically, they are moving toward optimal solutions in the search space. The set of solutions, however, converges to an optimum solution according to the principle of the Darwinian theory of evolution.
The quantum genetic algorithm (QGA) was identified by Narayanan and Moore [
20]. The QGA, with even a smaller population, presents a great ability of global optimization and good robustness. Therefore, as compared with the common genetic algorithm, QGA has greater effectiveness [
21,
22]. QGAs are mostly constructed based on qubits (or quantum bits) and state superposition in quantum mechanics. In contrast to classical representations of chromosomes (a binary string, for instance), here they are represented by vectors of qubits (quantum registers).
In this paper, for controlling the quantum systems with uncertainties, a hybrid method based on the SLC method and QGA is used. Specially, artificial samples are generated by sampling the uncertainty parameters in the system model and an augmented system is constructed by using these samples in the training step. Then, to train a control law with the desired performance for the augmented system, QG (quantum genetic) learning and optimization algorithms are used. In the process of testing, a set of selected uncertainty samples is tested to evaluate the control performance. Additionally, an improvement of QGA is conducted to attain better results. In [
22] an adding quantum mutation operation in the conventional quantum genetic algorithm is used as an improving device. Quantum mutation, by swapping the value of the probability amplitude of qubits (
α,
β), can completely reverse the individual’s evolutionary direction. In this paper the mutation operation is implemented on measured qubits (bit strings), which is more effective than adding quantum mutation. Reduction of learning iterations, test error and training error, and also increasing the fidelity index are advantages of the proposed method.
This paper is organized as follows:
Section 2 represents the quantum control model and formulates the control problem; A quantum genetic learning ensemble control algorithm is designed in
Section 3; Simulation results and control performance are illustrated in
Section 4; Conclusions are presented in
Section 5.
2. Problem Formulation
In this paper, a finite-dimensional (
-level) closed quantum system with a state in an underlying Hilbert space is considered. The states can be written as a superposition of eigenstates as follows:
where complex numbers
satisfy
and
are the eigenstates of the
N-level quantum system [
23]. Usually, the states of two-level quantum systems are considered as arrows from the origin to points on the Bloch sphere [
24].
The dynamical equation can be described as the following Schrödinger equation:
where
is Plank constant (assume
in this paper),
is the system Hamiltonian and
The dynamics of the system are governed under the following Hamiltonian:
where
is the free Hamiltonian of the system and
is the time-dependent control Hamiltonian that represents the interaction of the system with the external control fields
(scalar functions). Additionally,
for
are Hermitian operators.
In practical applications, there exist external disturbances affecting the control fields. Assume that the system Hamiltonian is disturbed as follows:
where functions
,
characterize uncertainty functions and
To compare and indicate the advantages of the proposed method, it is assumed that the situations and assumptions are similar to the system described in [
19]. Therefore, let
for
be continuous functions and the parameters
could be time-dependent. For simplicity, one can assume that the uncertainty bounds
are all equal in this paper. Additionally, let the nominal values of
are 1 and the fluctuations of the uncertainty parameters
be
(where
).
The objective is to design the controls
to steer the quantum system with uncertainties from an initial state
to a target state
with high fidelity. The fidelity between two pure quantum states
and
is defined as [
25]:
Suppose that a similar ensemble’s members with different Hamiltonians are given. The main objective is to drive the members from an initial state to a desired state. To control the ensemble, one can select a set of samples instead of all ensemble members and create an augmented system to be controlled. Let
be the Hamiltonian of the selected samples, where
is the number of the training samples. The augmented system is constructed as follows:
where
and
is number of the training samples. The task is to find the best control
such that the performance function
for each control strategy in
, is maximized. Thus, the control problem can be formulated as a maximization problem as follows:
when
,
and
.
Note that depends on the control signal , implicitly, subject to Schrödinger equation, be satisfied.
4. Simulation Results
In this section two examples are simulated. Assume that all of the control signals are bounded in a known interval .
Objective and protocols of simulation are explained as follows:
Let the initial state in real coordinates the time interval () is divided by and time slices . Additionally, the quantum genetic populations are the input control signals. The evolution generation number, the size of the population and the length of the each quantum chromosome are 200, 100, and 24, respectively. The mutation rate is 0.05 and the selection percentage of individuals is The stop condition for the iterative algorithm is considered as ( = 0.001). The objective of the problem-solving is transferring all of the initial states to the target state , with maximum fidelity. The value of is set .
Example 1: Consider the following two level quantum systems:
where
is the free Hamiltonian and
,
Additionally,
,
, and
are Pauli matrices. Assume that the system’s state is written as
where
is the orthonormal basis of the corresponding Hilbert space. Let
, where
are complex time depended coefficients. Therefore, Equation (17) is equivalent to
In this example, let
) be the Mexican hat wavelet functions for
and
on
for
E = 0.21. After sampling the uncertainty parameters, every sample can be described as follows:
where
and
∈
. Additionally,
is the complex conjugate of
. To construct an augmented system for the training step of the SLC method, consider
N training samples that are selected through sampling the uncertainties, as follows:
The results of simulation are illustrated in
Figure 2.
Figure 2a illustrates the control signals
,
obtained in the training step.
Figure 2b illustrates the mean of fidelity function of any states as a fitness function of the QGA. Finally,
Figure 2c illustrates simultaneously steering ensemble members to the desired state. As simulation results indicate, 25 training samples are steered to the target state with a fidelity amplitude 0.9986 and error = 0.001. After running the control system, with founded control signals of the training step, for 200 test samples the fidelity amount is 0.9968 and the corresponding error is 0.003.
Example 2: The second example is a three-level quantum system with uncertainties in Hamiltonian parameters that are found widely in natural and artificial atoms. Some atoms can be explained by a V-type three-level quantum system model. It is important to reach a robust preparation of this class of states for practical applications of quantum technology. The SLC, contributed with QGA, is used for a V-type quantum control system. Assume the initial state is:
with
, the orthonormal basis of the corresponding Hilbert space. Let
, where
are complex numbers. Then we have
We take
= diag(1.5, 1, 0) as the free Hamiltonian and choose
,
,
, and
as follows:
After sampling the uncertainty parameters, every sample can be described as follows:
where
,
, and
∈ [1 −
E, 1 +
E].
E ∈ [0,1] is a given constant and
is the complex conjugate of
. Comparing the results with previous works, uncertainty coefficients are chosen the same as what is given in [
19], that is,
and
have uniform distributions over
To construct an augmented system for the training step of the SLC design, we choose
training samples (denoted as
) through sampling the uncertainties as follows:
where
and
. Now, the objective is to find a robust control strategy
to drive the quantum system from
(i.e.,
to
(i.e.,
). The general conditions here are similar to ones mentioned in previous example but
. Apart from the initial values, the results are always converged and it is more precise than the gradient method as shown in
Table 2.
The training error is computed as
in which
is the fidelity function for training samples. For calculating the test error, optimal control
is implemented to the test samples, which are selected randomly. Additionally, the amount of
is computed for test samples, as a test error index. The method presented in this paper always converges and does not depend on initial choices of
Figure 3a–c demonstrate the control signals
,
and fidelity function for steering training samples simultaneously to the target state.
The training samples are steered to the target state with a fidelity of 0.9982. The control values found in training step are applied on 200 testing samples. The fidelity amount of 0.9954 is achieved with a test error equal to 0.005.
Figure 3a,b show the control signals
,
,
, and
through the time interval
and
Figure 3c illustrates the mean of the fidelity function of all of the states as a fitness function of QGA.