Quantum learning control using differential evolution with equally-mixed strategies

Hailan MA ,Daoyi DONG ,Chuan-Cun SHU ,Zhangqing ZHU ,Chunlin CHEN ?

1.School of Management and Engineering,Nanjing University,Nanjing Jiangsu 210093,C hina;

2.School of Engineering and Information Technology,University of New South Wales,Canberra,ACT 2600,Australia

Quantum learning control using differential evolution with equally-mixed strategies

Hailan MA1,Daoyi DONG2,Chuan-Cun SHU2,Zhangqing ZHU1,Chunlin CHEN1?

1.School of Management and Engineering,Nanjing University,Nanjing Jiangsu 210093,C hina;

2.School of Engineering and Information Technology,University of New South Wales,Canberra,ACT 2600,Australia

Learning control has been recognized as a powerful approach in quantum information technology.In this paper,we extend the application of differential evolution(DE)to design optimal control for various quantum systems.Various DE methods are introduced and analyzed,and EMSDE featuring in equally mixed strategies is employed for quantum control.Two classes of quantum control problems,including control of four-level open quantum ensembles and quantum superconducting systems,are investigated to demonstrate the performance of EMSDE for learning control of quantum systems.Numerical results verify the effectiveness of the EMSDE method for various quantum systems and show the potential for complex quantum control problems.

Differentialevolution with equally-mixed strategies(EMSDE),quantumlearning control,superconducting circuits,quantum control

1 Introduction

One of the important objectives in quantum information technology is to manipulate the dynamics of quantum systems by appropriate external fields such that the resultant evolution closely approximates the desired evolution[1-3].To accomplish the required control tasks on quantum systems,scientist have proposed various control methods such as optimal control techniques[2,4,5],Lyapunov control methods[6,7],quantum feedback control including measurement-based feedback and coherent feedback[8,9],incoherent control[10-14]and learning control approaches[15].In particular,learning control provides a powerful approach to obtain an optimal(usually sub-optimal)control[15].Gradient based methods are usually employed to generate a good solution in a local search way,in which the control design is based on system models and gra-dient information[5,16-18].Several results on quantum control using reinforcement learning[19-21]have also presented.Evolutionary algorithms employ stochastic learning in a specified searching space,and have been widely applied to many quantum optimal control problems[22].In particular,genetic algorithm(GA)has achieved a greatsuccess in the closed-loop learning control of quantum systems in the laboratory[16].In this paper,we focus on the application of differential evolution(DE)for quantum control.

Differential evolution is another competitive form of evolutionary computation,which was proposed by Storn and Price in 1990s[23].Since then,a wealth of variants of the basic DE algorithm emerge as powerful optimization methods[24],including jDE[25],oppositionbased DE(ODE)[26],DEGL[27]and JADE[28].Motivated by its good performance for complex optimization problems,researchers have adopted DE as an approach for quantum manipulations and have obtained several results[29-34].In[32-34],DE methods have been adopted to fulfil a desired state transition by designing optimal control fields for open quantum systems,while the numerical results in[32]indicated that the DE method should be further improved for complex problems such as ensemble control of open systems.According to[35,36],DE with mixed strategies exhibited improved learning performance for complex optimization problems.Hence,we employ an DE algorithm with equally-mixed strategies(EMSDE)aiming at enhancing the leaning control performances over complex quantum systems.In[37],an improved DE algorithm has been proposed for three classes of quantum robust control problems including the control of two-level inhomogeneous quantum ensembles,robust control of quantum networks[38-42]and control of fragmentation of molecules using femtosecond laser pulses.In this paper,we further extend the application of DE algorithms for quantum control problems and the EMSDE is employed for two classes of quantum control problems:control of four-level open quantum ensembles,and control of quantum superconducting circuits.

Quantum ensemble control[43]is an important issue among quantum information technology and has wide applications in fields ranging from long-distance quantum communication to magnetic-resonance imaging[44,45].Related achievements include the manipulation of a randomly oriented ensemble of molecules in physical chemistry[46],the design of compensating pulse sequences in NMR spectroscopy[47],and the correction of systematic errors in quantum information processing[43].Chen et al.[48]achieved high fidelity control of closed quantum ensemble by means of gradient-based method.The existence of dissipation might irreversibly transfer the information into the environment[49].In that case,the control problem of open quantum ensembles may have local optimal solutions.Therefore,we employ the EMSDE method for open quantum ensembles to enhance good control performance.

Quantum superconducting circuits based on Josephson junctions are one promising candidate for investigating both computational power and possible experimental implementations of quantum computers.Those macroscopic circuits can behave quantum mechanically like artificial atoms that can be used to test the laws of quantum mechanics on macroscopic systems[50-54].Manipulation ofsuperconducting qubits can be achieved by adjusting external parameters such as currents and voltages or by tuning the coupling between two superconducting qubits[53].In practical applications,there exist different uncertainties and fluctuations in quantum superconducting circuits.Robustcontrolofquantum superconducting circuits has been an important task for the development of practical quantum superconducting applications.A sampling-based learning control(SLC)method has been proposed for achieving robust control of quantum superconducting systems where a gradientbased algorithm was employed to find an optimal control field[54].Here,we adopt EMSDE to design optimalcontrolfields forquantum superconducting systems where it is not necessary to know the gradient information.

The paper is organized as follows.Section 2 introduces the basic steps of the conventional DE and gives a systematic description of the EMSDE algorithm.The EMSDE algorithm is applied to the control problem of four-level open quantum ensembles in Section 3.We employ the EMSDE method to design an optimal control law for quantum superconducting systems in Section 4.Conclusions are presented in Section 5.

2 Differentialevolution with equally-mixed strategy

DE was initially proposed in 1995 by Storn and Price[23]and has been successfully applied in diverse fields such as mechanical engineering,pattern recognition,signal processing and power systems[24].In this sec-tion,we first introduce the procedures of the standard DE algorithm,then analyze DE with variant strategies and control parameters,finally elaborate an improved DE algorithm featuring in equally-mixed strategies for quantum control.

2.1 DE algorithm

Differential evolution(DE)is a parallel direct search method which utilizes NPD-dimensional parameter vectors,termed as individuals,which encode the candidate solutions,i.e.,

where rand(0,1)isa uniformly distributed random number.After initialization,DE works through a cycle of stages consisting ofmutation,crosser and selection.The iterations continue until a termination criterion(such as exhaustion of maximum functional evaluations)is satisfied.

a)Mutation

In the context of differential evolution, “mutation”generates difference vectors and produces a mutantvectorVi,Gwith respect to each individualXi,G,so called target vector,in the current population.In one of the simplest DE-mutation forms,in order to create a mutant vector for eachith target vector from the current population,three other distinct parameter vectors,sayXr1,Xr2,Xr3,are sampled randomly to form the current population.The indicesr1,r2,r3are mutually exclusive integers randomly generated within the range[1,NP],which are also differentfrom the base vectorindexi.The mutation strategy is expressed in the following form

where the scaling factorFis a positive controlparameter typically lying in the interval[0.4,1].

b)Crossover

After the mutation phase,crossover operation is applied to each pair of the targetXi,Gand its corresponding mutant vectorVi,Gto generate a trail vector:

The DE family has two types of crossover operations,i.e.,exponential and binomial.In the basic version,DE employs the binomial(uniform)crossover outlined as

wherej=1,2,...,Dand rand(j)∈[0,1]is a uniform random number.CR is a user-specified constant within the range[0,1),which controls the fraction of parameter values copied from the mutant vector.rand(1,D)∈{1,2,...,D}is a randomly chosen index,and the conditionj=jrandensures thatUi,Ginherits at least one parameter fromVi,G.

c)Selection

To measure how far the “best”performance we have achieved,an objective function(or fitness function)is designed.The selection step aims at comparing the trail vectorUi,Gagainst the target vectorXi,Gwith respect to their objective function values and determining whether the target vector or the trial vector survives to the next generation.The selection operation is described as

wheref(x)is the objective function to be minimized.If the new trial vector yields an equal or lower value of the objective function(equal or larger value for maximization problem),it replaces the corresponding target vector in the next generation;otherwise the target vector survives.Hence,the population either gets better or remains the same in fitness status,but never deteriorated.

Actually,it is the process of mutation that demarcates one DE scheme from another,and numerical results indicates that DE variants employing different mutation strategies usually performs differently when solving different optimization problems[24].For example,“DE/rand/1/bin”is a most commonly used strategy and it usually shows slow convergence speed and bears stronger exploration capacity.Strategies relying on the best solution found so far such as“DE/randto-best/1/bin”, “DE/best/2/bin”,usually have the fast convergence speed and perform well when solving unimodal problems.However,they are more likely to get stuck at a local optimum and thereby lead to a premature convergence when solving multimodal problems.Two-difference-vectors-based strategies such as “DE/rand/2/bin”tends to generate better perturbations than one-difference-vector-based strategies.To balance the exploration and exploitation abilities of DE,“DE/target-to-best/1/bin”scheme with the concept of neighborhood of each population member was proposed[27].“DE/current-to-rand/1”being a rotation variantstrategy can solve rotated problems betterthan other strategies[24].

For control parameters of DE,Storn and Price[55]indicated that a good initial choice ofFwas 0.5 and the effective range ofFis usually set as[0.4,1],while CR=0.1 is a good initial choice and CR=0.9 can be tried to increase the convergence speed.In[56],a good choice for CR was suggested between 0.3 and 0.9.The authors in[57]suggested that typically 0.4＜F＜0.95 withF=0.9 can serve as a good first choice.They also pointed out that CR should lie in(0,0.2)when the function is separable,while CR∈(0.9,1)is effective when the parameters in the function are dependent.Furthermore,some researchers considered developing techniques such as self-adaptations to automatically find an optimal setofcontrolparameters[24].For example,Qin et al.[35]provided a parameter rule where bothFand CR are gradually self-adapted according to a Gaussian distribution.

2.2 Test of different DE strategies

When implementing the DE,users need to determine the appropriate mutation-trial strategies and parameter settings to ensure the success of the algorithm[35].It is a high-cost practice to perform a trial-and-error search for the most appropriate trial vector generation strategy and fine-tune its associated control parameter values,i.e.,the values of CR,Ffor a given problem.Moreover,during different stages of evolution,different trial vector generation strategies coupled with specific control parameter values can be more effective than others and single strategy may result in premature convergence thus leading to a failure in complex problems such as nonseperable and multimodal functions[24,58,59].Several variants of DE utilizing the idea of mixed strategies including SDE[60],SaDE[35]as well as EPDE[36]have been proposed and exhibited good performances.

To achieve a better understanding of strategies and control parameters of DE[61],we investigate different mutation strategies of DE and decide on five DE variants listed in Table 1.As we can see,the first four variants focus on single strategy and choose the traditional scheme(i.e.,DE/rand/1/bin),while DE5 employs mixed strategies with four strategies to be chosen.

Table 1 Features of five DE variants.

As we know,it is the process of mutation that demarcates one DE scheme from another,therefore we provide the mutation descriptions of four candidate strategies as follows.

DE/rand/1:DE/rand to best/2:

DE/current-to-rand/1:

The indicesr1,r2,r3,r4andr5are mutually exclusive integers randomly chosen from the range[1,NP]and all are different from the base indexi.Xbestis the best individual vector with the best fitness(i.e.,the lowest objective function value for a minimization problem)in the population.The control parameterKin the strategy“DE/current-to-rand/1”is setasK=0.5 to eliminate one additional parameter.

Experiments were conducted on a suite of 10 benchmark functions(see Table 2)to evaluate the above five variants,both 10-dimensional(10D)and 30-dimensional(30D)functions were tested,with the maximum number of function evaluations(FEs)set as 50 0000.All experiments were run 20 times independently.The population size is setas NP=20 for10-dimensional case and NP=50 for 30-dimensional case.Tables 3 and 4 report the the success rate and mean as well as variance function values(row order)by applying the five algorithms to optimize the 10D and 30D numerical functionsf1-f10,respectively.The best results are typed in bold.The success of an algorithm means that this algorithm can result in a function value very near to the prespecified optimal minimized value,i.e.,f(x)?f(x＊)+10-5,for all problems with the number of FEs less than the prespecified maximum number.The success rate is calculated as the number of successful runs divided by the total number of runs.

Table 2 Descriptions of benchmark functions.

?

Remark 1Functionsf1-f4are unimodal and functionsf5-f10are multimodal.The functionsf9andf10are constructed on the basis of periodic sinusoidal functions and have multiple optimal solutions.Considering the fact that some benchmark functions have the same parameter values for different dimensions at the global optimum because of their symmetry,we shift the conventional benchmark in a random way,i.e.,

As we can see,f1andf2are easily optimized by five DE variants for both cases ofD=10 andD=30,with best results as 100%success rates and zero error(i.e.,both mean and variance function values are equal to 0.00E+00),whilef3andf4are less easily optimized.DE1 with parametersF=0.9,CR=0.1 ranks first four times for 10D functions and two times for 30D functions.

Table 3 Results for 10D functions.

However,DE5 variants with mixed strategies achieves 80%success rate for functionf3(D=30),while single DE variants(i.e.,DE2)can achieve 100%success rate.Multimodal functionsf5-f8are more difficult to optimize.Among them,f6andf8can be optimized to zero error for both 10D functions and 30D functions.However,f7tends to be complicated,since DE variants with single strategies fail to achieve 100%success rate,even for DE1,which exhibits excellent performance for functionsf1-f6.Remarkably,DE5 with mixed strategies and variational parameters obtains the best results on functionf7,with 100%success rate for both 10D function and 30D function.For periodic functionsf9andf10,the optimization procedures turn to be easier than multimodal functions.WhenD=10,DE5 obtains the best results on both functions(with the least mean values).WhenD=30,DE5 obtains the best optimization performance on functionf9,and DE1 ranks first on functionf10.

Table 4 Results for 30D functions.

Actually,the five DE variants have similaroptimization efficiency exceptDE2 with parameterF=0.9,CR=0.9.It is concluded that DE with single strategy such as“DE/rand/1/bin”is good enough for simple optimization problems,DE with mixed strategies(i.e.,DE5)performs more efficiently for multimodal functions which could not be well solved by a single strategy.

2.3 EMSDE for quantum control

The discovery of optimal solutions for manipulating quantum phenomena is a significant goal of quan-tum control.Early studies investigated conditions under which optimal solutions exist.Underlying the search for optimal controls is the landscape that specifies the physical objective as a function of the control variables,denoted asJ(u),which can be explained as an optimization problem.Evolutionary algorithms[23]have been a good candidate for solving such an optimization problem.Among these evolutionary algorithms,DE is a powerful approach for numerous optimization problems in diverse fields.

Based on the observations in Section 2.2,we attempt to adopt an equally-mixed strategy DE method(EMSDE)to generate a good solution for quantum control problems.Firstly,we choose one mutation scheme from a strategy candidate pool where several mutation schemes with effective yet diverse characteristics are equally distributed.Subsequently,crossover operation is performed on the corresponding mutant vector to generate the trial vector.Furthermore,various values ofFand CR are assigned for each individual to increase the diversity of the population.To construct the candidate pool,we investigate different mutation-crossover strategies commonly used in the DE literatures[24,59]and choose four strategies: “DE/rand/1”, “DE/rand to best/2”,“DE/rand/2”and “DE/current-to-rand/1”,which demonstrate distinct capabilities at different stages of evolution.

Actually,it is a difficult task to figure out proper parameter values for a given problem which greatly influence behaviors of the algorithm.As we know,smallFvalues aim at exploitation,while largerFvalues aim at exploration.In the proposed EMSDE algorithm,the parameterFis approximated by a normal distribution with mean value 0.5 and standard deviation 0.3,denoted by N(0.5,0.3).It is clear that values ofFfall into the range[-0.4,1.4]with probability of 0.997,which helps maintain a balance between exploitation and exploration.Furthermore,we accept some cases of extraordinary value(far from[-0.4,1.4])to increase its diversity.Similarly,we assume CR obeys a normal distribution denoted by N(0.5,0.1),and the small standard deviation 0.1 helps guarantee that most values of CR lies in[0,1][24].A key point is that values falling beyond[0,1]is unacceptable,since CR has probabilistic meaning for the chance of survival.Consequently,a set ofFand CR values are randomly sampled from normal distribution and assigned to each target vector in the current population.The algorithmic description of the EMSDE is presented in Algorithm 1.

Algorithm 1(Algorithmic description of EMSDE)Input:The parameter search range[Xmin,Xmax],the population size NP,the maximum generation Gmax Set G=0,initialize the population by x ji,0=x jmin+rand(0,1)·(x j max-x jmin)and evaluate the vectors and mark the best one as Xbest,G repeat(for each generation G=1,2,...,Gmax)repeat(for each vector Xi,i=1,2,...,NP)Set parameters F=N(0.5,0.3),CR=N(0.5,0.1)Randomly generate five exclusive integers ri∈ [1,NP],and a real number pp∈ (0,1]if pp∈(0,0.25]then flag=1,Vi,G=Xr1,G+F·(Xr2,G-Xr3,G)else if pp ∈ (0,0.25]then flag=2,Vi,G=Xi,G+F·(Xbest,G-Xi,G)+F·(Xr1,G-Xr2,G)+F·(Xr3,G-Xr4,G)else if pp∈(0.50,0.75]then flag=3,Vi,G=Xr1,G+F·(Xr2,G-Xr3,G+Xr4,G-Xr5,G)else flag=4,Vi,G=Xi,G+K ·(Xr1,G-Xi,G)+F·(Xr2,G-Xr3,G)end if Boundaries check and reinitialize mutant vector Vi,G as v ji,G=x jmin+rand(0,1)·(x jmax-x j min)if flag=4 then obtain the trail vector as Ui,G=Vi,G else perform the binomial crossover and generate the trial vector Ui,G end if Compute the fitness functions J(Ui,G)if J(Ui,G)?J(Xi,G)then Xi,G+1←Ui,G,J(Xi,G+1)←J(Ui,G)end if Renew the best vector Xbest,G and i←i+1 until i=NP G←G+1 until G=Gmax Output:The optimal control strategy u＊=Xbest,G

Remark 2After performing the mutation operation,we obtain new donor vectors,some of which might hopefully survive into the next generations and serve as parents.We should add a procedure where each vector is evaluated in view of boundary constraints.If any parameter of the vector falls beyond the pre-defined lower or upper bounds,we will replace it with a randomly chosen value from the allowed range.In Section 2.1,we considerthe minimization problem as an example,while exploring quantum control landscape we may need to solve is a maximization problem.Therefore,the goal of optimization is to increase the objective function towards the optimal value.

3 Optimal control of open quantum ensembles

3.1 Control of open quantum ensemble

A quantum ensemble consists of many single quantum systems(e.g.,identical spin systems or molecules),where each single system is referred to as a member of the ensemble in this paper.In practical applications,the members of a quantum ensemble could have variations in the parameters that characterize the system dynamics.For example,the spins of an ensemble in nuclear magnetic-resonance(NMR)experiments may encounter large dispersion in the strength of the applied radio frequency field(rf inhomogeneity)as well as the members exhibiting variations in their natural frequencies(Larmor dispersion)[43].Such quantum ensembles are called inhomogeneous quantum ensembles.Several methods have been presented to design control laws for inhomogeneous quantum ensembles when dissipation is not considered[45,48].

For a practical quantum ensemble,it may be unavoidable to interact with its environment.In this work,we consider an open ensemble consisting of many open quantum systems with parameter variations.For an open quantum system,its state can be described by a positive Hermitian density operator ρ satisfying tr(ρ)=1.Under the assumption of a short environmental correlation time permitting the neglect of memory effects,a Markovian master equation for ρ(t)can describe the dynamics of an open quantum system interacting with its environment[62].Markovian master equations in the Lindblad form are described as[63,64].

Letn×nmatrices iU1,iU2,...,iUm(m=n2-1)be orthogonal generators ofSU(n),the density operator can be written as

where the real numbersl=tr(Uiρ)is an element of the coherent Bloch vectors:=(s1,s2,...,sm)T.Substituting(10)into(9),we can obtain the evolution ofthe coherent vectorsas

where the LH0,LD,LHj(j=1,2,...,M)are super operators,f0is the inhomogeneous source term,detailed explanations see in literature[18].

Let Γ =(γ1,γ2,...,γM),the Hamiltonian of inhomogeneous open quantum ensemble can be described as

where the functionsg j(γj)(j=0,1,...,M)characterize possible inhomogeneities.For simplicity,we assumeg j(γj)= γj,and the parameters γjare randomly distributed over[1-Δ,1+Δ].The constant Δ ∈[0,1]represents the bounds of the parameter dispersion.The objective is to design the controlu={u j(t),j=1,2,...,M}with the purpose of simultaneously driving the members(with different parameters)of the quantum ensemble from an initial states0to a desired final states(T)regarding a target statesf.The control outcome is described by a performance functionJ(u)for each control strategy.The control problem can be formulated as a maximization problems as follows:

where E[JΓ(u)]denotes the average performance function regarding parameter inhomogeneities Γ.And the performance functionJ(u)takes the following form[18]:

3.2 Numerical results

We consider a specific inhomogeneous open fourlevel ensemble with parameter bound Δ=0.2.Assume the four energy levels are

and 〈1|=(1,0,0,0),〈2|=(0,1,0,0),〈3|=(0,0,1,0),〈4|=(0,0,0,1).Members of the ensemble are governed by the following Hamiltonian:

whereu1(t),u2(t),u3(t)∈[-10,10],and we take

And the Lindblad operators are given by[64]

with τ12=0.4,τ13=0.3,τ14=0.2.The average performance function is given as

whereNΓis the total number of the chosen samples,we choose three samples for each parameter,and here we haveNΓ=9.As an example,we consider a control task,driving the ensemble from the initial state|φ0〉=(1,0,0,0)to the target state|φf〉=(0,0,0,1)within the target timeT=2.During implementation,the total time interval[0,T]are equally divided into 200 time steps(namelyD=200).

To demonstrate the performance of the EMSDE algorithm,we make comparisons between different variants of DE including the basic DE(labeled as DE),DE with various parameters(denoted as NDE for its normal distribution)and the proposed EMSDE.To begin with,we discuss the case of the traditional DE scheme,i.e.,“DE/rand/1/bin”with three typical set of control parameters considered.Judging from Fig.1(a),“DE1”(F=0.9,CR=0.1)achieve the best performance,i.e.,Jmax=0.9483 among three DE schemes with constantparameters,and“DE3”(F=0.9,CR=0.9)are not far behind with maximum value of 0.9350,“DE2”(F=0.5,CR=0.3)have the worst optimization performance withJmax=0.8986.We then compare the training performances of DE1, “NDE”(F=N(0,5,0.3),CR=(0.5,0.1)),EMSDE,with results illustrated in Fig.1(b).As we can see,the EMSDE method converges to maximum average valueJmax=0.9858,while NDE converge to a maximum value of 0.9488.A comparison of testing performances between those algorithms in Fig.2 shows that EMSDE is superior to otherversions ofDE(NDE and basic DE)featuring in single mutation strategy,suggesting that mixed strategies tremendously enhance the learning effect.And we summarize the training and testing performances in Table 5.The optimal control fields for the quantum ensemble learned by EMSDE are demonstrated in Fig.3.

Fig.1 Training performance of 4-level open quantum ensemble.

Fig.2 Testing performance of four-level open quantum ensemble.

Table 5 Summary of training and testing.

Fig.3 The optimal control fields learned by EMSDE.

4 Controlofquantum superconducting circuits

4.1 Controlling quantum superconducting circuits

In superconducting circuit,the charging energyECand the Josephson coupling energyEJare two significant energy level,which have significant effect on the quantum mechanical behavior of a Josephson-junction circuit.Differentkinds of superconducting qubits can be realized according to the regimes ofEJ/EC.For example,a charge qubit can form whenEC?EJ[51].It is based on a small superconducting island,i.e.,Cooperpair box(CPB)coupled to the outside world through a weak Josephson and driven by a voltage source through a gate capacitance within the charge regime.In practical applications,the Josephson junction in the charge qubit is usually replaced by a dc superconducting quantum interference device(SQUID)with low inductance and a magnetic flux[51].When the box’s offset charge,induced by the gate voltageVg,is about the same as the charge of a single electron,the system can be described like any two-level atomic-physics-like system with the reduced Hamiltonian

whereFz(Vg)can be adjusted through external voltageVg,Fx(Φ)corresponds to a tunable effective coupling with the external magnetic flux Φ in the SQUID and

In this case,σzand σxare system Hamiltonian operators for a charge qubit,whileFz(Vg)andFx(Φ)are external control fields(namelyu)that we are seeking for.Since we do not consider possible dissipation here,we use wavefunction to describe the quantum state.For such a quantum system,the control performance is defined as[65]

withF(|ψ(T)〉,|ψtarget〉)=|〈ψ(T)|ψtarget〉|measuring the fidelity between the finalstate|ψ(T)〉and the targetstate|ψtarget〉[1].For an optimal control problem of quantum superconducting systems,the objective is to find a control field to maximizeJ(u).

In practical applications,the existence of noise and fluctuations(e.g.,fluctuations in magnetic fields and electric fields)in superconducting quantum circuits is unavoidable.We assume that the Hamiltonian with uncertainties can be written into(12).When we consider to accomplish a state transition for a quantum superconducting system with fluctuations,the problem can be formulated as

where E[JΓ(u)]denotes the average performance function with respect to the parameter fluctuations Γ,and Δ∈[0,1]represents the bounds of the parameter fluctuations.

4.2 Numerical results

We consider a coupled superconducting circuitwhere a symmetric dc SQUID with two sufficient large junctions is used to couple two charge qubits.Each qubit is realized by a Cooper-pair box with Josephson coupling energyEJjand capacitanceCJj.Each Cooper-pair box is biased by an applied voltageVithrough a gate capacitanceC j(j=1,2).The superconducting circuits are described in Fig.4.Denote?as the tensor product,and we define

The Hamiltonian of the coupled charge qubits can be described as

Fig.4 Schematic of the superconducting circuits under consideration.

where γj∈ [1-Δ,1+ Δ],j= 1,2,...,5.Letu3(t)=u4(t)=9.1 GHz,the control termsu1(t),u2(t)∈[0,40]GHz,|u5(t)|?0.5 GHz.For simplicity,we assume γ1= γ2,γ3= γ4,γ5=1.When we consider the fluctuations,the average performance function to guide the learning process is given

we choose three samples for each parameter,andNΓ=9 when considering the existence of fluctuations.As an example,we consider the task of driving the system from the initial state|ψ0〉=|g,g〉to the target state|ψtarget〉=|e,e〉with parameter fluctuations Δ =0.05.The time internal[0,2]ns is equally divided into 200 smaller time intervals(i.e.,D=200).And the population size for EMSDE is set as NP=50.

To demonstrate the effectiveness of the EMSDE algorithm,we also implement GA algorithm(with crossover probabilityPc=0.8,mutation probabilityPm=0.05,and population size NP=50)to find solutions for superconducting circuits.We consider the optimal control problem(no fluctuations)as well as the robust control result(with fluctuations).Comparisons of GA and EMSDE in view of training performance are illustrated in Fig.5.As we can see,the EMSDE method exhibits wonderful performance for superconducting circuits in both cases,while the performance of GA is less satisfactory especially for the case with fluctuations.To be more specific,EMSDE is able to achieve high fidelities almost 1 for the optimal control problem andJmax=0.9997 for the case with fluctuations.GA can achieve the fidelitiesJmax=0.9992 for the optimal control task andJmax=0.9566 for the case with fluctuations.The testing performances of two methods are shown in Fig.6,and the control performance for samples achieved by EMSDE approximately keeps stable nearJ(u)=1,while the fidelities for samples with fluctuations achieved by GA fluctuate a little.Furthermore,the optimal control fields learned by EMSDE are demonstrated in Fig.7.

Fig.5 The training performance of two-qubit superconducting circuits via EMSDE and GA.(a)corresponds to the results of EMSDE,(b)corresponds to the results of GA.

Fig.6 The testing performance of two-qubit superconducting circuits via EMSDE and GA.

Fig.7 The optimal control fields learned by EMSDE.

The superiorities of EMSDE over GA for this problem lie in the mechanism of generating the new population from the old population.In GA,parents are selected based on probabilities that lead to individuals with better fitness.The crossover operation combines partial parts of two parents to generate a new offspring around some good solution candidates rather than the entire set of populations.Beside,GA possibly fails to locate the global optimum as a result of low mutation probability.The crossover operation in DE generates offspring individuals from the entire set of populations so that newly generated offsprings are always different from parentindividuals.The higher mutation probability enables DE to explore the search space more efficiently and therefore reduces the chance of getting trapped in local minima.On the other hand,GA fails to achieve a high fidelity for system with fluctuations,and the possible explanations are that GA converts candidate solutions into a binary format limiting its searching ranges in the sense of resolution.By contrast,DE constructs candidate solutions in forms of real numbers thus successfully avoiding this problem.

5 Conclusions

In order to solve two classes of quantum learning control problems,we have employed the improved DE variants,i.e.,EMSDE,featuring in mixed strategies and various control parameters.Judging from the numerical results,the EMSDE algorithm has good performance.Compared to the conventional DE with single mutation strategy(including DE1,DE2,DE3,NDE),EMSDE exhibits excellent learning performance for open quantum ensembles and has improved performance over GA for quantum superconducting systems.In future research,we will extend the application of improved DE for emerging quantum control technology,and make more effort into exploring efficient algorithms for laboratory applications,where optimal control fields may be further required in accordance to specific circumstances to comply with a plurality of implicit constraints(e.g.,low peak intensity for avoiding sample damage,and short enough pulse length for combating decoherence.To this end,a general quantum optimal control method that has been developed by incorporating multiple constraints into optimization algorithms would be employed for more closely aligning experimental and theoretical results[66-68].Moreover,we will try to hybridize DE with other learning algorithms(e.g.,reinforcement learning algorithm[69])to design powerful approaches for solving complex quantum control problems.

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25 May 2017;revised 27 June 2017;accepted 27 June 2017

DOI 10.1007/s11768-017-7069-y

?Corresponding author.

E-mail:clchen@nju.edu.cn.

This paper is dedicated to Professor Ian R.Petersen on the occasion of his 60th birthday.This work was supported by the National Natural Science Foundation of China(Nos.61374092,61432008),the National Key Research and Development Program of China(No.2016YFD0702100)and the Australian Research Council’s Discovery Projects funding scheme under Project DP130101658.

?2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Hailan MAwas born in Xiangyang,China,in 1992.She received the B.E.degree in Automation and the M.Sc.degree in Control Science and Engineering from Nanjing University,Nanjing,China,in 2014 and 2017,respectively.Her research interests include machine learning and quantum control.E-mail:646265112@qq.com.

Daoyi DONGwas born in Hubei,China.He received the B.E.degree in Automatic Control and the Ph.D.degree in Pattern Recognition and Intelligent Systems from the University of Science and Technology of China,Hefei,China,in 2001 and 2006,respectively.He was as a Post-Doctoral Fellow with the Institute of Systems Science,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing,China,from 2006 to 2008.He was with the Institute of Cyber-Systems and Control,Zhejiang University,Zhejiang,China.He held visiting positions with Princeton University,Princeton,NJ,U.S.A.,the University of Hong Kong,Hong Kong,and the City University of Hong Kong,Hong Kong.

He is currently a Senior Lecturer with the University of New South Wales,Canberra,Australia.His current research interests include quantum control,reinforcement learning,and intelligent systems and control.Dr.Dong is a recipientofan International Collaboration Award and an Australian Post-Doctoral Fellowship from the Australian Research Council,a K.C.Wong Post-Doctoral Fellowship,and a President Scholarship from the Chinese Academy of Sciences.He is also a co-recipient of 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(WCICA).He serves as an Associate Editor of IEEE Transactions on Neural Networks and Learning Systems.E-mail:daoyidong@gmail.com.

Chuan-Cun SHUgraduated from Dalian University of Technology(DUT),China in 2010,earning his Ph.D.in Atomic and Molecular Physics.After obtained his Ph.D.,he joined Prof.Niels E.Henriksen’s group at Technical University of Denmark(DTU)by HC φrsted Postdoctoral Program,cofunded by Marie Curie Actions.After finished his research project at DTU in 2012,he had three years in Prof.Herschel Rabitz’s group at Princeton University as a full-time postdoctoral research associate.In May 2015,he joined Prof.Ian Petersen’s group at University of New South Wales Canberra as a Vice-Chancellor Postdoctoral Fellow.His current research interest focus on multiple constraint frequency domain quantum optimal control theory and its application to quantum systems.He has published more than 30 papers in peer reviewed international journals,including Journal of Physical Chemistry Letters,Optics Letters,Physical Review A and The Journal of Chemical Physics.E-mail:C.Shu@adfa.edu.au.

Zhangqing ZHUwas born in Wuwei,Anhui province,China,in 1967.He received the Ph.D.degree in Control Science and Engineering from Nanjing University of Science and Technology,Nanjing,China,in 2006.He is currently an associate professor in the Department of Control and Systems Engineering of Nanjing University,Nanjing,China.His research interests include network control and nonlinear system.E-mail:zzqing@nju.edu.cn.

Chunlin CHENwas born in Anhui,China,in 1979.He received the B.E.degree in Automatic Control and Ph.D.degree in Pattern Recognition and Intelligent Systems from the University of Science and Technology of China,Hefei,China,in 2001 and 2006,respectively.He was a Visiting Scholar with Princeton University,Princeton,NJ,U.S.A.,from September2012 to August 2013.He is currently a full Professor with the Department of Control and Systems Engineering,Nanjing University,Nanjing,China.His current research interests include machine learning,mobile robotics,and quantum control.E-mail:clchen@nju.edu.cn.