Performance Analysis and Coherent Guaranteed Cost Control for Uncertain Quantum Systems Using Small Gain and Popov Methods

# Performance Analysis and Coherent Guaranteed Cost Control for Uncertain Quantum Systems Using Small Gain and Popov Methods

## Abstract

This paper extends applications of the quantum small gain and Popov methods from existing results on robust stability to performance analysis results for a class of uncertain quantum systems. This class of systems involves a nominal linear quantum system and is subject to quadratic perturbations in the system Hamiltonian. Based on these two methods, coherent guaranteed cost controllers are designed for a given quantum system to achieve improved control performance. An illustrative example also shows that the quantum Popov approach can obtain less conservative results than the quantum small gain approach for the same uncertain quantum system.

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## 1 Introduction

Due to recent advances in quantum and nano-technology, there has been a considerable attention focusing on research in the area of quantum feedback control systems; e.g., [1]-[24]. In particular, robust control has been recognized as a critical issue in quantum control systems, since many quantum systems are unavoidably subject to disturbances and uncertainties in practical applications; e.g.,[1], [2] and [3]. A majority of papers in the area of quantum feedback control only consider the case in which the controller is a classical system. In this case, analog and digital electronic devices may be involved and quantum measurements are required in the feedback loop. Due to the limitations imposed by quantum mechanics on the use of quantum measurement, recent research has considered the design of coherent quantum controllers to achieve improved performance. In this case, the controller itself is a quantum system; e.g., [1], [4] and [5]. In the linear case, the quantum system is often described by linear quantum stochastic differential equations (QSDEs) that require physical realizability conditions in terms of constraints on the system matrices to represent physically meaningful systems.

As opposed to using QSDEs, the papers [6],[7] have introduced a set of parameterizations to represent a class of open quantum systems, where is a scattering matrix, is a vector of coupling operators and is a Hamiltonian operator. The matrix , together with the vector , describes the interface between the system and the field, and the operator represents the energy of the system. The advantage of using a triple is that this framework automatically represents a physically realizable quantum system. Therefore, in this paper, a coherent guaranteed cost controller is designed based on and the physical realizability condition does not need to be considered.

The small gain theorem and the Popov approach are two of the most important methods for the analysis of robust stability in classical control. The paper [8] has applied the small gain method to obtain a robust stability result for uncertain quantum systems. This result gives a sufficient condition for robust stability in terms of a strict bounded real condition. The small gain method has also been extended to the robust stability analysis of quantum systems with different perturbations and applications (e.g., see [9], [10], [11] and [12]). The paper [13] has introduced a quantum version of the Popov stability criterion in terms of a frequency domain condition which is of the same form as the classical Popov stability condition (e.g., see [25]). The Popov approach has also been used to analyze the robust stability of an optical cavity containing a saturated Kerr medium [14]. Also, the paper [13] has shown that the frequency domain condition obtained in [13] is less conservative than the stability result using the small gain theorem [8].

In this paper, we extend the quantum small gain method in [8] and the Popov type approach in [13] from robust stability analysis to robust performance analysis for uncertain quantum systems. We assume that the system Hamiltonian can be decomposed in terms of a known nominal Hamiltonian and a perturbation Hamiltonian , i.e., . The perturbation Hamiltonian is contained in a set of Hamiltonians . We consider uncertain quantum systems where the nominal system is a linear system and the perturbation Hamiltonian is quadratic. Moreover, a coherent controller is designed using the small gain approach and the Popov approach for the uncertain quantum system, where a guaranteed bound on a cost function is derived in terms of linear matrix inequality (LMI) conditions. Although preliminary versions of the results in this paper have been presented in the conference papers [22] and [23], this paper presents complete proofs of the main results and modifies the example in [22] for a consistent performance comparison of the proposed two methods.

The remainder of the paper proceeds as follows. In Section 2, we define the general class of quantum systems under consideration and specify the nominal system as a linear quantum system. We then present a set of quadratic Hamiltonian perturbations in Section 3. In Section 4, a performance cost function for the given system is defined. Moreover, a small gain approach and a Popov type method are used to analyze the performance of the given system. In Section 5, a quantum controller is added to the original system to stabilize the system and also to achieve improved performance. Also, corresponding approaches are used to construct a coherent guaranteed cost controller in terms of LMI conditions. In Section 6, an illustrative example is provided to demonstrate the method that is developed in this paper. A performance comparison between these two methods is also shown in the illustrative example. We present some conclusions in Section 7.

## 2 Quantum Systems

In this section, we describe the general class of quantum systems under consideration, which is defined by parameters . Here , is a known self-adjoint operator on the underlying Hilbert space referred to as the nominal Hamiltonian and is a self-adjoint operator on the underlying Hilbert space referred to as the perturbation Hamiltonian contained in a specified set of Hamiltonians ; e.g.,[6], [7]. The set can correspond to a set of exosystems (see, [6]). The corresponding generator for this class of quantum systems is given by

 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. Based on a quantum stochastic differential equation, the triple , together with the corresponding generators, defines the Heisenberg evolution of an operator [6]. The results presented in this paper will build on the following results from [13].

###### Lemma 1

[13] 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 [13] and [6].

In this paper, the nominal system is considered to be a linear quantum system. 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 canonical commutation relations between annihilation and creation operators are described in the following way

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

where [2].

The coupling vector is assumed to be of the form

 L=[N1N2][laa#]=~N[aa#] (7)

where and . We also write

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

When the nominal Hamiltonian is a quadratic function of the creation and annihilation operators as shown in (4) and the coupling operator vector is a linear function of the creation and annihilation operators, the nominal system corresponds to a linear quantum system (see, [2]).

We consider 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 a 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

In Section 2, we introduced the nominal linear quantum system. This section defines the quadratic Hamiltonian perturbations (e.g., see [8], [18]) for the quantum system under consideration. We first define two general sets of Hamiltonians in terms of a commutator decomposition, and then present two specific sets of quadratic Hamiltonian perturbations.

### 3.1 Commutator Decomposition

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

### 3.2 Alternative Commutator Decomposition

Given a set of non-negative operators , a self-adjoint operator , a coupling operator , real parameters , and a set of Popov scaling parameters , we define a set of perturbation Hamiltonians in terms of the commutator decompositions [13]

 [V−θH1,H2]=[V−θH1,z†]w−w†[z,V−θH1],L(H2)≤L(z†)w+w†L(z)+β[z,L]†[z,L] (15)

for and , where and are given vectors of operators. Note that (12) and (15) correspond to a general quadratic perturbation of the Hamiltonian. This set is then defined in terms of the sector bound condition

 (w−1γz)†(w−1γz)≤1γ2z†z. (16)

We define

 Unknown environment '% (17)

### 3.3 Quadratic Hamiltonian Perturbation

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

 Unknown environment 'array% (18)

where and is a Hermitian matrix of the form

 Δ=[Δ1Δ2Δ#2Δ#1] (19)

with and .

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

 z=[ζζ#]=[E1E2E#2E#1][aa#]=E[aa#]. (20)

Then

 H2=12[a†aT]E†ΔE[aa#]. (21)

When the matrix is subject to the norm bound

 ∥Δ∥≤2γ, (22)

where refers to the matrix induced norm, we define

 W3={H2 of the form (???) and (???) such that condition (???) is satisfied}. (23)

In [8], it has been proven that for any set of self-adjoint operators ,

 W3⊂W1. (24)

When the matrix is subject to the bounds

 0≤Δ≤4γI, (25)

we define

 W4={H2 of the form (???) and (???) such that condition (???) is satisfied}. (26)

In [13], it has been proven that if is a constant vector, then for any set of self-adjoint operators ,

 W4⊂W2. (27)

## 4 Performance Analysis

In this section, we provide several results on performance analysis for quantum systems subject to a quadratic perturbation Hamiltonian. Also, the associated cost function is defined in the following way:

 J=limsupT→∞1T∫T0⟨[a†aT]R[aa#]⟩dt (28)

where . We denote

 Unknown environment '% (29)

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

###### Lemma 2

(See Lemma 4 of[8]) Suppose is of the form (4) and is of the form (7). Then

 [V,H1]=[aa#]†(PJM−MJP)[aa#], (30)
 L(V)= −12[aa#]†(N†JNJP+PJN†JN)[aa#] +Tr(PJN†[I000]NJ), (31)
 [[aa#],[aa#]†P[aa#]]=2JP[aa#]. (32)
###### Lemma 3

For and defined in (20),

 [z,V]=2EJP[aa#], (33)
 [V,z†][z,V]=4[aa#]†PJE†EJP[aa#], (34)
 z†z=[aa#]†E†E[aa#]. (35)

Proof: The proof follows from Lemma 2.

###### Lemma 4

(See Lemma 5 of [13]) For defined in (20) and being of the form (7),

 [z,L]=[E[aa#],~N[aa#]]=EJΣ~NT (36)

is a constant vector, where

 Σ=[0II0]. (37)
###### Lemma 5

(See Lemma 6 of [13]) For defined in (20), defined in (4) and being of the form (7), we have

 Unknown environment 'array% (38)

and

 i[z,V]=2iEJP[aa#]. (39)

Now we present two theorems (Theorem 1 and Theorem 2) which can be used to analyze the performance of the given quantum systems using a quantum small gain method and a Popov type approach, respectively.

### 4.1 Performance Analysis Using the Small Gain Approach

###### Theorem 1

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

 ⎡⎣F†P+PF+E†Eγ2τ2+R2PJE†2EJP−I/τ2⎤⎦<0 (40)

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 (41)

where

 ~λ=Tr(PJN†[I000]NJ). (42)

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

###### Lemma 6

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, (43)

then

 limsupT→∞1T∫T0⟨W(t)⟩dt≤~λ+δτ2,∀t≥0. (44)

Proof: Since and ,

 G(V)=−i[V,H1]+L(V)−i[V,z†]w+iw†[z,V]. (45)

Also,

 0≤(τ[V,z†]−iτw†)(τ[V,z†]−iτw†)†=τ2[V,z†][z,V]+i[V,z†]w−iw†[z,V]+w†wτ2. (46)

Substituting (45) into (46) 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. (47)

Hence,

 G(V)+W≤~λ+δτ2. (48)

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

Proof of Theorem 1: Using the Schur complement [26], the inequality (40) is equivalent to

 F†P+PF+4τ2PJE†EJP+E†Eγ2τ2+R<0. (49)

If the Riccati inequality (49) has a solution of the form (10) and , according to Lemma 2 and Lemma 3, we have

 Unknown environment 'array% (50)

Therefore, it follows from (40) that condition (43) is satisfied with

 ~λ=Tr(PJN†[I000]NJ)≥0. (51)

Then, according to the relationship (24) and Lemma 6, we have

 limsupT→∞1T∫T0⟨W(t)⟩dt=limsupT→∞1T∫T0⟨[a†aT]R[aa#]⟩dt≤~λ+δτ2. (52)

### 4.2 Performance Analysis Using the Popov Approach

###### Theorem 2

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

 [PF+F†P+R−2iPJE†+E†+θF†E†2iEJP+E+θEF−γI]<0 (53)

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

 J =limsupT→∞1T∫T0⟨W(t)⟩dt =limsupT→∞1T∫T0⟨[a†aT]R[aa#]⟩dt≤λ (54)

where

 λ=Tr(PJN†[I000]NJ)+4θγ~N#ΣJE†EJΣ~NT. (55)

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

###### Lemma 7

(See Theorem 1 of [13]) Consider a set of non-negative self-adjoint operators , an open quantum system and an observable , where and defined in (17). Suppose there exists a and real constants , such that

 −i[V,H1]+L(V)+1γ(i[z,V−θH1]+θL(z)+z)† ×(i[z,V−θH1]+θL(z)+z)+θβ[z,L]†[z,L]+W≤λ. (56)

Then

 limsupT→∞1T∫T0⟨W(t)⟩dt≤λ. (57)

Here denotes the Heisenberg evolution of the operator and denotes quantum expectation.

Proof of Theorem 2: Using the Schur complement, (53) is equivalent to

 PF+F†P+1γ(−2iPJE†+E†+θF†E†)×(2iEJP+E+θEF)+R<0. (58)

According to Lemma 2 and Lemma 5, we have

 −i[V,H1]+L(V)+1γ(i[z,V−θH1]+θL(z)+z)† ×(i[z,V−θH1]+θL(z)+z)+4θγ[z,L]†[z,L]+W =[aa#]†⎛⎜ ⎜⎝PF+F†P+1γ(−2iPJE†+E†+θF†E†)×(2iEJP+E+θEF)+R⎞⎟ ⎟⎠[aa#] +Tr(PJN†[I000]NJ)+4θγ~N#ΣJE†EJΣ~NT. (59)

From this and using the relationship (27), Lemma 4 and Lemma 7, we obtain

 limsupT→∞1T∫T0⟨W(t)⟩dt=limsupT→∞1T∫∞0⟨[aa#]†R[aa#]⟩dt≤λ (60)

where

 λ=Tr(PJN†[I000]NJ)+4θγ~N#ΣJE†EJΣ~NT. (61)

## 5 Coherent Guaranteed Cost Controller Design

In this section, we design a coherent guaranteed cost controller for the uncertain quantum system subject to a quadratic perturbation Hamiltonian to make the control system not only stable but also to achieve an adequate level of performance. The coherent controller is realized by adding a controller Hamiltonian . is assumed to be in the form

 H3=12[a†aT]K[aa#] (62)

where is a Hermitian matrix of the form

 K=[K1K2K#2K#1] (63)

and , . Associated with this system is the cost function

 J=limsupT→∞1T∫∞0⟨[aa#]†(R+ρK2)[aa#]⟩dt (64)

where is a weighting factor. We let

 W=[aa#]†(R+ρK2)[aa#]. (65)

The following theorems (Theorem 3 and Theorem 4) present our main results on coherent guaranteed cost controller design for the given quantum system using a quantum small gain method and a Popov type approach, respectively.

### 5.1 Coherent Controller Design Using the Small Gain Approach

###### 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 (62). With , and , if there exist a matrix ( is a constant scalar and is the identity matrix), a Hermitian matrix and a constant , such that

 ⎡⎢ ⎢ ⎢ ⎢ ⎢⎣A+4τ2JE†EJYqR12qE†Y−I/ρ00qR120−I0qE00−γ2τ2I⎤⎥ ⎥ ⎥ ⎥ ⎥⎦<0 (66)

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 (67)

where

 ~λ=Tr(PJN†[I000]NJ). (68)

Proof: Suppose the conditions of the theorem are satisfied. Using the Schur complement, (66) is equivalent to

 ⎡⎢ ⎢ ⎢⎣A+4τ2JE†EJ+ρYYqR12qE†qR12−I0qE0−γ2τ2I⎤⎥ ⎥ ⎥⎦<0. (69)

Applying the Schur complement again, it follows that (69) is equivalent to

 [A+4τ2JE†EJ+ρYY+q2RqE†qE−γ2τ2I]<0 (70)

and (70) is equivalent to

 Unknown environment '%' (71)

Substituting into (71), we obtain

 q(F−iJK)†+(F−iJK)q+4τ2JE†EJ+q2(E†Eγ2τ2+R+ρK2)<0. (72)

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

 (F−iJK)†P+P(F−iJK)+4τ2PJE†EJP+E†Eγ2τ2+R+ρK2<0. (73)

It follows straightforwardly from (73) that is Hurwitz. 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). (74)

According to the relationship (24) and Lemma 6, we have

 limsupT→∞1T∫T0⟨W(t)⟩dt=limsupT→∞1T∫∞0⟨[aa#]†(R+ρK2)[aa#]⟩dt≤~λ+δτ2 (75)

where

 ~λ=Tr(PJN†[I000]NJ). (76)

###### Remark 1

In order to design a coherent controller which minimizes the cost bound (67) in Theorem 3, we need to formulate an inequality

 Tr(PJN†[I000]NJ)+δτ2≤ξ. (77)

We know that and apply the Schur complement to inequality (77), so that we have

 ⎡⎢⎣−ξ+δτ2B12B12−q⎤⎥⎦≤0 (78)

where Applying the Schur complement again, it is clear that (78) is equivalent to

 ⎡⎢ ⎢ ⎢ ⎢⎣−ξδ12B12δ12−τ20B120−q⎤⎥ ⎥ ⎥ ⎥⎦≤0. (79)

Hence, we minimize subject to (79) and (66) in Theorem 3. This is a standard LMI problem.

### 5.2 Coherent Controller Design Using the Popov Approach

###### Theorem 4

Consider an uncertain quantum system , where , is in the form of (4), is of the form (7), , the controller Hamiltonian is in the form of (62). With , and , if there exist a matrix ( is a constant scalar and is the identity matrix), a Hermitian matrix and a constant , such that

 ⎡⎢ ⎢ ⎢ ⎢ ⎢⎣AB†YqR12B−γI00Y0−I/ρ0qR1200−I⎤⎥ ⎥ ⎥ ⎥ ⎥⎦<0 (80)

where and , then the associated cost function satisfies the bound

 limsupT→∞1T∫T0⟨W(t)⟩dt=limsupT→∞1T∫∞0⟨[aa#]†(R+ρK2)[aa#]⟩dt≤λ (81)

where

 λ=Tr(PJN†[I000]NJ)+4θγ~N#ΣJE†EJΣ~NT. (82)

Proof: Suppose the conditions of the theorem are satisfied. Using the Schur complement, (80) is equivalent to

 ⎡⎢ ⎢ ⎢⎣A+1γB†BYqR12Y−I/ρ0qR120−I⎤⎥ ⎥ ⎥⎦<0. (83)

We then apply the Schur complement to inequality (83) and obtain

 ⎡⎢⎣A+1γB†B+ρYYqR12qR12−I⎤⎥⎦<0. (84)

Also, (84) is equivalent to

 Fq+qF†−iJY+iYJ+1γ(−2iJE†+qE†+θqF†E†+iθYJE†)×(2iEJ+Eq+θEFq−iθEJY)+q2R+ρYY<0. (85)

Substituting into (85), we obtain

 (F−iJK)q+q(F−iJK)†+1γ(−2iJE†+qE†+θq(F−iJK)†E†)×(2iEJ+Eq+θE(F−iJK)q)+q2(R+ρK2)<0. (86)

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

 P(F−iJK)+(F−iJK)†P+1γ(−2iPJE†+E†+θ(F−iJK)†E†)×(2iEJP+E+θE(F−iJK))+R+ρK2<0. (87)

It follows straightforwardly from (87) that is Hurwitz. We also know that

 −i[V,H1+H3]+L(V) +1γ(i[z,V−θ(H1+H3)]+θL(z)+z)† ×(i[z,V−θ(H1+H3)]+θL(z)+z) +4θγ[z,L]†[z,L]+W =[a