Performance Analysis and Coherent Guaranteed Cost Control for Uncertain Quantum Systems

# Performance Analysis and Coherent Guaranteed Cost Control for Uncertain Quantum Systems

## Abstract

This paper presents several results on performance analysis for a class of uncertain linear quantum systems subject to either quadratic or non-quadratic perturbations in the system Hamiltonian. Also, coherent guaranteed cost controllers are designed for the uncertain quantum systems to achieve improved control performance. The coherent controller is realized by adding a control Hamiltonian to the quantum system and its performance is demonstrated by an example.

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## 1 Introduction

Recent years have seen a rapid development of quantum technology and consequently there has been a considerable amount of research focusing on the area of quantum feedback control systems; e.g., see [1]-[7], [10]-[12]. In this research, robustness plays a vital role; e.g., see [5], [11], [12]. Several robust control methods that are widely used in classical systems have been adopted in quantum control areas. For example, control theory has been used to solve a robust feedback controller synthesis problem for quantum systems [5]. A transfer function approach has also been used to analyse robustness in the feedback control of quantum systems[11], [12]. The small gain theorem has been used to analyse the stability and robustness of quantum feedback networks [14]. In this paper, we extend some results in classical control on system performance analysis and guaranteed cost control design to quantum systems.

A majority of existing papers in quantum feedback control only consider the case where the controller is a classical system. That is, the controller may be implemented via analog or digital electronics and quantum measurements are involved. However, some recent results have shown that the controller itself can be a quantum system, which is often referred to as coherent quantum control [5], [11], [12], [14], [15]. The advantage of using coherent quantum control is its ability to achieve improved system performance, since quantum measurements inherently involve the destruction of quantum information. Such controllers are often defined by linear quantum stochastic differential equations (QSDEs) and require physical realizability conditions so that they represent physically implementable systems; e.g., [5], [15], [16]. The coherent quantum controller, designed in this paper uses the framework involving triples , where is a scattering matrix, is a vector of coupling operators and is a Hamiltonian operator [7]. The matrix , together with the vector , specifies the interface between the system and the fields and the parameter describes the self-energy of the system. To control such a quantum system, we add a controller Hamiltonian to the system. In this approach, we do not need to consider physical realizability conditions for the controller system, since a triple automatically represents a physically realizable quantum system. This paramerization for open quantum systems is used in some recent papers, e.g., see [4], [5], [6] and [13], but few controller design methods have been established based on the approach.

The paper [4] presents conditions of dissipativity and stability for this class of quantum systems. Then, the paper [1] built on the result of [4] to provide stability conditions for a class of uncertain quantum systems subject to unknown perturbations. Based on [6] and [4], we extend the guaranteed cost control method to the quantum domain and provide a performance guarantee for the given system when the system Hamiltonian is in the form of , where is a known nominal Hamiltonian and is a perturbation Hamiltonian. Furthermore, motivated by [8] and [9], we add a quantum controller in the system Hamiltonian of the given system not only to guarantee that the system is stable but also to obtain an adequate level of performance.

We begin in Section 2 by presenting the general class of uncertain quantum system models under consideration. In particular, we specify the underlying systems as linear quantum systems. In Section 3, we present a class of quadratic perturbation Hamiltonians and a general class of non-quadratic perturbation Hamiltonians. In Section 4, we present the performance analysis problem for the given systems in terms of a strict bounded real condition. In Section 5, we add a quantum controller to the original system to achieve stability and a guaranteed performance level. In Section 6, we provide an example to illustrate the theory which has been developed in this paper. Conclusions are presented in Section 7.

## 2 Quantum Systems

The open quantum systems under consideration are defined by parameters where the system Hamiltonian is decomposed as . Here is a known nominal Hamiltonian and is a perturbation Hamiltonian contained in a specified set of Hamiltonians ; e.g., [1], [4] and [7]. We define the corresponding generator operator

 G(X)=−i[X,H]+L(X) (1)

where . Here, describes the commutator between two operators and the notation refers to the adjoint transpose of a vector of operators. and are two self-adjoint operators on the underlying Hilbert space. By introducing a quantum stochastic differential equation, the Heisenberg evolution of an operator is defined by the triple together with the corresponding generators[4]. The results presented in this paper will build on the following results from [3].

###### Lemma 1

[3] Consider an open quantum system defined by and suppose there exist non-negative self-adjoint operators and on the underlying Hilbert space such that

 G(V)+W≤λ (2)

where is a real number. Then for any plant state, we have

 limsupT→∞1T∫T0⟨W(t)⟩dt≤λ. (3)

Here denotes the Heisenberg evolution of the operator and denotes quantum expectation; e.g., see [3] and [4].

In this paper, we consider nominal systems corresponding to linear quantum systems. We assume that is in the following form

 H1=12[a†aT]M[aa#] (4)

where is a Hermitian matrix and has the following form with and

 M=[M1M2M#2M#1]. (5)

Here is a vector of annihilation operators on the underlying Hilbert space and is the corresponding vector of creation operators. In the case of matrices, the notation refers to the complex conjugate transpose of a matrix. In the case of vectors of operators, the notation refers to the vector of adjoint operators and in the case of complex matrices, this notation refers to the complex conjugate matrix. The commutation relations between annihilation and creation operators are described as follows

 [[aa#],[aa#]†]=[aa#][aa#]†−([aa#]#[aa#]T)T= J (6)

where [6].

The coupling vector is assumed to be of the form

 L=[N1N2][laa#] (7)

where and . We also write

 [LL#]=N[aa#]=[N1N2N#2N#1][aa#]. (8)

We consider a self-adjoint ”Lyapunov” operators of the form

 Unknown environment '% (9)

where is a positive definite Hermitian matrix of the form

 P=[P1P2P#2P#1]. (10)

We then consider the set of non-negative self-adjoint operators defined as

 P={V  of the form (???) such that P>0 is aHermitian matrix of the form (???)}. (11)

## 3 Perturbations of the Hamiltonian

For the set of non-negative self-adjoint operators and given real parameters , a particular set of perturbation Hamiltonians is defined in terms of commutator decomposition

 [V,H2]=[V,z†]w−w†[z,V] (12)

for , where and are given vectors of operators. is then defined in terms of sector bound condition:

 w†w≤1γ2z†z+δ. (13)

We define

 W1={H2:∃ w,z such % that (???) is satisfiedand (???) is satisfied ∀ V∈P}. (14)

We then consider a set of quadratic perturbation Hamiltonians that is in the form of

 Unknown environment 'array% (15)

where and is a Hermitian matrix of the form

 Δ=[Δ1Δ2Δ#2Δ#1] (16)

with and . The matrix is subject to the norm bound

 ∥Δ∥≤2γ (17)

where refers to the matrix induced norm.

We define

 W2={H2 of the form (???) such that conditions (???) and (???) are satisfied}. (18)

Since the nominal system is linear, we use the relationship:

 z=[ζζ#]=[E1E2E#1E#2][aa#]=E[aa#]. (19)

Then

 H2=12[a†aT]E†ΔE[aa#] (20)

In [1], it has been proven that for any set of self-adjoint operators ,

 W2⊂W1. (21)

For the set of non-negative self-adjoint operators and given real parameters , and , a particular set of perturbation Hamiltonians is defined in terms of commutator decomposition

 [V,H2]=[V,z]w∗1−w1[z∗,V]+12[z,[V,z]]w∗2−12w2[z,[V,z]]∗ (22)

for , where and are given scalar operators. Here, the notation refers to the adjoint of an operator. The set is defined in terms of the sector bound condition

 w1w∗1≤1γ2zz∗+δ1 (23)

and the condition

 w2w∗2≤δ2. (24)

We define

 W3={H2:∃ w1,w2,z such that (???) and\ (???)are satisfied and (???) is satisfied ∀ V∈P}. (25)

We consider a set of non-quadratic perturbation Hamiltonians . For the set of non-negative self-adjoint operators and given real parameters , , , a set of non-quadratic perturbation Hamiltonians is defined in terms of the following power series.

 H2=f(ζ,ζ∗)=∞∑k=0∞∑l=0Sklζk(ζ∗)l=∞∑k=0∞∑l=0SklHkl (26)

where , , and is a scalar operator on the underlying Hilbert space. Also

 H∗2=∞∑k=0∞∑l=0S∗klζl(ζ∗)k=∞∑l=0∞∑k=0Slkζl(ζ∗)k=H2. (27)

Hence, is self-adjoint operator. We define

 f′(ζ,ζ∗)=∞∑k=1∞∑l=0kSklζk−1(ζ∗)l, (28)
 f′′(ζ,ζ∗)=∞∑k=1∞∑l=0k(k−1)Sklζk−2(ζ∗)l. (29)

We consider the sector bound condition

 f′(ζ,ζ∗)∗f′(ζ,ζ∗)≤1γ2ζζ∗+δ1, (30)

and the condition

 f′′(ζ,ζ∗)∗f′′(ζ,ζ∗)≤δ2. (31)

We define

 W4={H2 of the form (???) such that (???) and  (???) are satisfied}. (32)

From [1], we have the fact that for any set of self-adjoint operators ,

 W4⊂W3. (33)

In the nominal linear system, we define

 z=ζ=E1a+E2a#=[E1E2][aa#]=~E[aa#]. (34)

The following result has been proven in [1].

###### Lemma 2

(See Lemma 5 of[1]) Given any ,

 μ=[z,[V,z]]=−~EΣJPJ~ETis a constant, (35)

where .

## 4 Performance Analysis

In this section, we present several results on performance analysis for the two classes of quantum systems defined above. We define the associated cost function for a quantum system as

 J=limsupT→∞1T∫T0⟨[[]cca†aT]R[aa#]⟩dt (36)

where . We denote that

 Unknown environment '% (37)

In order to prove the following theorems on performance analysis, we require some algebraic identities.

###### Lemma 3

(See Lemma 3 of[1]) Consider is of the form (4) and is of the form (7). Then

 [V,H1]=[aa#]†(PJM−MJP)[aa#], (38)
 L(V)= −12[aa#]†(N†JNJP+PJN†JN)[aa#] +Tr(PJN†[I000]NJ), (39)
 [[aa#],[aa#]†P[aa#]]=2JP[aa#]. (40)

Now we present two theorems which can be used to analyse the performance of quantum systems subject to quadratic Hamiltonian perturbations and non-quadratic Hamiltonian perturbations, respectively.

###### Theorem 1

Consider an uncertain quantum system , where , is in the form of (4), is of the form (7) and . If is Hurwitz, and

 ⎡⎣F†P+PF+E†Eγ2τ2+R2PJE†2EJP−I/τ2⎤⎦<0 (41)

has a solution in the form of (10) and , then

 J =limsupT→∞1T∫T0⟨W(t)⟩dt =limsupT→∞1T∫T0⟨[a†aT]R[aa#]⟩dt≤~λ+δτ2 (42)

where

 ~λ=Tr(PJN†[I000]NJ). (43)

In order to prove this theorem, we need the following two lemmas.

###### Lemma 4

Consider an open quantum system where and , and the set of non-negative self-adjoint operators . If there exists a and real constants , such that

 −i[V,H1]+L(V)+τ2[V,z†][z,V]+1γ2τ2z†z+W≤~λ, (44)

then

 limsupT→∞1T∫T0⟨W(t)⟩dt≤~λ+δτ2,∀t≥0. (45)

Proof: Since and ,

 G(V)=−i[V,H1]+L(V)−i[V,z†]w+iw†[z,V]. (46)

Also,

 0≤(τ[V,z†]−iτw†)(τ[V,z†]−iτw†)†=τ2[V,z†][z,V]+i[V,z†]w−iw†[z,V]+w†wτ2. (47)

Substituting (46) into (47) and using the sector bound condition (13), the following inequality is obtained:

 G(V)≤−i[V,H1]+L(V)+τ2[V,z†][z,V]+1γ2τ2z†z+δτ2, (48)

Hence,

 G(V)+W≤~λ+δτ2. (49)

Consequently, the conclusion in the lemma follows from Lemma 1.

###### Lemma 5

For and defined in (19),

 [z,V]=2EJP[aa#], (50)
 [V,z†][z,V]=4[aa#]†PJE†EJP[aa#], (51)
 z†z=[aa#]†E†E[aa#]. (52)

Proof: The proof follows from Lemma 3.
Proof of Theorem 1: Using the Schur complement, the inequality (41) is equivalent to

 F†P+PF+4τ2PJE†EJP+E†Eγ2τ2+R<0. (53)

If the Riccati inequality (53) has a solution of the form (10) and , according to Lemma 3 and Lemma 5, we have

 Unknown environment 'array% (54)

Therefore, it follows from (41) that condition (44) will be satisfied with

 ~λ=Tr(PJN†[I000]NJ)≥0. (55)

Then, according to the relationship (21) and Lemma 4, we have

 limsupT→∞1T∫T0⟨W(t)⟩dt=limsupT→∞1T∫T0⟨[a†aT]R[aa#]⟩dt≤~λ+δτ2. (56)

###### Theorem 2

Consider an uncertain quantum system , where , is in the form of (4), is of the form (7) and . If is Hurwitz, and

 ⎡⎢⎣F†P+PF+Σ~ET~E#Σγ2τ2+R2PJΣ~ET2~E#ΣJP−I/τ2⎤⎥⎦<0 (57)

has a solution of the form (10) and , then

 J=limsupT→∞1T∫T0⟨W(t)⟩dt=limsupT→∞1T∫T0⟨[a†aT]R[aa#]⟩dt≤~λ+δ1τ2+μμ∗/4+δ2 (58)

where

 ~λ=Tr(PJN†[I000]NJ), (59)

and is defined as in (35).

In order to prove this theorem, we need the following two lemmas.

###### Lemma 6

Consider an open quantum system where and , and the set of non-negative self-adjoint operators . For any , is a constant. If there exist real constants and such that

 −i[V,H1]+L(V)+τ2[V,z][z∗,V]+1γ2τ2zz∗+W≤~λ, (60)

then

 limsupT→∞1T∫T0⟨W(t)⟩dt≤~λ+δ1τ2+μμ∗/4+δ2,∀t≥0. (61)

Proof: Since and ,

 G(V)= −i[V,H1]+L(V)−i[V,z]w∗1+iw1[z∗,V] −i2μw∗2+i2w2μ∗. (62)

Also,

 0≤(τ[V,z]−iτw1)(τ[V,z]−iτw1)∗=τ2[V,z][z∗,V]+i[V,z]w∗1−iw1[z∗,V]+1τ2w1w∗1. (63)

Furthermore,

 0≤(12μ−iw2)(12μ−iw2)∗=14μμ∗−i2w2μ∗+i2μw∗2+w2w∗2. (64)

Substituting (63) and (64) into (4.2), the following inequality is obtained.

 G(V)≤ −i[V,H1]+L(V)+τ2[V,z][z∗,V]+1γ2τ2zz∗ +δ1τ2+14μμ∗+δ2. (65)

Hence, using (60), it follows that

 G(V)+W≤~λ+δ1τ2+μμ∗/4+δ2. (66)

Consequently, the conclusion in the lemma follows from Lemma 1.

###### Lemma 7

For and defined in (34),

 [z∗,V]=2~E#ΣJP[aa#], (67)
 [V,z][z∗,V]=4[aa#]†PJΣ~ET~E#ΣJP[aa#], (68)
 Missing or unrecognized delimiter for \left (69)

Proof: The proof follows from Lemma 3.
Proof of Theorem 2: Using the Schur complement, the inequality (57) is equivalent to

 F†P+PF+4τ2PJΣ~ET~E#ΣJP+Σ~ET~E#Σ/(γ2τ2)+R<0. (70)

If the Riccati inequality (70) has a solution in the form of (10) and , according to Lemma 3 and Lemma 7, we have

 −i[V,H1]+L(V)+τ2[V,z][z∗,V]+1γ2τ2zz∗+W= [aa#]†(F†P+PF+4τ2PJΣ~ET~E#ΣJP+Σ~ET~E#Σ/(γ2τ2)+R)[aa#] +Tr(PJN†[I000]NJ). (71)

Therefore, it follows from (57) that condition (60) will be satisfied with

 ~λ=Tr(PJN†[I000]NJ)≥0. (72)

According to the relationship (33) and Lemma 6, we have

 limsupT→∞1T∫T0⟨W(t)⟩dt=limsupT→∞1T∫T0⟨[a†aT]R[aa#]⟩dt≤~λ+δ1τ2+μμ∗/4+δ2. (73)

## 5 Coherent Guaranteed Cost Controller Design

In some applications, it is desirable to design a quantum control system which is not only stable but also guarantees an adequate level of performance. In this section, we design a coherent guaranteed cost controller for quantum systems subject to quadratic or non-quadratic Hamiltonian perturbations. The coherent controller is realized by adding a term to the nominal system Hamiltonian. Assume that the controller Hamiltonian is of the form

 H3=12[a†aT]K[aa#] (74)

where is a Hermitian matrix of the form

 K=[K1K2K#2K#1] (75)

and , . Associated with this system is the cost function

 J=limsupT→∞1T∫∞0([aa#]†(R+ρK2)[aa#])dt (76)

where is a weighting factor. We let

 W=[aa#]†(R+ρK2)[aa#]. (77)

The following sections present our main results on coherent guaranteed cost controller design for quantum systems subject to quadratic and non-quadratic Hamiltonian perturbations, respectively.

###### Theorem 3

Consider an uncertain quantum system , where , is in the form of (4), is of the form (7), and the controller Hamiltonian is in the form of (74). With , and , if there exist a matrix ( is a constant and is identity matrix), a Hermitian matrix and a constant , such that

 ⎡⎢ ⎢ ⎢ ⎢ ⎢⎣A+4τ2JE†EJYqR12qE†Y−I/ρ00qR120−I0qE00−γ2τ2I⎤⎥ ⎥ ⎥ ⎥ ⎥⎦<0 (78)

where , then the associated cost function satisfies the bound

 limsupT→∞1T∫T0⟨W(t)⟩dt=limsupT→∞1T∫∞0([aa#]†(R+ρK2)[aa#])dt≤~λ+δτ2 (79)

where

 ~λ=Tr(PJN†[I000]NJ). (80)

Proof: Suppose the conditions of the theorem are satisfied. Using the Schur complement, (78) is equivalent to

 ⎡⎢ ⎢ ⎢⎣A+4τ2JE†EJ+ρYYqR12qE†qR12−I0qE0−γ2τ2I⎤⎥ ⎥ ⎥⎦<0. (81)

Applying the Schur complement again, it is clear that (81) is equivalent to

 [A+4τ2JE†EJ+ρYY+q2RqE†qE−γ2τ2I]<0 (82)

and (82) is equivalent to

 Unknown environment '%' (83)

Substituting into (83), we obtain

 q(F−iJK)†+(F−iJK)q+4τ2JE†EJ+q2(E†Eγ2τ2+R+ρK2)<0. (84)

Since , premultiplying and postmultiplying this inequality by the matrix , we have

 (F−iJK)†P+P(F−iJK)+4τ2PJE†EJP+E†Eγ2τ2+R+ρK2<0. (85)

We know and . Hence

 4τ2PJE†EJP+E†Eγ2τ2+R+ρK2>0,(F−iJK)†P+P(F−iJK)<0. (86)

Therefore, we have the following fact

 ~F=−iJ(M+K)−12JN†JN is Hurwitz. (87)

We also know that

 −i[V,H1+H3]+L(V)+τ2[V,z†][z,V]+1γ2τ2z†z+W =[aa#]†⎛⎜ ⎜⎝(F−iJK)†P+P(F−iJK)+4τ2PJE†EJP+E†E/(γ2τ2)+R+ρK2⎞⎟ ⎟⎠[aa#] +Tr(PJN†[I000]NJ). (88)

According to the relationship (21) and Lemma 4, we have

 limsupT→∞1T∫T0⟨W(t)⟩dt=limsupT→∞1T∫∞0([aa#]†(R+ρK2)[aa#])dt≤~λ+δτ2 (89)

where

 ~λ=Tr(PJN†[I000]NJ). (90)

###### Remark 1

In order to design a coherent controller which minimizes the cost bound (79) in the above theorem, we need to formulate an inequality

 Tr(PJN†[I000]NJ)+δτ2≤ξ. (91)

We know that and apply Schur complement to inequality (91), so that we have

 ⎡⎢⎣−ξ+δτ2B12B12−q⎤⎥⎦≤0 (92)

where