A proposal of adaptive parameter tuning for robust stabilizing control of $N$–level quantum angular momentum systems

The contributions of this study are summarized as follows:

• •

  An adaptive parameter tuning algorithm for robust quantum stabilizing control is proposed (Equation (6)).

• •

  An asymptotic property of the estimate under steady-state conditions is derived (Proposition 4).

• •

  Local convergence of the proposed method is evaluated under certain assumptions (Theorem 6).

The rest of this paper is organized as follows. The problem is stated in Section II. In Sec. III, we propose an adaptive parameter tuning algorithm and the analytical results are shown. The proposed method is evaluated numerically and compared with the application of [21] in Sec. IV. We conclude the paper in Sec. V.

$\mathbb{N}$, $\mathbb{R}$ and $\mathbb{C}$ are natural, real, and complex numbers, respectively, and $\mathrm{i}:=\sqrt{-1}$. $\mathbb{R}^{n\times m}$ and $\mathbb{C}^{n\times m}$ are real and complex $n\times m$ matrices, respectively. $X^{\ast}$ implies the Hermitian conjugate of matrix $X$. We use $I_{n}\in\mathbb{C}^{n\times n}$ as the identify matrix. For $X=X^{\ast}\in\mathbb{C}^{n\times n}$, $X>0$ ($X\geq 0$) indicates that $X$ is a positive-(semi)definite matrix. When two positive-semidefinite matrices $X$ and $Y$ satisfy $X=Y^{2}$, we denote $Y=\sqrt{X}$. The absolute value of a square matrix is defined as $|X|:=\sqrt{X^{\ast}X}$ and for $X\in\mathbb{C}^{n\times n}$, the trace norm is defined as $\|X\|_{\rm Tr}:=\mathrm{Tr}[|X|]$. $\mathcal{S}(\mathbb{C}^{n}):=\{\rho\in\mathbb{C}^{n\times n}\ |\ \rho=\rho^{\ast}\geq 0,\ \mathrm{Tr}[\rho]=1\}$. Denote $[X,Y]_{-}:=XY-YX$ $\forall X,Y\in\mathbb{C}^{n\times n}$. $\mathbb{E}_{w}$ indicates the expectation in terms of a random variable or a stochastic process $w$. $O(\varepsilon)$ is Landau’s $O$ as $\varepsilon\to 0$.

Let $J\in\mathbb{N}$ and $N:=2J+1$, and let us consider the following quantum stochastic differential equation [2, 9, 21].

	$\displaystyle d\rho(t)=$	$\displaystyle\mathrm{i}[H_{\omega}(u(t)),\rho(t)]_{-}dt-\frac{M}{2}[J_{z},[J_{z},\rho(t)]_{-}]_{-}dt$
		$\displaystyle+\sqrt{\eta M}\left(J_{z}\rho(t)+\rho(t)J_{z}-2\mathrm{Tr}[J_{z}\rho(t)]\rho(t)\right)$
		$\displaystyle\quad\quad\times\left(dy(t)-2\sqrt{\eta M}\mathrm{Tr}[J_{z}\rho(t)]dt\right)$		(1)

with an initial state $\rho(0)\in\mathcal{S}(\mathbb{C}^{N})$, where $\rho(t)\in\mathcal{S}(\mathbb{C}^{N})$ is a conditional state of the system, $u(t)$ is the control input, $y(t)$ is the measurement output, $H_{\omega}(u):=\omega J_{z}+uJ_{y}$,

	$\displaystyle J_{z}:=$	$\displaystyle\mathrm{diag}(J,J-1,\ \dots,\ -J+1,-J),$
	$\displaystyle J_{y}:=$	$\displaystyle\begin{bmatrix}0&-\mathrm{i}c_{1}&0&\cdots&0\\ \mathrm{i}c_{1}&\ddots&\ddots&&\vdots\\ 0&\ddots&\ddots&\ddots&0\\ \vdots&&\ddots&\ddots&-\mathrm{i}c_{N-1}\\ 0&\cdots&0&\mathrm{i}c_{N-1}&0\end{bmatrix},$

$c_{m}=\frac{1}{2}\sqrt{(2J+1-m)m}$, $m=1,\dots,N-1$, and $\omega>0$, $M>0$ is the coupling constant, and $\eta\in(0,1]$ denotes measurement efficiency [25]. $u(t)$ is control input and throughout this paper, $u$ is assumed bounded. $\rho(t)$ is called the state, which is a quantum counterpart of (conditional) probability law. Equation (1) is called stochastic master equation having $N$ different equilibrium points if the control input $u(t)=0$. We denote each equilibrium point as $\rho_{n}\in\mathcal{S}(\mathbb{C}^{N})$, $n=0,\dots,2J$, and the target state is described by $\rho_{\bar{n}}$. Note that $\rho_{n}$ consists of an eigenvector of $J_{z}$, i.e.,

$\displaystyle J_{z}\rho_{n}=\rho_{n}J_{z}=(J-n)\rho_{n}\quad\forall n\in\{0,\dots,2J\}.$

A stabilization problem of (1) is to ensure that the state converges to a desirable equilibrium point. Therefore, model uncertainty must be considered in practical situations as different uncertainties exist in the model, initial state, and parameters of (1). In this paper, we consider parametric uncertainty and unknown initial condition. Then, the nominal model is described as follows.

	$\displaystyle d\hat{\rho}(t)=$	$\displaystyle\mathrm{i}[H_{\hat{\omega}}(u(t)),\hat{\rho}(t)]_{-}dt-\frac{\hat{M}}{2}[J_{z},[J_{z},\hat{\rho}(t)]_{-}]_{-}dt$
		$\displaystyle+\sqrt{\hat{\eta}\hat{M}}\left(J_{z}\hat{\rho}(t)+\hat{\rho}(t)J_{z}-2\mathrm{Tr}[J_{z}\hat{\rho}(t)]\hat{\rho}(t)\right)$
		$\displaystyle\quad\quad\times\left(dy(t)-2\sqrt{\hat{\eta}\hat{M}}\mathrm{Tr}[J_{z}\hat{\rho}(t)]dt\right)$		(2)

with its initial state $\hat{\rho}(0)\in\mathcal{S}(\mathbb{C}^{N})$. The differences from (1) are the initial state $\hat{\rho}(0)$ and the parameters $(\hat{\omega},\hat{M},\hat{\eta})$. Because the accessible state is $\hat{\rho}(t)$, the goal of the stabilization problem is to find a feedback controller $u(t)=u_{FB}(\hat{\rho}(t))$ that ensures $\lim_{t\to\infty}\rho(t)=\rho_{\bar{n}}$ as one of the stochastic convergences.

Liang et. al. [21] found certain sufficient conditions when the nominal state stabilization becomes the true state stabilization. One of their main results is that if the ratio of $\hat{\eta}\hat{M}$ and $\eta M$ is close to $1$, then there exists a stabilizing controller. For convenience, we write $\hat{\theta}:=\sqrt{\hat{\eta}\hat{M}}$ and $\theta:=\sqrt{\eta M}$, and then the following result holds [21].

Note that Theorem 1 is only part of their results. See [21] for the details.

Before stating our problem, we present a minor modification of Theorem 1.

Corollary 2 implies that if we can set the parameter $\hat{\theta}(t)$ appropriately, the stabilization is achieved even if the initial parameter $\hat{\theta}(0)$ does not satisfy the condition (3). Therefore, the problem we deal with is how to estimate $\theta$ while stabilizing the state $\rho(t)$. Adaptive parameter tuning $\hat{\theta}(t)$ with stabilizing control is a simple and useful solution for the problem, as shown in the next section.

For convenience, we use $\hat{\theta}(t):=\sqrt{\hat{\eta}\hat{M}(t)}$, $\hat{x}(t):=\mathrm{Tr}[J_{z}\hat{\rho}(t)]$, and $x(t):=\mathrm{Tr}[J_{z}\rho(t)]$. Then, we propose the following parameter tuning algorithm.

	$\displaystyle d\hat{\theta}(t)=$	$\displaystyle f(t)\left\{-\hat{x}(t)^{2}\hat{\theta}(t)dt+\frac{1}{2}\hat{x}(t)dy(t)\right\},$		(6)
	$\displaystyle f(t):=$	$\displaystyle(Kt+1)^{-p},\quad t\geq 0,$		(7)

where $p\in(0,1]$ and $K>0$. Note that we update $\hat{M}(t)$ as $\hat{M}(t)=\hat{\theta}(t)^{2}/\hat{\eta}$. From the filtering theory [25, 26], $dy(t)$ can be replaced by $dw(t)+2\theta x(t)$, where $w(t)$ is a standard Wiener process, and (6) then gives

$\displaystyle d\hat{\theta}(t)=$

$\displaystyle f(t)\hat{x}(t)\left\{(\theta x(t)-\hat{\theta}(t)\hat{x}(t))dt+\frac{1}{2}dw(t)\right\}.$

If the noise $w(t)$ is removed, updating $\hat{\theta}(t)$ by Eq. (6) implies the same as instant gradient method of the cost function $|\theta x(t)-\hat{\theta}(t)\hat{x}(t)|^{2}$ with the weight $f(t)$. This is a type of Robbins-Monro algorithm for continuous-time problems [27, 28, 29]. Clearly, if $x(t)=\hat{x}(t)\neq 0$ for all $t\geq 0$, then the parameter tuning law is the continuous-time Robbins-Monro algorithm, which is guaranteed to converge to the true parameter. Unfortunately, the assumption $x(t)=\hat{x}(t)\neq 0$ $\forall t\geq 0$ may not hold; therefore, we need to seek the condition when the parameter $\hat{\theta}(t)$ converges to the region described by (4). Note that the true parameter $\theta$ cannot be an equilibrium point of the system (6). Thus, the noise $w$ is unavoidable, so we examine how to choose the parameter $(K,p)$ and obtain the accurate estimate asymptotically.

Here, we describe that the choice of the parameters $p$ and $K$ of (7) is valid from the following proposition. For convenience, we write $\mathbb{E}_{w}[\bullet]\equiv\mathbb{E}_{w}[\bullet|\rho(0),\hat{\rho}(0)]$.

From Proposition 4, if $\rho(t)$ and $\hat{\rho}(t)$ are in the same equilibrium state, the parameter $\hat{\theta}(t)$ updated by (6) converges to the true value with probability one. Unfortunately, since the true state $\rho(t)$ is not accessible, we cannot confirm whether $\rho(t)$ and $\hat{\rho}(t)$ are practically in the same equilibrium state. To avoid being trapped in different equilibrium points before learning the parameter accurately, we employ feedforward control in the following subsection.

In this subsection, we evaluate our tuning algorithm (6) with the following control input.

$\displaystyle u(t)=$

$\displaystyle u_{FB}(\hat{\rho}(t))+u_{FF}(t),$

where $u_{FF}(t)\in[0,\infty)$ is a strictly decreasing, bounded continuous function with $\lim_{t\to\infty}u_{FF}(t)=0$, and $u_{FB}$ is a stabilizing feedback control law if $\hat{\theta}(t)$ satisfies the condition of Corollary 2 (e.g., of (4.22) or (4.23) in [21].) The role of $u_{FF}$ is to eliminate the $\hat{\rho}(t)$ from the target state $\rho_{\bar{n}}$ before learning the parameter accurately. Then, our proposed method ensures local convergence under certain assumptions. For convenience, we write $\mathbb{E}_{w}^{\prime}[\bullet]:=\mathbb{E}_{w}[\bullet|\rho(t_{0}),\hat{\rho}(t_{0})]$.

Some readers may think the assumptions of Theorem 6 are too strong to be valid in practice; however, several numerical experiments support that they may hold in many cases, one of which is demonstrated in the following section.