# Strict dissipativity for generalized linear-quadratic problems in infinite
dimensions

Lars Grüne1, Friedrich Philipp2, and Manuel Schaller2

###### Abstract.

We analyze strict dissipativity of generalized linear quadratic optimal
control problems on Hilbert spaces. Here, the term “generalized” refers to
cost functions containing both quadratic and linear terms. We characterize
strict pre-dissipativity with a quadratic storage function via coercivity of a
particular Lyapunov-like quadratic form. Further, we show that under an
additional algebraic assumption, strict pre-dissipativity can be strengthened
to strict dissipativity. Last, we relate the obtained characterizations of
dissipativity with exponential detectability.

Keywords. Optimal control, strict dissipativity, infinite-dimensional
problems, linear-quadratic problems

1Department of Mathematics, Faculty of Mathematics, Physics and Computer
Sciences, University of Bayreuth, Bayreuth, Germany lars.gruene@uni-
bayreuth.de

2Institute for Mathematics, Faculty of Mathematics and Natural Sciences,
Technische Universität Ilmenau, Ilmenau, Germany {friedrich.philipp,
<EMAIL_ADDRESS>
Acknowledgments: F. Philipp was funded by the Carl Zeiss Foundation within the
project DeepTurb—Deep Learning in und von Turbulenz. M. Schaller and L. Grüne
were funded by the DFG (project numbers 289034702 and 430154635 for MS and
244602989 for LG).

##  1\. Introduction

The notions dissipativity and strict dissipativity of dynamical systems,
introduced by Jan Willems in the seminal works [20, 21], are central in
analyzing the behavior of dynamical systems as they ensure an energy balance
for trajectories in terms of a storage function and a supplied energy. Whereas
dissipativity states that the stored energy can not increase by more than the
supplied energy, strict dissipativity as its strengthened version includes an
additional dissipation term.

While passivity, a particular case of dissipativity has been well-established
for decades as a foundation for designing stabilizing controllers [3, 19], the
role of dissipativity for analyzing economic model predictive control (MPC),
where the cost functional is not positive definite in the state was recognized
quite recently [5, 1, 7, 12]. There, a particular form of strict dissipativity
with the supply rate given by the cost functional, renders the optimal value
function a Lyapunov function that itself can be leveraged to prove stability
of the MPC-closed loop. Besides direct applications in MPC stability analysis,
dissipativity plays an important role for optimally operated steady states see
[14, 15] and [16], and to the so-called turnpike property at an optimal
equilibrium, see [11].

Despite these various applications of strict dissipativity in optimal control,
the first characterization of dissipativity of finite-dimensional generalized
linear-quadratic optimal control problems (OCPs), that is, OCPs with quadratic
and linear terms in the cost and linear dynamics, was provided in [8, 9]. In
these works, a close relationship between detectability, where the output
corresponds to the quadratic cost in the state, and strict dissipativity were
given. Recently, in [10], we presented preliminary steps towards a
characterization of strict dissipativity in the context of infinite-
dimensional systems, however under very strong spectral assumptions on the
underlying semigroup. In this work, we first establish a result for general
dynamics that characterizes strict pre-dissipativity by means of a Lyapunov-
like form inequality and then link the latter to detectability properties.

This paper is organized as follows. After providing some notation and the
problem statement in Section 2, we establish alternative representations of
strict dissipativity by means of an integral inequality or strict
dissipativity of a problem with rotated stage cost in Section 3. In Section 4,
we focus on quadratic storage functions and provide in Theorem 9 a
characterization of dissipativity for a given steady state by means of a
Lyapunov-like form inequality. Whereas this result for a given steady state
involves an algebraic condition, we show in Theorem 11 that this condition can
be omitted if one aims for dissipativity at some steady state. Last, in
Section 5, we prove in Theorem 17 that the obtained form inequality is implied
by exponential detectability and that the converse implication holds under an
additional algebraic assumption on the Lyapunov operator.

##  2\. Generalized linear-quadratic optimal control problems

In this part we define the class of optimal control problems (OCPs) we shall
study in this paper. Before, however, we require some notation.

Notation. Let $(\mathcal{H},\langle\,\cdot\,,\,\cdot\,\rangle)$ be a real or
complex Hilbert space. A self-adjoint operator $T$ in $\mathcal{H}$ will be
called

  * •

strictly positive, if there exists $c>0$ such that $\langle Tx,x\rangle\geq
c\|x\|^{2}$ for all $x\in\operatorname{dom}T$;

  * •

positive, if $\langle Tx,x\rangle>0$ for all
$x\in\operatorname{dom}T\backslash\\{0\\}$;

  * •

non-negative, if $\langle Tx,x\rangle\geq 0$ for all
$x\in\operatorname{dom}T$.

The square root of a non-negative self-adjoint operator $T$ will be denoted by
$T^{1/2}$. For the existence and properties of this square root we refer to
[13, Theorem V.3.35].

The space of all bounded linear operators from a Hilbert space
$\mathcal{H}_{1}$ into another one $\mathcal{H}_{2}$ will be denoted by
$L(\mathcal{H}_{1},\mathcal{H}_{2})$. As usual, we set
$L(\mathcal{H}):=L(\mathcal{H},\mathcal{H})$. An operator $K\in
L(\mathcal{H})$ is called coercive, if $K^{*}K$ is strictly positive. If
$M\subset\mathcal{H}$ is a closed linear subspace, then $\mathbb{P}_{M}$
denotes the orthogonal projection onto $M$. The closure of the range
$\operatorname{ran}T$ of a linear operator $T$ will be denoted by
$\overline{\operatorname{ran}}\,T$.

We say that a function $V:\mathcal{H}\to\mathbb{R}$ is differentiable at
$x_{0}\in\mathcal{H}$ if there exists a continuous $\mathbb{R}$-linear
functional $V^{\prime}(x_{0}):\mathcal{H}\to\mathbb{R}$ (the derivative of $V$
at $x_{0}$) such that

$\lim_{x\to
x_{0}}\frac{V(x)-V(x_{0})-V^{\prime}(x_{0})(x-x_{0})}{\|x-x_{0}\|}=0.$

As usual, $V$ is called differentiable if it is differentiable at each
$x\in\mathcal{H}$. Due to Riesz’ representation theorem, the derivative has
the form $V^{\prime}(x)h=\operatorname{Re}\langle h,g(x)\rangle$,
$h\in\mathcal{H}$, with a unique $g(x)\in\mathcal{H}$. We will frequently
identify $g(x)$ with $V^{\prime}(x)$. If $V$ is differentiable, we say that
$V$ is $C^{1}$ if the function $g:\mathcal{H}\to\mathcal{H}$ is continuous. If
$V(x)=\langle Px,x\rangle+2\operatorname{Re}\langle x,w\rangle$ with a self-
adjoint operator $P\in L(\mathcal{H})$ and a vector $w\in\mathcal{H}$, then
$V$ is $C^{1}$ with $V^{\prime}(x)h=2\operatorname{Re}\langle h,Px+w\rangle$.

The OCP. Throughout this paper, we assume that $A$ is a generator of a
$C_{0}$-semigroup on the Hilbert space $\mathcal{H}$. The control space
$\mathcal{U}$ is also assumed to be a Hilbert space
$(\mathcal{U},\langle\cdot,\cdot\rangle)$ with the same notation for the
scalar product. We consider the infinite-dimensional linear quadratic problem

(2.1) $\min_{u\in
L^{2}(0,T;\mathcal{U})}\int_{0}^{T}\ell(x,u)\,dt\,\,\,\mathrm{s.t.}\,\dot{x}=Ax+Bu,\quad
x(0)=x_{0}$

with cost function $\ell:\mathcal{H}\times\mathcal{U}\to\mathbb{R}$, defined
by

$\ell(x,u):=\|Cx\|^{2}+\|Ku\|^{2}+2\operatorname{Re}\langle
z,x\rangle+2\operatorname{Re}\langle v,u\rangle.$

Solutions of $\dot{x}=Ax+Bu$ will be considered in the mild sense. Here, $C\in
L(\mathcal{H},\mathcal{Y})$, $K\in L(\mathcal{U},\mathcal{Z})$, $B\in
L(\mathcal{U},\mathcal{H})$, $z\in\mathcal{H}$, and $v\in\mathcal{U}$, where
$\mathcal{Y}$ and $\mathcal{Z}$ are Hilbert spaces. We assume that there is
some $c_{K}>0$ such that

(2.2) $\|Ku\|\geq c_{K}\|Bu\|,\qquad u\in\mathcal{U}.$

This condition is equivalent to
$\operatorname{ran}B^{*}\subset\operatorname{ran}K^{*}$, see [22, Theorem
IV.2.2] and is trivially satisfied if $K$ is coercive.

A pair $(x_{e},u_{e})\in\operatorname{dom}A\times\mathcal{U}$ will be called a
controlled equilibrium (or steady state) of the control problem
$\dot{x}=Ax+Bu$, if $Ax_{e}+Bu_{e}=0$. It is easy to see that the OCP (2.1) is
equivalent to the following one:

(2.3) $\min_{u\in
L^{2}(0,T;U)}\int_{0}^{T}\tilde{\ell}(x,u)\,dt\,\,\,\mathrm{s.t.}\,\dot{x}=Ax+Bu,\,\,x(0)=x_{0}-x_{e},$

where

$\displaystyle\tilde{\ell}(x,u)=\|Cx\|^{2}+\|Ku\|^{2}+2\operatorname{Re}\langle
z+C^{*}Cx_{e},x\rangle+2\operatorname{Re}\langle v+K^{*}Ku_{e},u\rangle.$

By equivalence of (2.1) and (2.3) it is meant that $(x,u)$ is optimal for
(2.1) if and only if $(x,u)-(x_{e},u_{e})$ is optimal for (2.3). The optimal
values might differ from each other, however.

##  3\. Strict pre-dissipativity

In this section, we shall introduce and discuss the notion of strict
(pre-)dissipativity of the OCP (2.1), which involves the notion of storage
functions and the class $\mathcal{K}$. The latter is defined by

$\displaystyle\mathcal{K}:=\big{\\{}\alpha:[0,\infty)\to[0,\infty)\,|\,\alpha\text{
is continuous}$ and strictly increasing with
$\displaystyle\alpha(0)=0\big{\\}}.$

A storage function on $\mathcal{H}$ is a $C^{1}$-function
$V:\mathcal{H}\to\mathbb{R}$.

The following definition of strict dissipativity in the realm of infinite-
dimensional optimal control is directly adopted from the finite-dimensional
setting, see, e.g., [6, 10].

######  Definition 1.

We say that the linear-quadratic OCP (2.1) is strictly pre-dissipative at a
controlled equilibrium $(x_{e},u_{e})$, if there exist a storage function $V$
on $\mathcal{H}$ and a dissipation rate $\alpha\in\mathcal{K}$ such that for
all $x\in\operatorname{dom}A$ and all $u\in\mathcal{U}$ we have

(3.1)
$V^{\prime}(x)(Ax+Bu)\,\leq\,\ell(x,u)-\ell(x_{e},u_{e})-\alpha(\|x-x_{e}\|).$

The OCP (2.1) is called strictly dissipative at $(x_{e},u_{e})$ if it is
strictly pre-dissipative with a storage function $V$ that is bounded from
below.

We close this section by providing two characterizations of strict
dissipativity.

######  Lemma 2.

The OCP (2.1) is strictly pre-dissipative (dissipative) at $(x_{e},u_{e})$
with storage function $V$ if and only if the OCP (2.3) is strictly pre-
dissipative (dissipative) at $(0,0)$ with storage function
$\tilde{V}=V(\,\cdot\,+x_{e})$.

###### Proof.

Let OCP (2.1) be strictly pre-dissipative at $(x_{e},u_{e})$ with storage
function $V$. Then for all $x\in\operatorname{dom}A$ and $u\in\mathcal{U}$ we
have

$\displaystyle\tilde{V}^{\prime}(x)(Ax+Bu)$
$\displaystyle=V^{\prime}(x+x_{e})\big{(}A(x+x_{e})+B(u+u_{e})\big{)}$
$\displaystyle\leq\ell(x+x_{e},u+u_{e})-\ell(x_{e},u_{e})-\alpha(\|x\|)$
$\displaystyle=\|C(x+x_{e})\|^{2}+\|K(u+u_{e})\|^{2}+2\operatorname{Re}\langle
z,x+x_{e}\rangle$ $\displaystyle\quad+2\operatorname{Re}\langle
v,u+u_{e}\rangle-\|Cx_{e}\|^{2}-\|Ku_{e}\|^{2}-2\operatorname{Re}\langle
z,x_{e}\rangle$ $\displaystyle\quad-2\operatorname{Re}\langle
v,u_{e}\rangle-\alpha(\|x\|)$
$\displaystyle=\|Cx\|^{2}+2\operatorname{Re}\langle
Cx,Cx_{e}\rangle+\|Ku\|^{2}+2\operatorname{Re}\langle Ku,Ku_{e}\rangle$
$\displaystyle\quad+2\operatorname{Re}\langle
z,x\rangle+2\operatorname{Re}\langle v,u\rangle-\alpha(\|x\|)$
$\displaystyle=\|Cx\|^{2}+\|Ku\|^{2}+2\operatorname{Re}\langle
z+C^{*}Cx_{e},x\rangle$ $\displaystyle\quad+2\operatorname{Re}\langle
v+K^{*}Ku_{e},u\rangle-\alpha(\|x\|)$
$\displaystyle=\tilde{\ell}(x,u)-\alpha(\|x\|).$

Since $\tilde{\ell}(0,0)=0$, it follows that OCP (2.3) is strictly pre-
dissipative at $(0,0)$ with storage function $\tilde{V}$. The proof of the
opposite implication follows similar lines. ∎

In what follows, we denote the mild solution to

(3.2) $\dot{x}=Ax+Bu,\quad x(0)=x_{0}\in\mathcal{H}$

by $x_{u}(\,\cdot\,;x_{0})$, i.e.,

$x_{u}(t;x_{0})=T(t)x_{0}+\int_{0}^{t}T(t-s)Bu(s)\,ds,$

where $T(\cdot)$ denotes the $C_{0}$-semigroup generated by $A$.

######  Lemma 3.

The OCP (2.1) is strictly pre-dissipative at $(x_{e},u_{e})$ with storage
function $V$ and a continuous dissipation rate $\alpha\in\mathcal{K}$ if and
only if for any $x_{0}\in\mathcal{H}$, any $u\in L^{2}_{\rm
loc}([0,\infty),\mathcal{U})$, and any $0\leq t_{1}<t_{2}$ we have

(3.3)
$\displaystyle\begin{split}V(x_{u}(t_{2};x_{0}))&-V(x_{u}(t_{1};x_{0}))\\\
&\leq\,\int_{t_{1}}^{t_{2}}\\!\\!\big{[}\ell(x_{u}(t;x_{0}),u(t))-\ell(x_{e},u_{e})-\alpha(\|x_{u}(t;x_{0})-x_{e}\|)\big{]}\,dt.\end{split}$

###### Proof.

$\Longrightarrow$: Assume that (2.1) is strictly pre-dissipative at
$(x_{e},u_{e})$ with storage function $V$ and $\alpha\in\mathcal{K}$. Set
$J:=[t_{1},t_{2}]$, $\tilde{J}:=[0,t_{2}]$, and denote the $L^{p}$-norm on
$\tilde{J}$ by $\|\cdot\|_{p}$, $p\in[1,\infty]$. Let $u\in L^{2}(J,U)$ and
$x_{0}\in\mathcal{H}$. Since $\operatorname{dom}A$ is dense in $\mathcal{H}$,
there exists a sequence $(z_{n})\subset\operatorname{dom}A$ such that
$z_{n}\to x_{0}$ as $n\to\infty$. Moreover, by the density of the step
functions in $L^{2}(\tilde{J},U)$ [23, Proposition 23.2, p407], see also [18,
Lemma 2.1], $C^{1}(\tilde{J},U)$ is dense in $L^{2}(\tilde{J},U)$ so that we
find a sequence $(u_{n})\subset C^{1}(\tilde{J},U)$ such that $u_{n}\to u$ in
$L^{2}(\tilde{J},U)$. We set $x(t):=x_{u}(t;x_{0})$ and
$x_{n}(t):=x_{u_{n}}(t;z_{n})$, $t\geq 0$. For $t\in J$ we have

$\displaystyle\|x_{n}(t)-x(t)\|$
$\displaystyle=\left\|T(t)(z_{n}-x_{0})+\int_{0}^{t}T(t-s)B(u(s)-u_{n}(s))\,ds\right\|$
$\displaystyle\leq M\left(\|z_{n}-x_{0}\|+\|B\|\|u_{n}-u\|_{1}\right),$

where $M=\sup\\{\|T(t)\|:t\in[0,t_{2}]\\}$. Hence, $x_{n}\to x$ uniformly on
$J$. Moreover, $x_{n}$ is a classical solution of (3.2) by [4, Theorem 3.1.3],
i.e., $x_{n}$ is a continuously differentiable curve, i.e., $x_{n}\in
C^{1}(0,T;X)\cap C(0,T;\operatorname{dom}(A))$ and
$\dot{x}_{n}(t)=Ax_{n}(t)+Bu_{n}(t)$ holds for every $t$. Therefore, $V\circ
x_{n}$ is continuously differentiable with

$\displaystyle\tfrac{d}{dt}V(x_{n}(t))$
$\displaystyle=V^{\prime}(x_{n}(t))(Ax_{n}(t)+Bu_{n}(t))\,$
$\displaystyle\leq\,\ell(x_{n}(t),u_{n}(t))-\ell(x_{e},u_{e})-\alpha(\|x_{n}(t)-x_{e}\|)$

by strict pre-dissipativity. Integrating this over $J$ gives

$\displaystyle V($ $\displaystyle
x_{n}(t_{2}))-V(x_{n}(t_{1}))\,\leq\,\int_{t_{1}}^{t_{2}}\big{[}\ell(x_{n},u_{n})-\ell(x_{e},u_{e})-\alpha(\|x_{n}(t)-x_{e}\|)\big{]}\,dt.$

By continuity of $V$, $\alpha$ and $\ell$, $\|x_{n}-x\|_{\infty}\to 0$ and
$\|u_{n}-u\|_{2}\to 0$ as $n\to\infty$, all terms in this inequality tend to
the corresponding ones in (3.3).

$\Longleftarrow$: Assume that the condition (3.3) holds and let
$x_{0}\in\operatorname{dom}A$ and $u_{0}\in U$ be arbitrary. Then the mild
solution $x:=x_{u}(\,\cdot\,,x_{0})$ corresponding to $u(t):=u_{0}$ for $t\geq
0$ is a classical solution satisfying (3.3) with $t_{1}=0$. Dividing (3.3) by
$t_{2}$ and letting $t_{2}\to 0$ yields

$\displaystyle V^{\prime}(x_{0})(Ax_{0}+Bu_{0})$
$\displaystyle=\tfrac{d}{dt}V(x(t))\big{|}_{t=0}\,\leq\,\ell(x_{0},u_{0})-\ell(x_{e},u_{e})-\alpha(\|x_{0}-x_{e}\|),$

which is strict pre-dissipativity of (2.1) at $(x_{e},u_{e})$. ∎

##  4\. Quadratic storage functions

In this paper, we are especially interested in quadratic storage functions for
strict (pre-)dissipativity. A function $V:\mathcal{H}\to\mathbb{R}$ will be
called quadratic if it is of the form

$V_{P,w}(x):=\langle Px,x\rangle+2\operatorname{Re}\langle w,x\rangle$

with a self-adjoint operator $P\in L(\mathcal{H})$ and $w\in\mathcal{H}$. With
regard to strict dissipativity it is of particular interest to characterize
the quadratic storage functions which are bounded from below.

######  Lemma 4.

The quadratic function $V_{P,w}$ is bounded from below if and only if $P$ is
non-negative and $w\in\operatorname{ran}P^{1/2}$. $V_{P,w}$ has a minimizer if
and only if $w\in\operatorname{ran}P$, in which case $P^{-1}\\{-w\\}$ is the
set of minimizers.

###### Proof.

Let $V=V_{P,w}$. If $P\geq 0$ and $w\in\operatorname{ran}P^{1/2}$, choose
$v\in\mathcal{H}$ such that $w=P^{1/2}v$. Then

(4.1) $\displaystyle
V(x)=\|P^{1/2}x\|^{2}+2\operatorname{Re}\big{\langle}P^{1/2}x,v\big{\rangle}=\|P^{1/2}x+v\|^{2}-\|v\|^{2}\geq-\|v\|^{2}.$

Conversely, assume that $V$ is bounded from below. Then there exists $c\geq 0$
such that $V+c\geq 0$. It is clear that $P\geq 0$. Write $w=w_{1}+w_{2}$ with
respect to the orthogonal decomposition

$\mathcal{H}=\ker P\oplus\overline{\operatorname{ran}}\,P.$

Then $w_{1}=0$. Indeed, otherwise
$V(-\frac{c+1}{2\|w_{1}\|^{2}}w_{1})+c=-1<0$. Let $\mathbb{P}_{n}$ denote the
spectral projection of $P$ corresponding to the interval $[1/n,\|P\|]$ (cf.
Appendix A) and set $w_{n}:=\mathbb{P}_{n}w$. Then
$w_{n}\in\operatorname{ran}P$ for any $n$ (see Lemma 18(a)) and
$w\in\overline{\operatorname{ran}}\,P$ implies that $w_{n}\to w$ as
$n\to\infty$. Let $x_{n}:=-P^{-1}w_{n}$. Then

$-c\leq
V(x_{n})=\|P^{1/2}x_{n}\|^{2}+2\operatorname{Re}\big{\langle}w_{n},x_{n}\big{\rangle}=-\|P^{-1/2}w_{n}\|^{2}.$

Hence, $(P^{-1/2}w_{n})$ is bounded and therefore possesses a weakly
convergent subsequence, i.e., $P^{-1/2}w_{n_{k}}\rightharpoonup z$ for some
$z\in\mathcal{H}$. But this implies $w_{n_{k}}\rightharpoonup P^{1/2}z$ and
therefore $w=P^{1/2}z\in\operatorname{ran}P^{1/2}$.

If $w\in\operatorname{ran}P$ and $v\in P^{-1}\\{-w\\}$, then
$V(x)=\|P^{1/2}x-P^{1/2}v\|^{2}-\|P^{1/2}v\|^{2}\geq-\|P^{1/2}v\|^{2}$ for all
$x\in\mathcal{H}$ and $x^{*}=v$ is a minimizer. Conversely, assume that $V$
has a minimizer $x^{*}$. Then $w=P^{1/2}v\in\operatorname{ran}P^{1/2}$ since
$V$ is bounded below. WLOG, we may assume that
$v\in\overline{\operatorname{ran}}P=\overline{\operatorname{ran}}P^{1/2}$, so
that there exists a sequence $(x_{n})\subset\mathcal{H}$ such that
$P^{1/2}x_{n}\to-v$. Hence, (4.1) shows that
$\inf\\{V(x):x\in\mathcal{H}\\}=-\|v\|^{2}$. Therefore, $V(x^{*})=-\|v\|^{2}$
and (4.1) implies $v=-P^{1/2}x^{*}$. Thus,
$w=P^{1/2}v=-Px^{*}\in\operatorname{ran}P$ and $x^{*}\in P^{-1}\\{-w\\}$. ∎

Note that the running cost

$\ell(x,u)=\left\langle\begin{bmatrix}C^{*}C&0\\\
0&K^{*}K\end{bmatrix}\begin{bmatrix}x\\\ u\end{bmatrix},\begin{bmatrix}x\\\
u\end{bmatrix}\right\rangle+2\operatorname{Re}\left\langle\begin{bmatrix}z\\\
v\end{bmatrix},\begin{bmatrix}x\\\ u\end{bmatrix}\right\rangle$

is itself a quadratic storage function. Hence, the following corollary is
readily derived from Lemma 4.

######  Corollary 5.

The running cost function $\ell$ is bounded from below if and only if
$z\in\operatorname{ran}C^{*}$ and $v\in\operatorname{ran}K^{*}$.

###### Proof.

The claim follows from
$\operatorname{ran}(C^{*}C)^{1/2}=\operatorname{ran}C^{*}$ and
$\operatorname{ran}(K^{*}K)^{1/2}=\operatorname{ran}K^{*}$, see [13, VI.2.7].
∎

######  Lemma 6.

The OCP (2.1) is strictly pre-dissipative at $(x_{e},u_{e})$ with storage
function $V_{P,w}$ if and only if the OCP (2.3) is strictly pre-dissipative at
$(0,0)$ with storage function $V_{P,\tilde{w}}$, where $\tilde{w}=w+Px_{e}$.

###### Proof.

For $x\in\mathcal{H}$ we have

$\displaystyle V_{P,w}(x+x_{e})$ $\displaystyle=\langle
P(x+x_{e}),x+x_{e}\rangle+2\operatorname{Re}\langle w,x+x_{e}\rangle$
$\displaystyle=\langle Px,x\rangle+2\operatorname{Re}\langle
Px_{e},x\rangle+\langle Px_{e},x_{e}\rangle+2\operatorname{Re}\langle
w,x\rangle+2\operatorname{Re}\langle w,x_{e}\rangle$
$\displaystyle=V_{P,\tilde{w}}(x)+(\langle
Px_{e},x_{e}\rangle+2\operatorname{Re}\langle w,x_{e}\rangle).$

Hence, the claim follows from Lemma 2. ∎

The next proposition shows in particular that strict pre-dissipativity is
subject to an algebraic constraint on the steady state and the linear factors
in the stage cost.

######  Proposition 7.

For $P=P^{*}\in L(\mathcal{H})$ and $w\in\mathcal{H}$ the following statements
are equivalent:

  1. (i)

The OCP (2.1) is strictly pre-dissipative at $(x_{e},u_{e})$ with storage
function $V_{P,w}$ and dissipation rate $\alpha\in\mathcal{K}$.

  2. (ii)

We have

(4.2) $\displaystyle\begin{split}&w+Px_{e}\in\operatorname{dom}A^{*},\\\
&A^{*}(w+Px_{e})=z+C^{*}Cx_{e},\\\ &B^{*}(w+Px_{e})=v+K^{*}Ku_{e}\end{split}$

and for all $x\in\operatorname{dom}A$ and all $u\in\mathcal{U}$,

(4.3) $2\operatorname{Re}\langle
Ax+Bu,Px\rangle\,\leq\,\|Cx\|^{2}+\|Ku\|^{2}-\alpha(\|x\|).$

###### Proof.

By Lemma 6, (i) is equivalent to

  1. (i’)

The OCP (2.3) is strictly pre-dissipative at $(0,0)$ with storage function
$V_{P,\tilde{w}}$, $\tilde{w}=w+Px_{e}$.

Moreover, $\tilde{\ell}(0,0)=0$, and hence (i’) means that for all
$x\in\operatorname{dom}A$ and $u\in\mathcal{U}$,

(4.4) $\displaystyle\begin{split}2\operatorname{Re}\langle
Ax+Bu,Px+\tilde{w}\rangle&\leq\,\|Cx\|^{2}+\|Ku\|^{2}+2\operatorname{Re}\langle
z+C^{*}Cx_{e},x\rangle\\\ &\qquad\quad\quad\,\,+2\operatorname{Re}\langle
v+K^{*}Ku_{e},u\rangle-\alpha(\|x\|).\end{split}$

(ii)$\Rightarrow$(i’). The combination of (4.2) and (4.3) gives (4.4).
Therefore, (i’) follows.

(i’)$\Rightarrow$(ii). By setting $x=0$ in (4.4) we obtain

$\|Ku\|^{2}+2\operatorname{Re}\langle
u,v+K^{*}Ku_{e}-B^{*}\tilde{w}\rangle\,\geq\,0$

for all $u\in\mathcal{U}$. But this is only possible if
$v+K^{*}Ku_{e}-B^{*}\tilde{w}=0$. Hence, (4.4) simplifies to

$\displaystyle 2\operatorname{Re}$ $\displaystyle\langle
Ax+Bu,Px\rangle+2\operatorname{Re}\langle Ax,\tilde{w}\rangle\,$
$\displaystyle\leq\,\|Cx\|^{2}+\|Ku\|^{2}+2\operatorname{Re}\langle
z+C^{*}Cx_{e},x\rangle-\alpha(\|x\|),$

which holds for all $x\in\operatorname{dom}A$ and all $u\in\mathcal{U}$. Thus,
setting $u=0$ yields

$2\operatorname{Re}\langle Ax,Px\rangle+2\operatorname{Re}\langle
Ax,\tilde{w}\rangle\,\leq\,\|Cx\|^{2}+2\operatorname{Re}\langle
z+C^{*}Cx_{e},x\rangle$

for all $x\in\operatorname{dom}A$. Let $a>0$ and replace $x$ by $ax$ in the
last inequality, i.e.,

$\displaystyle 2a^{2}\operatorname{Re}\langle Ax,Px\rangle$
$\displaystyle+2a\operatorname{Re}\langle
Ax,\tilde{w}\rangle\,\leq\,a^{2}\|Cx\|^{2}+2a\operatorname{Re}\langle
z+C^{*}Cx_{e},x\rangle.$

Dividing this by $a$ and letting $a\to 0$ implies that
$\operatorname{Re}\langle Ax,\tilde{w}\rangle\,\leq\,\operatorname{Re}\langle
x,z+C^{*}Cx_{e}\rangle$ for all $x\in\operatorname{dom}A$. Replacing $x$ by
$-x$ in this inequality shows that in fact $\operatorname{Re}\langle
Ax,\tilde{w}\rangle=\operatorname{Re}\langle x,z+C^{*}Cx_{e}\rangle$ holds for
all $x\in\operatorname{dom}A$. And as

$\displaystyle\operatorname{Im}\langle
x,z+C^{*}Cx_{e}\rangle=-\operatorname{Re}\langle
ix,z+C^{*}Cx_{e}\rangle=-\operatorname{Re}\langle
iAx,\tilde{w}\rangle=\operatorname{Im}\langle Ax,\tilde{w}\rangle,$

we obtain $\langle Ax,\tilde{w}\rangle=\langle x,z+C^{*}Cx_{e}\rangle$ for all
$x\in\operatorname{dom}A$. But this means that
$\tilde{w}\in\operatorname{dom}A^{*}$ and $A^{*}\tilde{w}=z+C^{*}Cx_{e}$. This
shows that (4.2) holds, and (4.3) follows by plugging (4.2) into (4.4). ∎

######  Remark 8.

As the proof above shows, the condition on $z$ and $v$ is even necessary for
“ordinary” (pre-)dissipativity (i.e. $\alpha=0$) with supply rate $\ell$ (and
output $y=x$).

######  Theorem 9.

For $P=P^{*}\in L(\mathcal{H})$ and a controlled equilibrium $(x_{e},u_{e})$
the following statements are equivalent:

  1. (i)

There exist $\gamma>0$ and $w\in\mathcal{H}$ such that the OCP (2.1) is
strictly pre-dissipative at $(x_{e},u_{e})$ with storage function $V_{\gamma
P,w}$.

  2. (ii)

We have

(4.5) $\begin{bmatrix}z+C^{*}Cx_{e}\\\
v+K^{*}Ku_{e}\end{bmatrix}\,\in\,\operatorname{ran}\begin{bmatrix}A^{*}\\\
B^{*}\end{bmatrix}$

and the following Lyapunov-like form inequality:

(4.6) $\displaystyle\begin{split}\exists\eta,m>0\,&\forall
x\in\operatorname{dom}A:\\\ &\|Cx\|^{2}-2\eta\operatorname{Re}\langle
Ax,Px\rangle\,\geq\,m\|x\|^{2}.\end{split}$

###### Proof.

(i)$\Rightarrow$(ii). Condition (4.2) in Proposition 7 implies (4.5) and
setting $u=0$ in (4.3) yields

$\|Cx\|^{2}-2\gamma\operatorname{Re}\langle
Ax,Px\rangle\,\geq\,\alpha(\|x\|),\qquad x\in\operatorname{dom}A.$

Now, replace $x$ by $x/\|x\|$ in this inequality to obtain (4.6) with
$\eta=\gamma$ and $m=\alpha(1)$.

(ii)$\Rightarrow$(i). It is no restriction to assume $\eta=1$ in (4.6). For
$\gamma\in(0,1)$, $x\in\operatorname{dom}A$, and $u\in\mathcal{U}$ we have

$\displaystyle\|Cx\|^{2}+\|Ku\|^{2}$ $\displaystyle-2\operatorname{Re}\langle
Ax+Bu,\gamma Px\rangle$
$\displaystyle=\|Cx\|^{2}-2\gamma\operatorname{Re}\langle
Ax,Px\rangle+\|Ku\|^{2}-2\gamma\operatorname{Re}\langle PBu,x\rangle$
$\displaystyle\geq(1-\gamma)\|Cx\|^{2}+\gamma
m\|x\|^{2}+c_{K}^{2}\|Bu\|^{2}-2\gamma\|P\|\|Bu\|\|x\|$
$\displaystyle\geq\left[c_{k}\|Bu\|-\frac{\gamma\|P\|}{c_{K}}\|x\|\right]^{2}-\frac{\gamma^{2}\|P\|^{2}}{c_{K}^{2}}\|x\|^{2}+\gamma
m\|x\|^{2}$
$\displaystyle\geq\gamma\left(m-\frac{\gamma\|P\|^{2}}{c_{K}^{2}}\right)\|x\|^{2}=c\|x\|^{2}$

with some $c>0$ is $\gamma$ is sufficiently small. If we set
$\alpha(t):=ct^{2}$, then this is (4.3) with $P$ replaced by $\gamma P$. Now,
due to (4.5), there exists $\tilde{w}\in\operatorname{dom}A^{*}$ such that
$A^{*}\tilde{w}=z+C^{*}Cx_{e}$ and $B^{*}\tilde{w}=v+K^{*}Ku_{e}$. Set
$w:=\tilde{w}-\gamma Px_{e}$. Then also (4.2) holds (with $P$ replaced by
$\gamma P$), and therefore Proposition 7 implies that the OCP (2.1) is
strictly pre-dissipative at $(x_{e},u_{e})$ with storage function $V_{\gamma
P,w}$. ∎

Our next aim is to show that (4.6) is essentially equivalent to strict pre-
dissipativity at a suitably chosen steady state $(x_{e},u_{e})$. In order to
find this steady state, we analyse the optimal steady state problem

(4.7)
$\displaystyle\min_{x\in\operatorname{dom}A,\,u\in\mathcal{U}}\ell(x,u)\qquad\text{s.t.
}Ax+Bu=0,$

which is certainly equivalent to

(4.8) $\displaystyle\min_{y\in\ker[A\;B]}\ell(y).$

######  Lemma 10.

Assume that $K$ is coercive and that (4.6) holds. Then the optimal steady
state problem (4.7) has a unique solution $y^{*}=(x_{e},u_{e})$. It satisfies

(4.9) $\mathbb{P}_{\ker[A\;B]}\begin{bmatrix}z+C^{*}Cx_{e}\\\
v+K^{*}Ku_{e}\end{bmatrix}=0.$

###### Proof.

As $A$ is a closed operator, so is $[A\;B]$, which implies that
$\mathcal{X}:=\ker[A\;B]$ is closed in $\mathcal{H}\times\mathcal{U}$. Also,
for $y\in\mathcal{X}$ we have $\ell(y)=\langle
Ty,y\rangle+2\operatorname{Re}\langle q,y\rangle$, where

$T=\mathbb{P}_{\mathcal{X}}\begin{bmatrix}C^{*}C&0\\\
0&K^{*}K\end{bmatrix}\bigg{|}_{\mathcal{X}}\qquad\text{and}\qquad
q=\mathbb{P}_{\mathcal{X}}\begin{bmatrix}z\\\ v\end{bmatrix}.$

We shall now show that the non-negative operator $T\in L(\mathcal{X})$ is in
fact strictly positive and thus boundedly invertible, so that the claim
follows from Lemma 4. Indeed, (4.9) is equivalent to $y^{*}=-T^{-1}q$. If $T$
was not strictly positive, there would exist a sequence
$(y_{n})\subset\mathcal{X}$ with $\|y_{n}\|=1$ for all $n\in\mathbb{N}$ and
$\langle Ty_{n},y_{n}\rangle\to 0$ as $n\to\infty$. If we write
$y_{n}=\left[\begin{smallmatrix}x_{n}\\\ u_{n}\end{smallmatrix}\right]$, this
means that $\|x_{n}\|^{2}+\|u_{n}\|^{2}=1$, $Cx_{n}\to 0$, and $Ku_{n}\to 0$.
Since $K$ is coercive, it follows that $u_{n}\to 0$. Further,
$Ax_{n}+Bu_{n}=0$ yields $Ax_{n}\to 0$, hence with (4.6) we conclude $x_{n}\to
0$ as $n\to\infty$. But this contradicts $\|x_{n}\|^{2}+\|u_{n}\|^{2}=1$. ∎

######  Theorem 11.

Let $P=P^{*}\in L(\mathcal{H})$, assume that $K$ is coercive and that
$\operatorname{ran}[A\;B]$ is closed. Then there exist $\gamma>0$ and
$w\in\mathcal{H}$ such that (2.1) is strictly pre-dissipative at some
controlled equilibrium with storage function $V_{\gamma P,w}$ if and only if
(4.6) holds. In this case, the controlled equilibrium $(x_{e},u_{e})$ can be
chosen as the minimizer of problem (4.7).

###### Proof.

If (2.1) is strictly pre-dissipative at some controlled equilibrium with
storage function $V_{\gamma P,w}$, then (4.6) holds by Theorem 9. Conversely,
assume that (4.6) is satisfied. By Lemma 10, the unique solution
$y^{*}=(x_{e},u_{e})\in\ker[A\;B]$ of the optimal steady state problem (4.7)
satisfies (4.9) and hence

$\begin{bmatrix}z+C^{*}Cx_{e}\\\
v+K^{*}Ku_{e}\end{bmatrix}\,\in\,\big{(}\\!\ker[A\;B]\big{)}^{\perp}=\operatorname{ran}\begin{bmatrix}A^{*}\\\
B^{*}\end{bmatrix}.$

Thus, (4.5) holds and Theorem 9 yields the claim. This also shows that
$(x_{e},u_{e})$ can always be chosen as the minimizer of problem (4.7). ∎

######  Remark 12.

If $A$ is bounded, the condition (4.6) can be written as an operator
inequality of Lyapunov type:

$PA+A^{*}P\,\leq\,\eta^{-1}(C^{*}C-mI).$

However, if $A$ is unbounded, the operator $T=PA+A^{*}P$ is in general not
self-adjoint. It is symmetric (i.e., $T\subset T^{*}$), but in general it is
not densely defined. Its domain might even be trivial, see [2].

We close this section with some corollaries.

######  Corollary 13.

The OCP (2.1) is strictly pre-dissipative at $(x_{e},u_{e})$ with a quadratic
storage function if and only if (4.5) holds and there exist a self-adjoint
operator $P\in L(\mathcal{H})$ and $m>0$ such that

$\|Cx\|^{2}-2\operatorname{Re}\langle Ax,Px\rangle\geq m\|x\|^{2}$

holds for all $x\in\operatorname{dom}A$.

######  Corollary 14.

The OCP (2.1) is strictly dissipative at $(x_{e},u_{e})$ with a quadratic
storage function if and only if there exist a non-negative operator $P\in
L(\mathcal{H})$ and $m>0$ such that

$[A\;\;B]^{-*}\begin{bmatrix}z+C^{*}Cx_{e}\\\
v+K^{*}Ku_{e}\end{bmatrix}\,\cap\,\operatorname{ran}P^{1/2}\neq\varnothing$

and

$\|Cx\|^{2}-2\operatorname{Re}\langle Ax,Px\rangle\geq m\|x\|^{2},\qquad
x\in\operatorname{dom}A.$

######  Corollary 15.

If $\operatorname{ran}[A\;B]$ is closed, then the OCP (2.1) is strictly pre-
dissipative at some controlled equilibrium with a quadratic storage function
if and only if there exist $P=P^{*}\in L(\mathcal{H})$ and $m>0$ such that

$\|Cx\|^{2}-2\operatorname{Re}\langle Ax,Px\rangle\geq m\|x\|^{2},\qquad
x\in\operatorname{dom}A.$

##  5\. Strict dissipativity and exponential detectability

In what follows, we shall call an operator $A$ exponentially stable if it
generates an exponentially stable $C_{0}$-semigroup $T(\cdot)$, i.e.,
$T(\cdot)x\in L^{2}([0,\infty),\mathcal{H})$ for each $x\in\mathcal{H}$. The
following theorem contains a weaker characterization for exponential stability
as in the standard literature. However, the proof remains essentially the
same.

######  Theorem 16.

Let $A$ be a generator of a $C_{0}$-semigroup in $\mathcal{H}$. Then the
following statements are equivalent:

  1. (i)

$A$ is exponentially stable.

  2. (ii)

There exists a positive operator $P\in L(\mathcal{H})$ such that

$2\operatorname{Re}\langle Ax,Px\rangle=-\|x\|^{2},\qquad
x\in\operatorname{dom}A.$

  3. (iii)

There exist a non-negative operator $P\in L(\mathcal{H})$ and $c>0$ such that

$2\operatorname{Re}\langle Ax,Px\rangle\,\leq\,-c\|x\|^{2},\qquad
x\in\operatorname{dom}A.$

###### Proof.

(i)$\Rightarrow$(ii) follows from Theorem 5.1.3 in [4] and
(ii)$\Rightarrow$(iii) is trivial. Assume that (iii) holds, let
$x\in\operatorname{dom}A$ and define $w(t):=\langle PT(t)x,T(t)x\rangle$,
$t\geq 0$. Differentiating $w$ with respect to time gives
$\dot{w}(t)=2\operatorname{Re}\langle
PAT(t)x,T(t)x\rangle\leq-c\|T(t)x\|^{2}$. Integrating this over $[0,t]$, we
obtain $0\leq w(t)\leq w(0)-c\int_{0}^{t}\|T(s)x\|^{2}\,ds$ and hence
$\int_{0}^{t}\|T(s)x\|^{2}\,ds\leq\frac{1}{c}w(0)=\frac{1}{c}\langle
Px,x\rangle$ for $x\in\operatorname{dom}A$ and $t\geq 0$. As
$\operatorname{dom}A$ is dense in $\mathcal{H}$, this inequality can be
extended to all $x\in\mathcal{H}$, which shows that $A$ is exponentially
stable. ∎

The pair $(A,C)$ is called exponentially detectable (cf. [4, Definition
5.2.1]), if there exists an operator $F\in L(\mathcal{Y},\mathcal{H})$ such
that $A+FC$ is exponentially stable.

In [6] it was proved that in finite dimensions strict pre-dissipativity (at
some steady state) with a quadratic storage function is equivalent to
detectability of the pair $(A,C)$. The next theorem partially extends this
statement to the infinite-dimensional case.

######  Theorem 17.

If $(A,C)$ is exponentially detectable, then there exists a non-negative
operator $P\in L(\mathcal{H})$ such that (4.6) holds. Conversely, if (4.6) is
satisfied with some $0\leq P\in L(\mathcal{H})$ and
$\operatorname{ran}C^{*}\subset\operatorname{ran}P$, then $(A,C)$ is
exponentially detectable.

###### Proof.

Assume that $(A,C)$ is exponentially detectable. By definition, there exists
$F\in L(\mathcal{Y},\mathcal{H})$ such that $A+FC$ is exponentially stable.
Hence, by Theorem 16 there exists a positive operator $P\in L(\mathcal{H})$
such that

$2\operatorname{Re}\langle(A+FC)x,Px\rangle=-\|x\|^{2},\qquad
x\in\operatorname{dom}A.$

Let $\eta>0$. For $x\in\operatorname{dom}A$ we obtain

$\displaystyle\|Cx\|^{2}-2\eta\operatorname{Re}\langle Ax,Px\rangle$
$\displaystyle=\|Cx\|^{2}+\eta\big{(}2\operatorname{Re}\langle
FCx,Px\rangle+\langle x,x\rangle\big{)}$
$\displaystyle=\left\langle\big{(}C^{*}C+\eta C^{*}F^{*}P+\eta PFC+\eta
I\big{)}x,x\right\rangle.$

Let $K:=PF$. In the remainder of this proof it will be shown that the bounded
self-adjoint operator

$T:=C^{*}C+\eta C^{*}K^{*}+\eta KC+\eta I$

is strictly positive for a suitable $\eta>0$. Then (4.6) follows. To this end,
let $\varepsilon\in(0,\frac{1}{4}\|K\|^{-2})$ and decompose the Hilbert space
as $\mathcal{H}=\mathcal{H}_{1}\oplus\mathcal{H}_{2}$ with
$\mathcal{H}_{1}=\operatorname{ran}E([0,\varepsilon])$ and
$\mathcal{H}_{2}=\operatorname{ran}E((\varepsilon,\infty))$, where $E$ denotes
the spectral measure of the non-negative self-adjoint operator $C^{*}C$. Set
$\mathbb{P}_{j}:=\mathbb{P}_{\mathcal{H}_{j}}$, $j=1,2$. With respect to this
decomposition write

$T=\begin{bmatrix}T_{11}&T_{12}\\\ T_{21}&T_{22}\end{bmatrix}$

where $T_{ij}=\mathbb{P}_{i}T|_{\mathcal{H}_{j}}$, $i,j=1,2$. For
$x\in\mathcal{H}_{1}$ we have

$\displaystyle\langle T_{11}x,x\rangle=\|Cx\|^{2}+\eta\langle
K^{*}x,Cx\rangle+\eta\langle
Cx,K^{*}x\rangle+\eta\|x\|^{2}\geq\eta\big{(}1-2\|K\|\sqrt{\varepsilon}\big{)}\|x\|^{2}$

as $\|Cx\|\leq\sqrt{\varepsilon}\|x\|$ (see Lemma 18(b)). With our choice of
$\varepsilon$ we conclude that $\kappa:=1-2\|K\|\sqrt{\varepsilon}>0$ so that
$T_{11}\geq\eta\kappa$.

Write $T=C^{*}C+\eta Z$ with $Z=C^{*}K^{*}+KC+I$. As both $\mathcal{H}_{1}$
and $\mathcal{H}_{2}$ are invariant under $C^{*}C$, it follows that

$T_{12}=\eta Z_{12},\quad T_{21}=\eta Z_{12}^{*},\quad\text{and}\quad
T_{22}=C^{*}C|_{\mathcal{H}_{2}}+\eta Z_{22},$

where $Z_{ij}=\mathbb{P}_{i}Z|_{\mathcal{H}_{j}}$. The second Schur complement
of the above decomposition of $T$ is the bounded operator in $\mathcal{H}_{2}$
given by

$S=T_{22}-T_{21}T_{11}^{-1}T_{12}=C^{*}C|_{\mathcal{H}_{2}}+\eta
Z_{22}-\eta^{2}Z_{12}^{*}T_{11}^{-1}Z_{12}.$

Now, $T_{11}\geq\eta\kappa$ implies $T_{11}^{-1}\leq\frac{1}{\eta\kappa}$ and
hence $Z_{12}^{*}T_{11}^{-1}Z_{12}\leq\frac{\|Z_{12}\|^{2}}{\eta\kappa}$.
Therefore,

$S\geq
C^{*}C|_{\mathcal{H}_{2}}+\eta\left(Z_{22}-\tfrac{1}{\kappa}\|Z_{12}\|^{2}\right),$

which is strictly positive for $\eta>0$ sufficiently small as
$C^{*}C|_{\mathcal{H}_{2}}\geq\varepsilon$. Therefore, the same holds for $T$
since

$\begin{bmatrix}T_{11}&T_{12}\\\
T_{21}&T_{22}\end{bmatrix}=\begin{bmatrix}I&0\\\
T_{21}T_{11}^{-1}&I\end{bmatrix}\begin{bmatrix}T_{11}&0\\\
0&S\end{bmatrix}\begin{bmatrix}I&T_{11}^{-1}T_{12}\\\ 0&I\end{bmatrix}.$

This proves (4.6).

Assume now that (4.6) holds with (WLOG) $\eta=1$ and that
$\operatorname{ran}C^{*}\subset\operatorname{ran}P$. Let
$\mathcal{H}_{0}:=\overline{\operatorname{ran}}P$ and
$P_{0}:=P|_{\mathcal{H}_{0}}\in L(\mathcal{H}_{0})$. Then $P_{0}$ is (possibly
not boundedly) invertible such that $PP_{0}^{-1}y=y$ for all
$y\in\operatorname{ran}P$. Next, the operator $F:=-\frac{1}{2}P_{0}^{-1}C^{*}$
is bounded by the closed graph theorem, i.e. $F\in
L(\mathcal{Y},\mathcal{H})$, and for $x\in\operatorname{dom}A$ we have

$\displaystyle 2\operatorname{Re}\langle(A+FC)x,Px\rangle$
$\displaystyle=2\operatorname{Re}\langle Ax,Px\rangle-\langle
P_{0}^{-1}C^{*}Cx,Px\rangle$ $\displaystyle=2\operatorname{Re}\langle
Ax,Px\rangle-\|Cx\|^{2}$ $\displaystyle\leq\,-m\|x\|^{2}.$

Hence, $A+FC$ is exponentially stable by Theorem 16. ∎

Combining Theorems 11 and 17 shows that exponential detectability implies
strict pre-dissipativity. If
$\operatorname{ran}C^{*}\subset\operatorname{ran}P$, then also the converse
holds. We note that invertibility (or, equivalently, strict positivity) of $P$
is sufficient for this condition to hold.

##  6\. Conclusion

We defined strict (pre-)dissipativity at controlled equilibria for linear-
quadratic optimal control problems in infinite dimensions and characterized
this notion in various ways. We investigated quadratic functions in terms of
boundedness from below and the existence of minimizers and characterized
strict pre-dissipativity with quadratic storage functions. It turned out that
this property depends on an algebraic condition (see (4.5)) and an analytic
one (see (4.6)) which is a form version of a Lyapunov-type operator
inequality. We also proved that this analytic condition is closely related to
exponential detectability of the pair $(A,C)$. This connection will be the
subject of future work. Moreover, we will investigate the differences between
conditions for strict dissipativity and strict pre-dissipativity.

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##  Appendix A Self-adjoint operators

Let $A$ be a self-adjoint operator in a Hilbert space $\mathcal{H}$. By the
spectral theorem for self-adjoint operators in Hilbert spaces (see, e.g., [17,
Theorem VIII.6]), there exists an orthoprojection-valued measure $E$ on the
Borel sigma algebra $\mathcal{B}$ of $\mathbb{R}$ which has the following
properties:

  * •

$E(\mathbb{R}\backslash\sigma(A))=0$ and $E(\sigma(A))=I$.

  * •

$E(\Delta_{1}\cap\Delta_{2})=E(\Delta_{1})E(\Delta_{2})=E(\Delta_{2})E(\Delta_{1})$
for $\Delta_{1},\Delta_{2}\in\mathcal{B}$.

  * •

$E(\Delta)\mathcal{H}$ is $A$-invariant for each $\Delta\in\mathcal{B}$.

  * •

$\sigma(A|_{E(\Delta)\mathcal{H}})\subset\sigma(A)\cap\Delta$ for closed
$\Delta\in\mathcal{B}$.

  * •

For each $x\in\operatorname{dom}A$,

$\langle
Ax,x\rangle=\int_{\mathbb{R}}t\,d\nu_{x}(t)\qquad\text{and}\qquad\|Ax\|^{2}=\int_{\mathbb{R}}t^{2}\,d\nu_{x}(t),$

where $\nu_{x}$ is the measure $\nu_{x}(\Delta):=\|E(\Delta)x\|^{2}$,
$\Delta\in\mathcal{B}$.

In the finite-dimensional case (i.e., when $A\in\mathbb{C}^{n\times n}$ is a
Hermitian matrix), the projection $E(\Delta)$ is the orthogonal projection
onto the sum of eigenspaces corresponding to the eigenvalues of $A$ in
$\Delta$. This is also the case if $A$ has only discrete spectrum in $\Delta$
(i.e., isolated eigenvalues of finite muliplicities). Here, we need the
following simple implications from the above.

######  Lemma 18.

The following statements hold:

  1. (a)

If $\Delta\in\mathcal{B}$ is compact and $0\notin\Delta$, then
$E(\Delta)\mathcal{H}\subset\operatorname{ran}A$.

  2. (b)

For $x\in E([-r,r])\mathcal{H}$, $r>0$, we have $\langle Ax,x\rangle\leq
r\|x\|^{2}$.

###### Proof.

The claim (a) follows from the fact that
$A_{\Delta}:=A|_{E(\Delta)\mathcal{H}}$ is boundedly invertible in
$E(\Delta)\mathcal{H}$. Hence
$E(\Delta)\mathcal{H}=\operatorname{ran}A_{\Delta}\subset\operatorname{ran}A$.
For the proof of (b) set $\Delta=[-r,r]$ and observe that

$\displaystyle\langle Ax,x\rangle$
$\displaystyle=\int_{\mathbb{R}}t\,d\nu_{x}(t)=\int_{\Delta}t\,d\nu_{x}(t)$
$\displaystyle\leq r\nu_{x}(\Delta)=r\|E(\Delta)x\|^{2}=r\|x\|^{2}$

for $x\in E(\Delta)\mathcal{H}$. ∎


††thanks: Corresponding author, email<EMAIL_ADDRESS>Corresponding author, email<EMAIL_ADDRESS>
# Quantum autoencoders for communication-efficient quantum cloud computing

Yan Zhu  QICI Quantum Information and Computation Initiative, Department of
Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong  HKU-
Oxford Joint Laboratory for Quantum Information and Computation  Ge Bai  QICI
Quantum Information and Computation Initiative, Department of Computer
Science, The University of Hong Kong, Pokfulam Road, Hong Kong  HKU-Oxford
Joint Laboratory for Quantum Information and Computation  Yuexuan Wang
College of Computer Science and Technology, Zhejiang University, Hangzhou,
China  Tongyang Li  Center on Frontiers of Computing Studies, Peking
University  School of Computer Science, Peking University  Center for
Theoretical Physics, Massachusetts Institute of Technology  Giulio Chiribella
QICI Quantum Information and Computation Initiative, Department of Computer
Science, The University of Hong Kong, Pokfulam Road, Hong Kong  Department of
Computer Science, University of Oxford, Parks Road, Oxford OX1 3QD, United
Kingdom  Perimeter Institute For Theoretical Physics, 31 Caroline Street
North, Waterloo N2L 2Y5, Ontario, Canada  The University of Hong Kong Shenzhen
Institute of Research and Innovation, Yuexing 2nd Rd Nanshan, Shenzhen 518057,
China

###### Abstract

In the model of quantum cloud computing, the server executes a computation on
the quantum data provided by the client. In this scenario, it is important to
reduce the amount of quantum communication between the client and the server.
A possible approach is to transform the desired computation into a compressed
version that acts on a smaller number of qubits, thereby reducing the amount
of data exchanged between the client and the server. Here we propose quantum
autoencoders for quantum gates (QAEGate) as a method for compressing quantum
computations. We illustrate it in concrete scenarios of single-round and
multi-round communication and validate it through numerical experiments. A
bonus of our method is it does not reveal any information about the server’s
computation other than the information present in the output.

##  1 Introduction

Over the past decade, quantum computing technology underwent a rapid series of
advances. Both the size and power of quantum computers have been steadily
increasing over the years, recently entering a new regime of "quantum
supremacy" [26], in which the output of the quantum computations can be barely
reproduced by the world’s best supercomputers. Milestone achievements are the
"quantum supremacy" demonstrations by Google [4] and USTC [34], using
superconducting qubits and photonic qubits, respectively.

A promising direction in quantum computing is the study of cloud computing
scenarios, where a client requests a remote server to perform some desired
quantum operations [11, 5, 23]. However, limits on the amount of quantum
communication between client and server constraint the size of the
computations that can be effectively implemented in quantum cloud computing.

In this paper, we address the problem of reducing the amount of communication
needed by the server to perform a quantum operation on quantum data provided
by the client. The operation belongs to a parametric family of quantum gates
known to both parties, but the specifications of the gate are known only to
the server. Our objective here is to maximize the accuracy of the executed
quantum operation when the capacity of the communication link and the total
number of qubits exchanged between the server and the client are limited.

Our main contribution is a method for compressing a parametric family of
quantum gates, turning it into another gate family acting on a smaller number
of qubits. Our method is based on autoencoders, a type of neural networks that
have been very successful in classical machine learning [28], and have been
recently used for the compression of quantum states [27]. We introduce a
quantum gate autoencoder (QAEGate), providing the first quantum machine
learning model that takes gates, rather than states, as inputs.

Our method also addresses the problem of minimizing the amount of information
revealed to the client. In many situations, the server would not like to
disclose any more information about the operation other than the unavoidable
information revealed by the application of the operation on the client’s
input. Our method achieves this feature by constructing a blind compression
protocol, independent of the specifications of the operation.

Technically, the training of our QAEGate model is based on stochastic gradient
descent [10]. We prove that the training is guaranteed to converge in
polynomial time in the size of the initial operation. We then conduct
numerical experiments that show the effectiveness of QAEGate in various
settings including single-round and multi-round communication between the
client and the server.

The remainder of the paper is organized as follows. In Section 2, we introduce
some related works about quantum cloud computing and quantum autoencoders.
Some preliminaries on autoencoders and quantum supermaps are provided in
Section 3. Section 4 is devoted to introduce the quantum cloud computing task
discussed in this paper. The structure and training details of our proposed
QAEGate model are covered in Section 5. Then, we explain how to apply our
method to concrete scenarios of quantum cloud computing in Section 6 and
conduct some numerical experiments in Section 7. We conclude this paper in
Section 8, with discussions on future directions for our method.

##  2 Related work

Classical cloud computing involves the execution of a computation on a remote
server [3]. In the quantum version of this scenario [5], a client provides an
input state and asks a remote server to perform a sequence of quantum gates on
it. Two important issues are (1) to cope with the limited amount of quantum
communication available in realistic scenarios, and (2) to minimize the
leakage of the information about the server’s computation. These two issues
have been previously approached with techniques from quantum Shannon theory
[32] and quantum cryptography [30]. Related issues have also been considered
in blind delegated quantum computation [11], where the goal is to guarantee
data confidentiality on the client side rather than the server side.

Quantum autoencoders for quantum states were proposed by [27] and has been
applied to quantum state compression and denoising quantum data [9, 1]. The
quantum model used in these works has a similar structure to classical
autoencoders, taking as input a quantum state represented by a fixed-length
vector. This model, however, cannot be extended from quantum states to quantum
gates, which in general are provided as black boxes. To convert a gate into a
quantum state, one would have to apply it on a fixed input state. However,
such conversion is not reversible due to the quantum No-programming theorem
[25, 33].

##  3 Preliminaries

#### Autoencoders.

An autoencoder is composed of an encoder and a decoder. One of its main
application is dimensionality reduction, where the encoder maps a high-
dimensional vector input $x$ to a low-dimensional representation $h$, and the
decoder maps $h$ back to a reconstructed high-dimensional vector $x^{\prime}$.
In the training process, the autoencoder is optimized so that the high-
dimensional vector $x^{\prime}$ is as close as possible to the original high-
dimensional vector $x$. If $x^{\prime}$ and $x$ are close enough, the overall
protocol provides a faithful compression of the original input $x$ into the
low dimensional vector $h$. The structure of a typical autoencoder is depicted
in Figure 1.

Figure 1: Structure of a typical autoencoder. In this specific example, the
autoencoder compresses a 4-dimensional vector into a 2-dimensional vector.

#### Quantum supermaps.

A general quantum operation is represented by a quantum channel $\mathcal{C}$,
which is a map from density operators to density operators [24]. A quantum
supermap [13, 14] $\tilde{\mathcal{S}}$ maps a quantum channel $\mathcal{C}$
into a quantum channel $\mathcal{C}^{\prime}$ as
$\mathcal{C}^{\prime}=\tilde{\mathcal{S}}(\mathcal{C})$. Every quantum
supermap can be represented by a quantum circuit in Figure 2 [13], where the
input quantum operation $\mathcal{C}$ sends states in $\mathcal{H}_{in}$ to
states in $\mathcal{H}_{out}$ and the output quantum operation
$\mathcal{C}^{\prime}$ sends states on $\mathcal{K}_{in}$ to states in
$\mathcal{K}_{out}$. The supermap is realized by two maps $\mathcal{V}$ and
$\mathcal{W}$ located at the input and at the output ports of the quantum
operation $\mathcal{C}$ respectively.

Figure 2: Quantum supermap.

##  4 The Quantum Cloud Computing Task

In this paper we will focus on a basic scenario of quantum cloud computing,
depicted in Figure 3. A parametric family of $n$-qubit quantum gates
$\\{U_{x}\\}_{x\in\mathsf{X}}$ with parameter $x$ represents a set of possible
quantum computations. A server is able to implement a gate $U_{x}$, unknown to
the client. The client chooses an $n$-qubit input state $\rho$ and asks the
server to apply the gate $U_{x}$ on it, thus obtaining the state
$\mathcal{U}_{x}(\rho):=U_{x}\rho U_{x}^{\dagger}$. At the same time, the
server wants to keep $x$ confidential, avoiding unnecessary information leaked
to the client during the communication.

A trivial way to achieve the above task is to have the sender send the quantum
state $\rho$ to the server, who performs the requirexx gate, anxx sends back
the output state $\mathcal{U}_{x}(\rho)$. However, the quantum communication
is an expensive resource, and it is often limited in realistic applications,
making it difficult to transmit a full $n$-qubit quantum state back and forth
between the client and the server. Here we consider the scenario where the
capacity of the quantum communication link is bounded by $a\leq n$ qubits.

In order to cope with the bottleneck on the amount of quantum communication,
we design a pair of quantum circuits, including an encoder $\mathcal{E}$
implemented by the server and a decoder $\mathcal{D}$ implemented by the
client. The encoder $\mathcal{E}$ serves as a quantum supermap transforming an
$n$-qubit quantum channel to an $a$-qubit quantum channel, while the decoder
$\mathcal{D}$ does the opposite, recovering the $n$-qubit quantum channel from
the output of the encoder. Since the decoder is implemented by the client, who
has no knowledge of the parameter $x$, the supermap $\mathcal{D}$ must be
independent of $x$. On the other hand, the encoder is implemented by the
server, and could in principle depend on $x$. However, a dependence on $x$ may
result into a leakage of information to the client. For this reason, we also
require the encoder $\mathcal{E}$ to be independent of $x$. We will later show
that this choice reaches the best possible confidentiality in the ideal
situation where $U_{x}$ is accurately implemented.

Figure 3: The basic scenario of quantum cloud computing. The client expects to
apply a quantum gate $U_{x}$ on its own state $\rho$ by communicating with a
server over a communication link of limited capacity.

##  5 Quantum Autoencoders for Quantum Gates

In this section we consider a basic scenario of quantum cloud computing with
one round of communication between client and server. In this scenario, we
develop a model of quantum autoencoders for quantum gates (QAEGate). In the
following, we will introduce the structure of the proposed model and
demonstrate its implementation and training through numerical experiments.
Furthermore, we prove that stochastic gradient descent has convergence
guarantee in the training of our QAEGate model. We will extend our model to
more scenarios in Section 6.

###  5.1 Structure

Our model consists of an encoder and a decoder. The encoder produces an
encoded quantum gate by inserting the gate $U_{x}$ into a suitable quantum
circuit, initializing $n-a$ input qubits to a fixed state $|0\rangle$, and
discarding $n-a$ output qubits. The result is a (generally noisy) quantum
channel acting on $a$ qubits. In turn, the decoder converts the $a$-qubit
channel back into an $n$-qubit channel, which aims to approximate the initial
gate. The structure of the encoder and the decoder are shown in Figure 4.

To construct the encoding and decoding circuits we use parameterized unitary
operators, which have been successfully employed to build variational quantum
circuits [16, 18]. Our construction is depicted in Figure 4. The encoder
consists of two parameterized unitary operators, denoted by
$U^{\boldsymbol{\theta_{\rm le}}}$ and $U^{\boldsymbol{\theta_{\rm re}}}$,
depending on parameters $\boldsymbol{\theta_{\rm
le}}=(\theta_{le_{1}},\theta_{le_{2}},...,\theta_{le_{m}})$ and
$\boldsymbol{\theta_{\rm
re}}=(\theta_{re_{1}},\theta_{re_{2}},...,\theta_{re_{m}})$. The gates
$U^{\boldsymbol{\theta_{\rm le}}}$ and $U^{\boldsymbol{\theta_{\rm re}}}$ are
placed on the left and right of the original quantum gate, respectively. The
decoder consists of two unitary operators $U^{\boldsymbol{\theta_{\rm ld}}}$
and $U^{\boldsymbol{\theta_{\rm rd}}}$, depending on parameters
$\boldsymbol{\theta_{\rm
ld}}=(\theta_{ld_{1}},\theta_{ld_{2}},...,\theta_{ld_{m}})$ and
$\boldsymbol{\theta_{\rm
rd}}=(\theta_{rd_{1}},\theta_{rd_{2}},...,\theta_{rd_{m}})$. The gates
$U^{\boldsymbol{\theta_{\rm ld}}}$ and $U^{\boldsymbol{\theta_{\rm rd}}}$ are
placed on the left and right of the encoded gate, combined with the identity
gate on the remaining $n-a$ qubits. Note that these parameterized unitary
operators are independent of the parameter $x$ of the original quantum gate
here.

Figure 4: Implementation details of QAEGate. The parameterized unitary
operators in yellow constitute the encoder while the parameterized unitary
operators in green constitute the decoder. In the encoding phase, we insert
the gate $U_{x}$ in the middle of the encoder, initialize $n-a$ input qubits
to a fixed state $|0\rangle$, and discard $n-a$ output qubits to obtain a
quantum channel acting on $a$ qubits. In the decoding phase, the decoder maps
the channel back to an approximation of the initial $n$-qubit quantum gate.

The choice of the parameters $\boldsymbol{\theta_{\rm le}}$,
$\boldsymbol{\theta_{\rm re}}$, $\boldsymbol{\theta_{\rm ld}}$, and
$\boldsymbol{\theta_{\rm rd}}$ is optimized in order to maximize the
similarity between the decoded quantum gate and the original quantum gate. As
a similarity measure, we use the overlap between the Choi operators [15, 19],
due to the relative ease of evaluating this quantity in numerical experiments.
As the optimization procedure, we will use a stochastic gradient descent
method, described in Subsection 5.2.

To run the optimization, one needs first to fix the parametrization of the
gates. The naive choice would be to pick a parametrization that can represent
arbitrary quantum gates. However, this approach has obvious difficulties:

  * •

Describing an arbitrary $n$-qubit unitary transformation requires an
exponential number of parameters, and therefore a full optimization of the
parameters in the autoencoder is only feasible for small values of $n$.

  * •

Due to hardware restrictions of the current quantum computers, it is difficult
to implement arbitrary $n$-qubit unitary operators. This is especially true
for the client, who may only have the ability to implement a small number of
basic quantum gates.

To address these problems, we propose an implementation of QAEGate based on an
approximation of generic $n$-qubit unitary transformations that uses a
polynomial number of parameters. We implement the parameterized unitary
operators for $n$ qubits by decomposing them into a sequence of parameterized
two-qubit unitary operators, where each pair of qubits corresponds to one such
operator. Figure 5 presents the case when $n=4$. For each two-qubit
parameterized unitary operator, it is further decomposed into a sequence of
basic quantum gates, as depicted in Figure 6.

Figure 5: Decomposition of a parameterized 4-qubit unitary gate into 6
parameterized two-qubit gates.

Figure 6: Decomposition of a two-qubit parameterized unitary gate.

Here $R_{X}(\theta)$ is the quantum gate defined as follows.

$\displaystyle
R_{X}(\theta):=\exp\left(i\theta\sigma_{x}\right)=\left(\begin{array}[]{cc}\cos\theta&-i\sin\theta\\\
-i\sin\theta&\cos\theta\\\ \end{array}\right)$

The other two gates $R_{Y}(\theta)$ and $R_{Z}(\theta)$ are similar to
$R_{X}(\theta)$ but for Pauli matrices $\sigma_{y}$ and $\sigma_{z}$.

The two-qubit gate $XX(\theta)$ is the Ising coupling gate that commonly
appears in real-world quantum computers with trapped-ion architecture [17]. It
is defined as $XX(\theta):=\exp(i\theta\sigma_{x}\otimes\sigma_{x})$ for the
Pauli matrix $\sigma_{x}$, and written in matrix form as:

$\displaystyle
XX(\theta)=\left[\begin{array}[]{cccc}\cos\theta&0&0&-i\sin\theta\\\
0&\cos\theta&-i\sin\theta&0\\\ 0&-i\sin\theta&\cos\theta&0\\\
-i\sin\theta&0&0&\cos\theta\end{array}\right].$

$YY(\theta)$ and $ZZ(\theta)$ are Ising coupling gates corresponding to Pauli
matrices $\sigma_{y}$ and $\sigma_{z}$ respectively.

###  5.2 Training

To find the optimal parameters of QAEGate, we adopt classical gradient-based
methods such as stochastic gradient descent [20].

We denote the parameters of QAEGate to be trained as $\boldsymbol{\theta}$.
The training set is composed of $k$ quantum gates sampled randomly from
$\\{U_{x}\\}$. In every iteration, we select a quantum gate $U$ from these $k$
gates randomly and insert it into the QAEGate. Then we calculate the overlap
$f$ between the Choi state $C$ of $U$ and the Choi state $C^{\prime}$ of the
decoded state from the QAEGate. We define the loss function as
$\mathcal{L}=1-f$ here and update the parameters of QAEGate according to the
gradients $\nabla_{\boldsymbol{\theta}}\mathcal{L}$ until the decoded quantum
gate is close enough to the original quantum gate or the maximum number of
iterations is reached. We describe the complete training process in Algorithm
1.

Data: $k$ quantum gates sampled randomly from $\\{U_{x}\\}$, maximum number of
iterations $K$, learning rate $\eta$, threshold $\delta$.

Initialize QAEGate parameters $\boldsymbol{\theta}$ randomly, $i=0$, $f=1$

while  _$i <K$ or $f>\delta$ _ do

Select the quantum gate $U$ randomly from $k$ input gates and insert it into
the QAEGate

Calculate the Choi state $C$ of $U$

Get the decoded state from the QAEGate and calculate its corresponding Choi
state $C^{\prime}$

$f={\rm Tr}(CC^{\prime})$, which is the overlap between $C$ and $C^{\prime}$

$\mathcal{L}=1-f$

Calculate $\nabla_{\boldsymbol{\theta}}\mathcal{L}$ and update
$\boldsymbol{\theta}$ as
$\boldsymbol{\theta}=\boldsymbol{\theta}-\eta\nabla_{\boldsymbol{\theta}}\mathcal{L}$

$i=i+1$

Algorithm 1 Quantum autoencoders for quantum gates (QAEGate).

###  5.3 Convergence Analysis

We further prove that our proposed QAEGate can be efficiently trained by the
stochastic gradient-based method in Algorithm 1. Specifically, we obtain the
following convergence guarantee:

######  Theorem 1 (Convergence guarantee).

If we perform SGD, as in Algorithm 1, to optimize $\boldsymbol{\theta}$ in the
training of the QAEGate model, then the convergence rate is
$T=\mathcal{O}(\frac{4\dim{\boldsymbol{\theta}}}{\epsilon^{4}})$ for achieving
the condition
$\mathbb{E}[\|\nabla_{\boldsymbol{\theta}}\mathcal{L}\|^{2}]\leq\epsilon^{2}$.

The proof is provided in the Appendix.

##  6 Applications

In this section we first apply QAGate to a basic scenario of the quantum cloud
computing, in which we just consider single-round communication between the
client and the server. Also, there is just one family of $n$-qubit quantum
gates $\\{U_{x}\\}_{x\in\mathsf{X}}$ stored in the server. Then, we extend the
structure of QAEGate to apply it to two more general scenarios. One allows
multiple-round communication between the client and the server. The other
enables the server to store different families of $n$-qubit quantum gates.

###  6.1 Basic scenario

Let us start from the basic scenario introduced in Figure 3 of Section 4. The
client starts from a generic $n$-qubit input $\rho$ and converts it into an
$a$-qubit state, by applying the gate $U^{\boldsymbol{\theta_{\rm ld}}}$ and
storing aside $n-a$ qubits in a quantum memory. Then, the $a$ qubits are sent
to the server through a quantum communication link. The server implements the
encoder supermap, converting the original $n$-qubit gate $U_{x}$ into a
quantum channel acting on $a$ qubits. The encoded channel is applied to an
$a$-qubit state received from the client, and produces an $a$-qubit state is
sent back to the client through the quantum communication link. Finally, the
client performs the gate $U^{\boldsymbol{\theta_{\rm rd}}}$ on the $a$ qubits
received from the server and on the $n-a$ qubits previously stored in the
quantum memory. Overall, the operations performed at the client’s end
implement the decoder, transforming the $a$-qubit channel implemented by the
server into an approximation of the target $n$-qubit gate.

An advantage of the structure of QAEGate is that it can fulfill the server’s
confidentiality. Specifically, the server wants to forbid the client from
obtaining the detailed implementation of the gates, which is described by $x$
in this scenario. For sure, some information about $x$ must be leaked to the
client since the client will be approximately granted one access to $U_{x}$,
which could be used to extract information about $x$, but we claim that an
arbitrary malicious client could not obtain more information than this minimal
amount. Formally, we obtain the following confidentiality guarantee for our
model:

######  Theorem 2 (Confidentiality guarantee).

In this protocol, the information about $x$ obtained by an arbitrary malicious
client is no larger than the information about $x$ obtainable by accessing
$U_{x}$ once.

###### Proof.

We first consider how much information about $x$ can be extracted given one
copy of $U_{x}$. We assume $x$ follows a certain distribution, and we denote
the random variable of $x$ as $X$. The most general way to extract information
from a gate $U_{x}$ is to insert it into a quantum circuit and make
measurements on the output of the circuit, as shown in Figure 7(a). Let the
output of the circuit to be $Y$, which is a random variable correlated with
$X$. We write $Y=\mathcal{F}(\mathcal{U}_{X})$ to denote the relationship
between $X$ and $Y$. The information one can know about $X$, by obtaining the
value of $Y$, is the mutual information $I(X;Y)$. The maximum amount of
information one can learn about $X$ is obtained by optimizing the circuit,
which is $I_{\max}:=\max_{\mathcal{F}}I(X;Y)$. Since any party who is able to
access $U_{x}$ could obtain this amount of information, the server, who grants
the client one use of $U_{x}$, would inevitably leak $I_{\max}$ amount of
information about $x$ to the client.

(a) A general quantum circuit $\mathcal{F}$ for extracting information about
$x$ from $U_{x}$.

(b) A general quantum circuit utilized by a malicious client to extract
information about $x$ by communicating with the server.

Figure 7:

Next, we consider a malicious client who wants to obtain the most information
about $x$. The server interacts with the client via the encoder, which is
equivalent to that the client has access to the encoded gate, which is a
channel $\mathcal{E}(\mathcal{U}_{x})$. The most general way for the client to
extract information about $x$ is to insert $\mathcal{E}(\mathcal{U}_{x})$ into
a circuit and measure the output of the circuit, as shown in Figure 7(b). Let
this circuit be $\mathcal{D}^{\prime}$ and its output be $Y^{\prime}$, which
is related to $X$ by
$Y^{\prime}=(\mathcal{D}^{\prime}\circ\mathcal{E})(\mathcal{U}_{X})$. The
maximum information the client could obtain is
$\max_{\mathcal{D}^{\prime}}I(X;Y^{\prime})$.

Now, since $\mathcal{E}$ is determined by the server, the circuit
$\mathcal{D}^{\prime}\circ\mathcal{E}$ with variable $\mathcal{D}^{\prime}$
forms a subset of the range of $\mathcal{F}$, which contains all possible
circuits taking $U_{x}$ as input. Therefore, the optimization over
$\mathcal{D}^{\prime}$ is upper bounded by the optimization over
$\mathcal{F}$, and thus we have

$\displaystyle\max_{\mathcal{D}^{\prime}}I(X;Y^{\prime})\leq I_{\max}\,.$ (1)

The equation above proves Theorem 2, since the left hand side is the maximal
information obtainable by an arbitrary malicious client, and the right hand
side is the inevitable information leakage given one access to $U_{x}$.

∎

###  6.2 Multi-round communication scenario

In this scenario, we relax the assumption of one-round communication and the
client can communicate with the server for multiple rounds. In each round, the
client can send to and receive from the sever an $a$-qubit quantum state
through the quantum communication link. For this multi-round communication
scenario, the server can exploit $U_{x}$ more than once. Here we depict the
case of two-round communication in Figure 8.

Figure 8: The multi-round communication scenario. The client can communicate
with the server for multiple rounds.

For the optimal performance, a general protocol may utilize different encoders
and decoders for each round of communication. Here we keep the remaining $n-a$
qubits of the output of the encoder in the first round and regard it as a part
of the input of the encoder in the second round. We depict such a variation of
the structure of the QAEGate model in Figure 9. The encoders and decoders of
the first and second round are separately parameterized.

Figure 9: QAEGate for multi-round communication scenario.

###  6.3 Sequence of gates scenario

In this scenario, we consider a more general situation that the client expects
the server to execute a sequence of gates from different parametric families.
In Figure 10, we depict the simplest case that the length of the sequence is
$2$. Here there are two classes of quantum gates,
$\\{U_{x}\\}_{x\in\mathsf{X}}$ and $\\{U_{y}\\}_{y\in\mathsf{Y}}$, stored in
the server. The client wants to obtain
$\mathcal{U}_{y}(\mathcal{U}_{x}(\rho))$ by communicating with the server.

Figure 10: Sequence of gates scenario. The client expects the server to
execute a sequence of gates from different parametric families.

In order to handle this scenario, we design the encoder and the decoder with
different structures and depict the modified QAEGate model in Figure 11.

Figure 11: QAEGate for sequence of gates scenario.

The encoder is composed of four parameterized unitary operators, denoted by
$U^{\boldsymbol{\theta_{le1}}}$, $U^{\boldsymbol{\theta_{re1}}}$,
$U^{\boldsymbol{\theta_{le2}}}$ and $U^{\boldsymbol{\theta_{re2}}}$. The first
two operators are placed on both sides of $U_{x}$ while the other two are
placed on both sides of $U_{y}$. The decoder is composed of three
parameterized parameterized unitary operators, denoted by
$U^{\boldsymbol{\theta_{\rm ld}}}$, $U^{\boldsymbol{\theta_{md}}}$ and
$U^{\boldsymbol{\theta_{\rm rd}}}$.

##  7 Numerical Experiments

In this section, we examine the performance of our proposed models by
numerical simulation. We show the effectiveness of our proposed models in
different settings for gates based on the Heisenberg model. All simulations
are implemented by Tensorflow quantum [12] and performed on a single GPU. We
train the models by stochastic gradient descent in all of the experiments.

###  7.1 Heisenberg model

Heisenberg model [6] is a famous statistical mechanical model, which describes
a magnetic system of half spins. It can be defined by the Hamiltonian

$\displaystyle\tilde{H}=$ $\displaystyle-\frac{1}{2}\sum_{j=1}^{n-1}(J_{\rm
x}\sigma_{j}^{x}\sigma_{j+1}^{x}+J_{\rm
y}\sigma_{j}^{y}\sigma_{j+1}^{y}+J_{\rm z}\sigma_{j}^{z}\sigma_{j+1}^{z})$
$\displaystyle-\frac{1}{2}\sum_{j=1}^{n}h\sigma_{j}^{z},$

where $J_{\rm x}$, $J_{\rm y}$, $J_{\rm z}$ and $h$ are constants representing
the strength of the coupling and the external magnetic field, respectively.
$\sigma^{x}$, $\sigma^{y}$, $\sigma^{z}$ are the Pauli matrices. $n$ is the
number of qubits in the quantum system. We define Heisenberg gates as the
quantum gates which represent evolution defined by a Hamiltonian operator of
Heisenberg model. These Heisenberg gates can be represented by

$\displaystyle U(t)=\exp(-i\tilde{H}t),$

where $\tilde{H}$ is the Hamiltonian of a Heisenberg model and $t$ is the
evolution time.

###  7.2 Simulation results

#### Basic Scenario.

In this scenario, we set $\\{U_{x}\\}$ stored in the sever as a class of
Heisenberg gates with $J_{\rm x}=J_{\rm y}=J_{\rm z}=0.1$ and $h=0.5$.

Here we define the evolution time $t$ as the parameter $x$, unknown to the
client. The client can only transmit $a=1$ qubit or $a=2$ qubits through the
quantum communication link. We consider three cases, $n=2,3,4$, in the
experiment. We exploit a training set composed of $50$ gates to optimize the
compression model and validate it on the test set composed of other $10$ gates
in each experiment.

We show the training curves of the QAEGate model in Figure 12. Here the x-axis
represents the epoch number of the training, and the y-axis represents the
overlap between the Choi states of the original gate and the decoded gate. We
can see that all of the training processes converge finally, which conforms to
our convergence analysis in Theorem 1.

Figure 12: Training curves of the QAEGate for the basic scenario. We train
three QAEGate models for three classes of Heisenberg gates, $n=2,3,4$,
respectively. All of training curves converge finally, which conforms to our
convergence analysis.

Furthermore, we present the performance of our QAEGate model on test sets in
Figure 13. The y-axis represents the average overlap values between the Choi
states of the decoded quantum gate and the original quantum gate. We can see
that our QAEGate model achieves satisfactory performance in all of the
experiments, which confirms the effectiveness of our proposed communication
model empirically. However, we can observe that QAEGate performs better when
the dimension of the class of Heisenberg gates $\\{U_{x}\\}$ is small.

Figure 13: The performance of our QAEGate model on test sets of the basic
scenario.

#### Multi-round communication scenario.

In this scenario, we allow two-round communication between the server and the
client while they can only transmit $a=1$ qubit through the quantum
communication link at a time.

We present the performance of our QAEGate model on test sets of the two-round
communication scenario in Figure 14. The results indicate that our proposed
model performs better when additional round of communication is available,
compared with the basic scenario with $a=1$. Especially for the case of $n=4$,
the average overlap value increases by $0.12$ compared with the basic single-
round scenario. We also compare the case of $a=2$ in the basic scenario and
the case of $a=1$ in the two-rounds communication scenario. In both cases, two
qubits are sent to and received from the server in total. We find that our
proposed models have similar performance.

Figure 14: The performance of our QAEGate model on test sets of the multi-
round communication scenario.

#### Sequence of gates scenario.

In this scenario, we set $\\{U_{x}\\}$ stored in the sever as a class of
Heisenberg gates with $J_{\rm x}=J_{\rm y}=J_{\rm z}=0.1$ and $h=0.5$ and
$\\{U_{y}\\}$ as a class of Heisenberg gates with $J_{\rm x}=J_{\rm y}=0.1$
and $J_{\rm z}=h=0.5$. Here We define the evolution times of them as $x$ and
$y$ whose values can be different. The client still can only transmit $a=1$
qubit through the quantum communication link. We also consider three cases,
$n=2,3,4$, in the experiment. The simulation results on test sets of this
scenario are shown in Figure 15 and our proposed model achieved satisfactory
performance in all three cases.

Figure 15: The performance of our QAEGate model on test sets of the sequence
of gates scenario.

##  8 Discussion and conclusions

We proposed a quantum machine learning model, QAEGate, for communication-
efficient quantum cloud computing, and applied it to single-round and multi-
round communication scenarios. In the single-round scenario, the protocol
avoids leakage of information about the server’s computation and achieves a
satisfactory performance in terms of overlap with the target gate versus the
number of qubits in its compressed version. By adjusting the structure of
QAEGate, we applied our method to other two scenarios, involving multiple
gates and multiple rounds of communication between the client and server.

The performance of our method has been tested numerically in a number of
examples. Due to the exponential complexity of simulating quantum gates on
classical computers, such analysis is necessarily limited to scenarios
involving a small number of qubits. On the other hand, our method is meant to
be eventually run on real quantum computers, on which the complexity of
QAEGate grows only polynomially. While the algorithm The current progress of
quantum hardware suggests that a real-world test may take place not too far in
the future.

An application of the techniques developed in this paper is learning unknown
quantum gates [8, 21, 29]. In this task, the goal is to imprint an unknown
gate into the state of a quantum memory from which the gate can be accessed at
a later time. Our QAEGate can be adapted to this task by adjusting the
structure of the encoder and the decoder so that quantum gates can be encoded
into quantum states. Referring to Figure 4, the idea is to choose different
number of qubits $a$ and $a^{\prime}$ for the input and output wires of the
encoder, respectively. In particular, one can set $a$ to zero, meaning that
the encoder produces a quantum state of $a^{\prime}$ qubits. The number of
qubits $a^{\prime}$ can be regarded as the size of the quantum memory of a
learning machine designed to learn a generic gate from a given parametric
family of quantum gates. An interesting feature of our method is that it
allows one to set the memory size (i.e. to fix $a^{\prime}$) and to achieve
approximate learning subject to constraints on the available quantum memory.

Another interesting direction of future research is assessing the power of
quantum processors [7, 22]. It has been often pointed out that the number of
qubits in a quantum processor is not an appropriate measure, because the
presence of noise generally limits the set of achievable computations. Hence,
it is important to have more inclusive measures, that take into account not
only the raw number of qubits, but also the size of the set of operations
implementable on them. The approach of quantum gate compression sheds light
into this problem by providing a rigorous notion of “effective size” of a
quantum processor, defined as the minimum number of qubits on which the
operations of the processor can be compressed with sufficiently high fidelity.

More specifically, the set of operations implemented by a given quantum
processor can be regarded as a parametric family of (generally noisy) quantum
gates. When the processor is not universal, such a family may be compressible
into operations acting on a smaller number of qubits, meaning that the
computations achieved by the processor can be in principle simulated with a
smaller quantum computer. The size of this minimal simulator can then be
regarded as a measure of the computing power of the original quantum
processor. Such a measure is independent of the hardware implementation, and
could be in principle evaluated by connecting the given processor to a
reference quantum computer that runs the QAEGate algorithm, or a
generalization of it working with noisy input gates.

A limitation of the above approach is that the evaluation of the minimum
number of qubits requires a trusted quantum computer of the same size of the
original processor. On the other hand, once a sufficiently large universal
quantum computer becomes available, it can be used as a standard reference for
assessing the power of other quantum processors, and to construct efficient
simulations that can be run on universal quantum computers of smaller size.

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##  Appendix A Proof of Theorem 1

According to [2], we have the following lemma about convergence of SGD for
nonconvex functions:

######  Lemma 1.

If a function $f(x)$ is $L_{1}$-smooth and we perform SGD update
$x_{t+1}\leftarrow x_{t}-\eta\nabla f_{i}(x_{t})$ each time for a random
$i\in[n]$, then the convergence rate is
$T=\mathcal{O}(\frac{L_{1}}{\epsilon^{4}})$ for finding $\mathbb{E}[\|\nabla
f(x)\|^{2}]\leq\epsilon^{2}$.

Therefore, we first discuss the smoothness property of the loss function
$\mathcal{L}(\boldsymbol{\theta})$ in the training of QAEGate model. Note that
the first- and second-order Lipschitz smoothness are equivalent to Lipschitz
continuity of the first- and second-order derivatives. So we start with a
lemma about the Lipschitz continuity of multivariate functions [31].

######  Lemma 2.

Given some function $f\colon\mathbb{R}^{M}\to\mathbb{R}$, if all partial
derivatives of $f$ are continuous, then for any $a,b\in\mathbb{R}$ the
function $f\colon[a,b]^{M}\to\mathbb{R}$ is $L$-Lipschitz continuous with

$\displaystyle
L=\sqrt{M}\max_{j\in\\{1,\dots,M\\}}\sup_{\boldsymbol{x}\in[a,b]^{M}}\left|\frac{\partial
f(\boldsymbol{x})}{\partial x_{j}}\right|.$ (2)

Next we prove a more general result about the smoothness of measurement
probability of quantum circuits, and then show that
$\mathcal{L}(\boldsymbol{\theta})$ fits into this result.

######  Lemma 3.

Consider a quantum circuit consisting of any number of fixed unitary gates and
$M$ variable unitary gates $U_{1}(\theta_{1}),\dots,U_{M}(\theta_{M})$, where
$U_{j}(\theta_{j}):=\exp(i\theta_{j}H_{j})$ for some Hermitian operator
$H_{j}$. Then the probability of any measurement outcome of the output of this
circuit is $L_{1}$-smooth and $L_{2}$-second-order smooth with respect to
$\boldsymbol{\theta}=(\theta_{1},\dots,\theta_{M})$, where

$\displaystyle L_{1}=4MH_{\max}^{2}$ (3) $\displaystyle
L_{2}=8M^{3/2}H_{\max}^{3}$ (4)

where $H_{\max}:=\max_{j}\|H_{j}\|_{2}$ and $\|\cdot\|_{2}$ is the induced
operator norm defined as $\|A\|_{2}:=\sup\frac{\|Ax\|_{2}}{\|x\|_{2}}$.

###### Proof.

Let $|\psi_{0}\rangle$ be the initial state of the circuit. Without loss of
generality, we assume the variable unitary gates are labeled as
$U_{1}(\theta_{1})$ to $U_{M}(\theta_{M})$ according to the order they are
applied. Then the output of this circuit can be written as
$|\psi_{\boldsymbol{\theta}}\rangle:=V_{M}U_{M}(\theta_{M})\dots
V_{1}U_{1}(\theta_{1})V_{0}|\psi_{0}\rangle$, where $V_{1},\dots,V_{M}$
denotes the fixed unitary gates. A measurement outcome can be described by a
positive operator-valued measure (POVM) element $P$ with the property that $P$
and $I-P$ are both positive semidefinite, where $I$ is the identity operator
of the output. The probability of this outcome is

$\displaystyle
f(\boldsymbol{\theta}):=\operatorname{Tr}[P|\psi_{\boldsymbol{\theta}}\rangle\langle\psi_{\boldsymbol{\theta}}|]=\langle\psi_{\boldsymbol{\theta}}|P|\psi_{\boldsymbol{\theta}}\rangle\,.$
(5)

To study the smoothness of $f(\boldsymbol{\theta})$, we take the partial
derivatives:

$\displaystyle\frac{\partial f(\boldsymbol{\theta})}{\partial\theta_{j}}$
$\displaystyle=\frac{\partial\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}}P|\psi_{\boldsymbol{\theta}}\rangle+\langle\psi_{\boldsymbol{\theta}}|P\frac{\partial|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{j}}=2\Re\left(\frac{\partial\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}}P|\psi_{\boldsymbol{\theta}}\rangle\right)$
$\displaystyle\frac{\partial^{2}f(\boldsymbol{\theta})}{\partial\theta_{j}\partial\theta_{k}}$
$\displaystyle=2\Re\left(\frac{\partial^{2}\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}\partial\theta_{k}}P|\psi_{\boldsymbol{\theta}}\rangle+\frac{\partial\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}}P\frac{\partial|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{k}}\right)$
$\displaystyle\frac{\partial^{3}f(\boldsymbol{\theta})}{\partial\theta_{j}\partial\theta_{k}\partial\theta_{l}}$
$\displaystyle=2\Re\left(\frac{\partial^{3}\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}\partial\theta_{k}\partial\theta_{l}}P|\psi_{\boldsymbol{\theta}}\rangle+\frac{\partial^{2}\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}\partial\theta_{k}}P\frac{\partial|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{l}}+\frac{\partial^{2}\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}\partial\theta_{l}}P\frac{\partial|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{k}}+\frac{\partial\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}}P\frac{\partial^{2}|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{k}\partial\theta_{l}}\right)$

These partial derivatives can be bounded with vector and operator norms. For a
unitary operator $U$, $\|U\|_{2}=1$. For the POVM element $P$, $\|P\|_{2}\leq
1$. The second-order partial derivatives can be bounded as:

$\displaystyle\left|\frac{\partial^{2}f(\boldsymbol{\theta})}{\partial\theta_{j}\partial\theta_{k}}\right|$
$\displaystyle\leq
2\left|\frac{\partial^{2}\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}\partial\theta_{k}}P|\psi_{\boldsymbol{\theta}}\rangle+\frac{\partial\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}}P\frac{\partial|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{k}}\right|$
$\displaystyle\leq
2\left\|\frac{\partial^{2}\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}\partial\theta_{k}}\right\|_{2}\|P\|_{2}\||\psi_{\boldsymbol{\theta}}\rangle\|_{2}+2\left\|\frac{\partial\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}}\right\|_{2}\|P\|_{2}\left\|\frac{\partial|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{k}}\right\|_{2}$
$\displaystyle\leq
2\max_{a,b}\left\|\frac{\partial^{2}|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{a}\partial\theta_{b}}\right\|_{2}+2\max_{a}\left\|\frac{\partial|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{a}}\right\|_{2}^{2}$
(6)

Similarly, for the third-order partial derivatives,

$\displaystyle\left|\frac{\partial^{3}f(\boldsymbol{\theta})}{\partial\theta_{j}\partial\theta_{k}\partial\theta_{l}}\right|$
$\displaystyle\leq
2\left|\frac{\partial^{3}\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}\partial\theta_{k}\partial\theta_{l}}P|\psi_{\boldsymbol{\theta}}\rangle+\frac{\partial^{2}\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}\partial\theta_{k}}P\frac{\partial|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{l}}+\frac{\partial^{2}\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}\partial\theta_{l}}P\frac{\partial|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{k}}+\frac{\partial\langle\psi_{\boldsymbol{\theta}}|}{\partial\theta_{j}}P\frac{\partial^{2}|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{k}\partial\theta_{l}}\right|$
$\displaystyle\leq
2\max_{a,b,c}\left\|\frac{\partial^{3}|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{a}\partial\theta_{b}\partial\theta_{c}}\right\|_{2}+6\left(\max_{a,b}\left\|\frac{\partial^{2}|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{a}\partial\theta_{b}}\right\|_{2}\right)\left(\max_{a}\left\|\frac{\partial|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{a}}\right\|_{2}\right)$
(7)

Therefore, to show that the second- and third-order partial derivatives are
bounded, it suffices to show that
$\max_{a}\left\|\frac{\partial|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{a}}\right\|_{2},\max_{a,b}\left\|\frac{\partial^{2}|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{a}\partial\theta_{b}}\right\|_{2}$
and
$\max_{a,b,c}\left\|\frac{\partial^{3}|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{a}\partial\theta_{b}\partial\theta_{c}}\right\|_{2}$
are all bounded. We observe that $\frac{\partial U_{j}(\theta_{j})}{\partial
j}=\frac{\partial\exp(i\theta_{j}H_{j})}{\partial j}=iH_{j}U_{j}(\theta_{j})$,
and we have

$\displaystyle\max_{a}\left\|\frac{\partial|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{a}}\right\|_{2}$
$\displaystyle=\max_{a}\left\|V_{M}U_{M}(\theta_{M})\dots
V_{a}iH_{a}U_{a}(\theta_{a})\dots
U_{1}(\theta_{1})V_{0}|\psi_{0}\rangle\right\|_{2}$
$\displaystyle\leq\max_{a}\left\|V_{M}U_{M}(\theta_{M})\dots
V_{a}\right\|_{2}\|H_{a}\|_{2}\left\|U_{a}(\theta_{a})\dots
U_{1}(\theta_{1})V_{0}|\psi_{0}\rangle\right\|_{2}$
$\displaystyle=\max_{a}\|H_{a}\|_{2}=H_{\max}$ (8)

With similar derivations, we can obtain
$\max_{a,b}\left\|\frac{\partial^{2}|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{a}\partial\theta_{b}}\right\|_{2}^{2}\leq
H_{\max}^{2}$ and
$\max_{a,b,c}\left\|\frac{\partial^{3}|\psi_{\boldsymbol{\theta}}\rangle}{\partial\theta_{a}\partial\theta_{b}\partial\theta_{c}}\right\|_{2}\leq
H_{\max}^{3}$. According to Eqs. (6) and (7), we obtain

$\displaystyle\left|\frac{\partial^{2}f(\boldsymbol{\theta})}{\partial\theta_{j}\partial\theta_{k}}\right|$
$\displaystyle\leq 4H_{\max}^{2}$ (9)
$\displaystyle\left|\frac{\partial^{3}f(\boldsymbol{\theta})}{\partial\theta_{j}\partial\theta_{k}\partial\theta_{l}}\right|$
$\displaystyle\leq 8H_{\max}^{3}$ (10)

Consider $\frac{\partial f(\boldsymbol{\theta})}{\partial\theta_{j}}$ as a
function of $\boldsymbol{\theta}$. Since its partial derivatives
$\frac{\partial^{2}f(\boldsymbol{\theta})}{\partial\theta_{j}\partial\theta_{k}}$
are bounded by $4H_{\max}^{2}$, according to Lemma 2, $\frac{\partial
f(\boldsymbol{\theta})}{\partial\theta_{j}}$ is $L_{1}^{\prime}$-continuous
with $L_{1}^{\prime}=4\sqrt{M}H_{\max}^{2}$. Then

$\displaystyle\|\nabla f(\boldsymbol{\alpha})-\nabla
f(\boldsymbol{\beta})\|_{2}$
$\displaystyle\leq\sqrt{M}\max_{j}\left|\frac{\partial
f(\boldsymbol{\alpha})}{\partial\theta_{j}}-\frac{\partial
f(\boldsymbol{\beta})}{\partial\theta_{j}}\right|$
$\displaystyle\leq\sqrt{M}L_{1}^{\prime}\|\alpha-\beta\|_{2}=4MH_{\max}^{2}\|\alpha-\beta\|_{2}$
(11)

and therefore $f$ is $L_{1}$-smooth with $L_{1}=4MH_{\max}^{2}$. Similarly,
consider
$\frac{\partial^{2}f(\boldsymbol{\theta})}{\partial\theta_{j}\partial\theta_{k}}$
as a function of $\boldsymbol{\theta}$. Since its partial derivatives
$\frac{\partial^{3}f(\boldsymbol{\theta})}{\partial\theta_{j}\partial\theta_{k}\partial\theta_{l}}$
are bounded by $8H_{\max}^{3}$, according to Lemma 2,
$\frac{\partial^{2}f(\boldsymbol{\theta})}{\partial\theta_{j}\partial\theta_{k}}$
is $L_{2}^{\prime}$-continuous with $L_{2}^{\prime}=8\sqrt{M}H_{\max}^{3}$.
Then

$\displaystyle\|\nabla^{2}f(\boldsymbol{\alpha})-\nabla^{2}f(\boldsymbol{\beta})\|_{2}$
$\displaystyle\leq
M\max_{j,k}\left|\frac{\partial^{2}f(\boldsymbol{\alpha})}{\partial\theta_{j}\partial\theta_{k}}-\frac{\partial^{2}f(\boldsymbol{\beta})}{\partial\theta_{j}\partial\theta_{k}}\right|$
$\displaystyle\leq
ML_{2}^{\prime}\|\alpha-\beta\|_{2}=8M^{3/2}H_{\max}^{3}\|\alpha-\beta\|_{2}$
(12)

and therefore $f$ is $L_{2}$-second-order smooth with
$L_{2}=8M^{3/2}H_{\max}^{3}$. ∎

Based on above lemma, we obtain the following theorem:

######  Theorem 3 (Smoothness).

The loss function $\mathcal{L}(\boldsymbol{\theta})$ in the training of
QAEGate model is $L_{1}$-smooth and $L_{2}$-second-order smooth.

###### Proof.

Let $p(\boldsymbol{\theta})$ be the probability of outcome $|+\rangle$ in the
SWAP test when the gates are parameterized with
$\boldsymbol{\theta=(\theta_{\rm le},\theta_{\rm re},\theta_{\rm
ld},\theta_{\rm rd})}$. Because of
$\mathcal{L}(\boldsymbol{\theta})=1-f(\boldsymbol{\theta})$ and
$f(\boldsymbol{\theta})=2p(\boldsymbol{\theta})-1$, we just need to discuss
the smoothness of $p(\boldsymbol{\theta})$. Since $p(\boldsymbol{\theta})$ is
the probability of a measurement outcome of a unitary circuit, and in this
circuit, every variable gate has the form of $U_{\theta}=\exp(i\theta H)$, we
can apply Lemma 3. Thus, $p(\boldsymbol{\theta})$ is $L_{1}$-smooth and
$L_{2}$-second-order smooth where
$L_{1}=4MH_{\max}^{2},L_{2}=8M^{3/2}H_{\max}^{3}$ and $M$ is the number of
variable gates in the circuit. To prove Theorem 3, we only need to show that
$H_{\max}$ is bounded.

The variable gates in our circuit only includes the following single- and two-
qubit gates:

$\displaystyle R_{X}(\theta):=\exp(i\theta\sigma_{x}),\leavevmode\nobreak\
R_{Y}(\theta):=\exp(i\theta\sigma_{y}),\leavevmode\nobreak\
R_{Z}(\theta):=\exp(i\theta\sigma_{z})$ (13) $\displaystyle
XX(\theta):=\exp(i\theta\sigma_{x}\otimes\sigma_{x}),\leavevmode\nobreak\
YY(\theta):=\exp(i\theta\sigma_{y}\otimes\sigma_{y}),\leavevmode\nobreak\
ZZ(\theta):=\exp(i\theta\sigma_{z}\otimes\sigma_{z})$ (14)

Therefore,
$H_{\max}=\max\\{\|\sigma_{x}\|_{2},\|\sigma_{y}\|_{2},\|\sigma_{z}\|_{2},\|\sigma_{x}\otimes\sigma_{x}\|_{2},\|\sigma_{y}\otimes\sigma_{y}\|_{2},\|\sigma_{z}\otimes\sigma_{z}\|_{2}\\}=1$.
∎

###### Proof.

Based on Theorem 3, we have proved that $\mathcal{L}(\boldsymbol{\theta})$ is
$L_{1}$-smooth, where $L_{1}=4M$. In our implementation, there is only one
variable for each variable unitary gate, so $M=\dim{\boldsymbol{\theta}}$
here. Then we can apply Lemma 1 and obtain that the convergence rate is
$T=\mathcal{O}(\frac{4\dim{\boldsymbol{\theta}}}{\epsilon^{4}})$ for finding
$\mathbb{E}[\|\nabla_{\boldsymbol{\theta}}\mathcal{L}(\boldsymbol{\theta})\|^{2}]\leq\epsilon^{2}$.

∎