Guaranteed Non-quadratic Performance for Quantum Systems with Nonlinear Uncertainties

# Guaranteed Non-quadratic Performance for Quantum Systems with Nonlinear Uncertainties

Ian R. Petersen This work was supported by the Australian Research Council (ARC) and the Air Force Office of Scientific Research (AFOSR). This material is based on research sponsored by the Air Force Research Laboratory, under agreement number FA2386-12-1-4075. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory or the U.S. Government. Ian R. Petersen is with the School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra ACT 2600, Australia. i.r.petersen@gmail.com
###### Abstract

This paper presents a robust performance analysis result for a class of uncertain quantum systems containing sector bounded nonlinearities arising from perturbations to the system Hamiltonian. An LMI condition is given for calculating a guaranteed upper bound on a non-quadratic cost function. This result is illustrated with an example involving a Josephson junction in an electromagnetic cavity.

## I Introduction

A number of papers have considered in recent years, the feedback control of systems governed by the laws of quantum mechanics rather than systems governed by the laws of classical mechanics; e.g., see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. In particular, the papers [10, 20] consider a framework of quantum systems defined in terms of a triple where is a scattering matrix of operators, is a vector of coupling operators and is a Hamiltonian operator. All operators are on an underlying Hilbert space.

The paper [21] considers a quantum system defined by a triple such that the quantum system Hamiltonian is written as . Here is a known nominal Hamiltonian and is a perturbation Hamiltonian, which is contained in a set of Hamiltonians . The paper [21] considers a problem of absolute stability for such uncertain quantum systems for the case in which the nominal Hamiltonian is a quadratic function of annihilation and creation operators and the coupling operator vector is a linear function of annihilation and creation operators. Such as nominal quantum system is said to be a linear quantum system; e.g., see [4, 5, 7, 8, 14]. However, the perturbation Hamiltonian is assumed to be contained in a set of non-quadratic Hamiltonians corresponding to a sector bounded nonlinearity. Then, the paper [21] obtains a frequency domain robust stability result. Extensions of the approach of [21] can be found in the papers [22, 23, 24, 25, 26, 27, 28] in which similar robust stability results are of obtain for uncertain quantum systems with different classes of uncertainty and different applications to specific quantum systems. Also, in the paper [24] a problem of robust performance analysis as well as robust stability analysis is considered.

In this paper, we extend the results of [21, 24, 25] by considering a problem of robust performance analysis with a non-quadratic cost functional for the class of uncertain quantum systems of the form considered in [21, 25]. The motivation for considering robust performance of a quantum system with a non-quadratic cost function arises from the fact that the presence of nonlinearities in the quantum system allows for the possibility of a non-Gaussian system state; e.g., see [29]. Such non-Gaussian system states include important non-classical states such as the Schrödinger cat state (also known as a superposition state, e.g., see [30]). These non-classical quantum states are useful in areas such as quantum information and quantum communications; e.g., see [31]. The presence of such non-classical states can be verified by obtaining a suitable bound on a non-quadratic cost function (such as the Wigner function, e.g., see [29, 30]). Our approach to obtaining a bound on the non-quadratic cost function is to extend the sector bound method considered in [21] to bound both the nonlinearity and non-quadratic cost function together. It is important that these two quantities are bounded together since the non-Gaussian state only arises due to the presence of the nonlinearity in the quantum system dynamics. Then, by applying a similar approach to that in [21, 24] we are able to derive a guaranteed upper bound on the non-quadratic cost function in terms of an LMI problem. In order to illustrate this result, it is applied to an example of a quantum system consisting of a Josephson junction in an electromagnetic cavity. The robust stability of a similar system was previously considered in the paper [25]. In this paper, we consider the robust performance of this system with respect to a non-quadratic cost functional.

A future application of the robust performance analysis approach proposed in this paper would be to use it to develop a method for the design of coherent quantum feedback controllers for quantum systems to achieve a certain closed loop performance bound in terms of a non-quadratic cost functional. In such a coherent quantum feedback control scheme both the plant and controller are quantum systems; e.g., see [5]. This would be useful in the generation of non-classical quantum states which are needed in areas of quantum computing and quantum information; e.g., see [31].

## Ii Quantum Systems with Nonlinear Uncertainties

The parameters will be considered to define an uncertain nonlinear quantum system. Here, is the scattering matrix, which is chosen as the identity matrix, L is the coupling operator vector and is the system Hamiltonian operator. is assumed to be of the form

 H=12[a†aT]M[aa#]+f(z,z∗). (1)

Here, is an -dimensional vector of annihilation operators on the underlying Hilbert space and is the corresponding vector of creation operators. Also, is a Hermitian matrix of the form

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

and , . In the case of vectors of operators, the notation refers to the transpose of the vector of adjoint operators and in the case of matrices, this 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. Also, the notation denotes the adjoint of an operator. The matrix is assumed to be known and defines the nominal quadratic part of the system Hamiltonian. Furthermore, we assume the uncertain non-quadratic part of the system Hamiltonian is defined by a formal power series of the form

 f(z,z∗) = ∞∑k=0∞∑ℓ=0Skℓzk(z∗)ℓ (3) = ∞∑k=0∞∑ℓ=0SkℓHkℓ,

which is assumed to converge in some suitable sense. Here , , and is a known scalar operator defined by

 z = E1a+E2a# (4) = [E1E2][aa#]=~E[aa#];

i.e., the vector is a known complex vector.

The term is referred to as the perturbation Hamiltonian. It is assumed to be unknown but is contained within a known set which will be defined below.

We assume the coupling operator vector is known and is of the form

 L=[N1N2][aa#]. (5)

Here, , are known matrices. Also, we write

 [LL#] = N[aa#] = [N1N2N#2N#1][aa#].

The annihilation and creation operators and are assumed to satisfy the canonical commutation relations:

 [[aa#],[aa#]†] \lx@stackrelΔ= [aa#][aa#]† (6) −([aa#]#[aa#]T)T = J

where ; e.g., see [6, 11, 14].

Also, we will consider a non-quadratic cost defined as

 C=limsupT→∞1T∫T0⟨W(z(t),z(t)∗)⟩dt (7)

where is a suitable non-quadratic function. Here , , denotes the Heisenberg evolution of the operators , and denotes quantum expectation; e.g., see [20]. The non-quadratic function is assumed to satisfy the following quadratic upper bound condition:

 W(z,z∗)≤1γ20zz∗+δ0, (8)

where , are given constants. will also be used in the definition of the set of allowable perturbation Hamiltonians .

To define the set of allowable perturbation Hamiltonians , we first define the following formal partial derivatives:

 ∂f(z,z∗)∂z\lx@stackrelΔ=∞∑k=1∞∑ℓ=0kSkℓzk−1(z∗)ℓ; (9)
 ∂2f(z,z∗)∂z2\lx@stackrelΔ=∞∑k=1∞∑ℓ=0k(k−1)Skℓzk−2(z∗)ℓ. (10)

and for given constants , , , , , we consider the sector bound conditions

 W(z,z∗)+∂f(z,z∗)∂z∗∂f(z,z∗)∂z≤1γ21zz∗+δ1, (11)
 ∂f(z,z∗)∂z∗∂f(z,z∗)∂z≤1γ22zz∗+δ2 (12)

and the condition

 ∂2f(z,z∗)∂z2∗∂2f(z,z∗)∂z2≤δ3. (13)

Then we define the set of perturbation Hamiltonians as follows:

 W={f(⋅) of the form (???) such that  conditions (???), (???) and (???) are % satisfied}. (14)

Note that the condition (13) effectively amounts to a global Lipschitz condition on the quantum nonlinearity.

Our main result, which gives an upper bound on the non-quadratic cost function (7), will be given in terms of the following LMI condition dependent on a parameter :

 ⎡⎢⎣F†P+PF+κΣ~ET~E#Σ2PJΣ~ET2~E#ΣJP−Iτ21⎤⎥⎦<0 (15)

where , and the quantity is defined as

 κ=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩1γ21+(1τ21−1) for τ21≤1;1τ21γ21+1γ20(1−1τ21) for τ21>1.
###### Theorem 1

Consider an uncertain open nonlinear quantum system defined by and a non-quadratic cost function such that is of the form (1), is of the form (5) and . Also, assume that defined in (7) is such that (8) is satisfied. Furthermore, assume that there exists a constant such that the LMI (15) has a solution . Then the cost satisfies the bound:

 C≤Tr(PJN†[I000]NJ)+ζ+√δ3|μ| (16)

where

 ζ=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩δ1+(1τ21−1)δ2 for τ21≤1;1τ21δ1+(1−1τ21)δ0 for τ21>1

and

 μ=−~EΣJPJ~ET. (17)

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

###### Lemma 1 (See Lemma 2 of [24])

Consider an open quantum system defined by and suppose there exists a non-negative self-adjoint operator on the underlying Hilbert space such that

 −ı[V,H]+12L†[V,L]+12[L†,V]L+W(z,z∗)≤λ (18)

where and are real numbers. Then for any system state, we have

 limsupT→∞1T∫T0⟨W(t)⟩dt≤λ.

We will consider quadratic “Lyapunov” operators of the form

 (19)

where is a positive-definite Hermitian matrix of the form

 P=[P1P2P#2P#1]. (20)

Hence, we consider a set of non-negative self-adjoint operators defined as

 P={V of the form (???) such that P>0 is a  Hermitian matrix of the form (???)}. (21)
###### Lemma 2 (See Lemma 5 in [21])

Given any , then

 [z,[z,V]]=[z∗,[z∗,V]]∗=μ (22)

where the constant is defined as in (17).

###### Lemma 3 (See Lemma 3 in [27] and Lemma 2 in [28])

Given any , then

 [V,f(z,z∗)] = [V,z]w∗1−w1[z∗,V] (23) +12μw∗2−12w2μ∗

where

 w1 = =∂f(z,z∗)∂z∗, w2 = =∂2f(z,z∗)∂z2∗,

and the constant is defined as in (17).

###### Lemma 4 (See Lemma 4 in [27])

Given and defined as in (5), then

 = [aa#]†[PJM−MJP][aa#].

Also,

 12L†[V,L]+12[L†,V]L= = Tr(PJN†[I000]NJ) −12[aa#]†(N†JNJP+PJN†JN)[aa#].

Furthermore,

 [[aa#],[a†aT]P[aa#]]=2JP[aa#].

Proof of Theorem 1. It follows from (4) that we can write

 z∗ = E#1a#+E#2a=[E#2E#1][aa#] = ~E#Σ[aa#].

Also, it follows from Lemma 4 that

 [z∗,V]=2~E#ΣJP[aa#].

Furthermore, and hence,

 [V,z][z∗,V]=4[aa#]†PJΣ~ET~E#ΣJP[aa#]. (25)

Also, we can write

 (26)

Hence using Lemma 4, we obtain

 −ı[V,12[a†aT]M[aa#]] (27) +12L†[V,L]+12[L†,V]L+τ21[V,z][z∗,V]+κzz∗ = [aa#]†⎛⎜ ⎜⎝F†P+PF+4τ21PJΣ~ET~E#ΣJP+κΣ~ET~E#Σ⎞⎟ ⎟⎠[aa#] +Tr(PJN†[ccI000]NJ)

where .

We now observe that applying the Schur complement to the LMI (15) implies that the matrix inequality

 F†P+PF+4τ21PJΣ~ET~E#ΣJP+κΣ~ET~E#Σ<0. (28)

will have a solution of the form (20). This matrix defines a corresponding operator as in (19). From this, it follows using (27) that

 −ı[V,12[a†aT]M[aa#]] +12L†[V,L]+12[L†,V]L+τ21[V,z][z∗,V] +κzz∗≤~λ

with

 ~λ=Tr(PJN†[[]ccI000]NJ)≥0.

Also, it follows from Lemma 3 that

 −ı[V,H]+12L†[V,L]+12[L†,V]L+W(z,z∗) (30) = −ı[V,f(z,z∗)]−ı[V,12[a†aT]M[aa#]] +12L†[V,L]+12[L†,V]L+W(z,z∗) = −ı[V,12[a†aT]M[aa#]] +12L†[V,L]+12[L†,V]L+W(z,z∗) −ı[V,z]w∗1+ıw1[z∗,V] −12ıμw∗2+12ıw2μ∗.

Furthermore, since is self-adjoint. Therefore, for

 0 ≤ (τ1[V,z]−1τ1ıw1)(τ1[V,z]−1τ1ıw1)∗ = τ21[V,z][z∗,V]+ı[V,z]w∗1 −ıw1[z∗,V]+1τ21w1w∗1

and hence

 −ı[V,z]w∗1+ıw1[z∗,V] (31) ≤ τ21[V,z][z∗,V]+1τ21w1w∗1.

Also, for

 0 ≤ (τ22μ−1τ2ıw2)(τ22μi−1τ2ıw2i)∗ = τ224μμ∗−ı2w2μ∗+ı2μw∗2 +1τ22w2w∗2

and hence

 ı2w2μ∗−ı2μw∗2 (32) ≤ τ224μμ∗+1τ22w2w∗2.

Also, it follows from (13) that

 w2w∗2≤δ3. (33)

If we let , it follows from (32) and (33) that

 ı2w2μ∗−ı2μw∗2≤12√δ3|μ|+12√δ3|μ|=√δ3|μ|. (34)

Furthermore, it follows from (11) and (12) that

 W(z,z∗)+w1w∗1≤1γ21zz∗+δ1 (35)

and

 w1w∗1≤1γ22zz∗+δ2. (36)

Combining these equations with (8), it follows that

 W(z,z∗)+1τ21w1w∗1 W≤⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩1γ21zz∗+δ1+(1τ21−1)(1γ22zz∗+δ2) for τ21≤1;1τ21(1γ21zz∗+δ1)+(1−1τ21)(1γ20zz∗+δ0) for τ21>1.

Substituting (31), (34), and (35) into (30), it follows that

 −ı[V,H]+12L†[V,L]+12[L†,V]L+W(z,z∗) (38) ≤ −ı[V,12[a†aT]M[aa#]] +12L†[V,L]+12[L†,V]L +τ21[V,z][z∗,V] +W(z,z∗)+1τ21w1w∗1+√δ3|μ|.

Hence, if , it follows from (II) that

 −ı[V,H]+12L†[V,L]+12[L†,V]L+W(z,z∗) −≤−ı[V,12[a†aT]M[aa#]] −=+12L†[V,L]+12[L†,V]L+τ21[V,z][z∗,V] −=+(1γ21+(1τ21−1))zz∗ −=+δ1+(1τ21−1)δ2+√δ3|μ|. (39)

Similarly, if , it follows from (II) that

 −ı[V,H]+12L†[V,L]+12[L†,V]L+W(z,z∗) −≤−ı[V,12[a†aT]M[aa#]] −=+12L†[V,L]+12[L†,V]L+τ21[V,z][z∗,V] −=+(1τ21γ21+1γ20(1−1τ21))zz∗ −=+1τ21δ1+(1−1τ21)δ0+√δ3|μ|. (40)

Hence,

 −ı[V,H]+12L†[V,L]+12[L†,V]L+W(z,z∗) −≤−ı[V,12[a†aT]M[aa#]] −=+12L†[V,L]+12[L†,V]L+τ21[V,z][z∗,V] −=+κzz∗ −=+ζ+√δ3|μ| (41)

where is defined in (II) and is defined in (1). Then it follows from (II) that

 −ı[V,H]+12L†[V,L]+12[L†,V]L+W(z,z∗) −≤~λ+ζ+√δ3|μ|.

From this, it follows from Lemma 1 with that the bound (16) is satisfied.

Note that the problem of minimizing the bound on the right hand side of (16) subject to the constraint (15) can be converted into a standard LMI optimization problem which can be solved using standard LMI software; e.g., see [32, 33].

## Iii Illustrative Example

To illustrate the main result of this paper, we consider an illustrative example consisting of a Josephson junction in an electromagnetic resonant cavity. This system was considered in the paper [25] using a model derived from a model presented in [34]. The system is illustrated in Figure 1.

In the paper [25], a model for this system of the form considered in Section II is derived and we consider the same model but with simplified parameter values for the purposes of this illustration. That is, we consider a Hamiltonian of the form (1) where

 M=⎡⎢ ⎢ ⎢⎣100001−0.500−0.5100000⎤⎥ ⎥ ⎥⎦

and

 f(z,z∗)=−cos(z+z∗)

where . Hence,

 ~E=[01√200].

Also, we consider a coupling operator vector of the form (5)

 L=[4a14a2].

In addition, we consider a non-quadratic cost function of the form (7) where

 W(z,z∗)=4zz∗−sin2(z+z∗)≤4zz∗.

Hence, we can set and in (8). A plot of the function versus for a real scalar is shown in Figure 2.

Furthermore, we calculate

 ∂f(z,z∗)∂z = sin(z+z∗) ∂2f(z,z∗)∂z2 = cos(z+z∗).

From this it follows that

 W(z,z∗)+∂f(z,z∗)∂z∗∂f(z,z∗)∂z W=4zz∗,

and hence, (11) is satisfied with and . Also,

 ∂f(z,z∗)∂z∗∂f(z,z∗)∂z=sin2(z+z∗)≤4zz∗,

and hence, (12) is satisfied with and . Moreover,

 ∂2f(z,z∗)∂z2∗∂2f(z,z∗)∂z2=cos2(z+z∗)≤1,

and hence (13) is satisfied with .

We now apply Theorem 1 to find a bound on the cost (7). This is achieved by solving the corresponding LMI optimization problem. In this case a solution to the LMI problem is found with

 P=⎡⎢ ⎢ ⎢⎣0.01200−0.000600.75−0.000600−0.00060.0120−0.0006000.75⎤⎥ ⎥ ⎥⎦

and . This leads to a cost bound (16) of .

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