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This article is cited in 2 scientific papers (total in 2 papers) On the detailed structure of quantum control landscape for fast single qubit phase-shift gate generation B. O. Volkovab, A. N. Pechenab a Steklov Mathematical Institute of Russian Academy of Sciences, Department of Mathematical Methods for Quantum Technologies, Moscow, Russia b University of Science and Technology MISIS, Moscow, Russia
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     § 1. IntroductionOptimal quantum control, which includes methods for manipulation of quantum systems, attracts now high attention due to various existing and prospective applications in quantum technologies [1]–[8]. Among important topics, one problem, which was posed in [9], is the analysis of quantum control landscapes, that is, local and global extrema of quantum control objective functionals. Various results have been obtained in this field, for example, in [10]–[23], for closed and open quantum systems. For open quantum systems, a formulation of completely positive trace preserving dynamics as points of complex Stiefel manifold (strictly speaking, of some factors of complex Stiefel manifolds over some equivalence relation) was proposed and theory of open system’s quantum control as gradient flow optimization over complex Stiefel manifolds was developed in details for two-level [24] and general $n$-level quantum systems [25] and applied to the analysis of quantum control landscapes. Control landscapes for open-loop and closed-loop control were analyzed in a unified framework [26]. A unified analysis of classical and quantum kinematic control landscapes was performed [27]. Computation of numbers of positive and negative eigenvalues of the Hessian at saddles of the control landscape is an important problem [11]. For numerical optimization in quantum control, various local and global search methods are used including such as based on Pontryagin maximum principle [28], [29], GRadient Ascent Pulse Engineering (GRAPE) [30], gradient flows [31], Krotov type methods [32], [33], gradient free CRAB optimisation [34], Hessian based methods such as the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm [35], geometric methods [36], genetic algorithms [37], machine learning [38], dual annealing [39], etc. The importance of mathematical analysis of extrema of quantum control objective functionals is motivated by the fact that local, but not global, maxima (if they would exist) would impede the search for globally optimal controls using local search algorithms, which could be more efficient otherwise. For this reason, points of local, but not global, extrema of the objective functional are called traps. In this work, we analytically study the detailed structure of the quantum control landscape around a special critical point for the problem of single-qubit phase shift gate generation on the fast time scale. The absence of traps for single-qubit gate generation was proved on long time scales in [13], [16] and on fast time scales [21], [22], where a single special control, which is a critical point, was studied and shown to be a saddle. However, the numbers of negative and positive eigenvalues of the Hessian at this saddle point control were not studied. At the same time, these numbers are important as they determine the numbers of directions towards decreasing and increasing of the objective and hence determine the level of difficulty for practical optimization starting in a vicinity of the saddle point. The numbers of positive and negative eigenvalues of the Hessian of the objective for some other examples of quantum systems were computed in [11]. In this work, we compute the numbers of negative and positive eigenvalues of the Hessian at this saddle point, give estimates on magnitude of the eigenvalues and also significantly simplify our previous proof of the theorem about the Hessian at this saddle point [22]. Numerical experiments for the problem of single-qubit phase shift gate generation on the fast time scale were performed in [22], [40]. In § 2, we summarize results of previous works which are relevant to our study. In § 3, the main theorem of this work is presented; its proof is provided in § 4. Conclusions in § 5 summarize this work. § 2. Previous resultsA single qubit driven by a coherent control $f\in L^2([0,T];\mathbb R)$, where $T>0$ is the final time, in the absence of the environment evolves according to the Schrödinger equation for the unitary evolution operator $U_t^f$,
$$
\begin{equation}
\frac{dU_t^f}{dt}=-i(H_0+f(t)V)U_t^f,\qquad U_0^f=\mathbb I,
\end{equation}
\tag{1}
$$
where $H_0$ and $V$ are the free and interaction Hamiltonians. Common assumption is that $[H_0,V]\ne 0$. This assumption guarantees controllability of the two-level system for large time; otherwise the dynamics is trivial. In this work, we consider time scale smaller than the controllability time. A single qubit quantum gate is a unitary $2\times 2$ operator $W$ defined up to a physically unrelevant phase, so that $W\in SU(2)$. The problem of single qubit gate generation can be formulated as
$$
\begin{equation}
J_W[f]=\frac14 \bigl|{\operatorname{Tr} \bigl(U_T^f W^\unicode{8224}\bigr)}\bigr|^2\to\max.
\end{equation}
\tag{2}
$$
Definition 1. A control $f^*$ is called a trap for the problem (2) if $f^*$ is a point of local, but not global, maximum of $J_W$, that is, $J(f^*)<\sup_f J_W[f]$. In [16], [21], and [22], results on the absence of traps for this problem were obtained. To explicitly formulate these results, consider the special constant control $f(t)=f_0$ and time $T_0$:
$$
\begin{equation}
f_0 := \frac{-\operatorname{Tr} H_0\operatorname{Tr} V +2\operatorname{Tr}(H_0V)}{(\operatorname{Tr} V^2)^2-2\operatorname{Tr}(V^2)},
\end{equation}
\tag{3}
$$
$$
\begin{equation}
T_0 := \frac {\pi}{\|H_0-\mathbb I \operatorname{Tr} H_0/2+f_0(V-\mathbb I \operatorname{Tr} V/2)\|},
\end{equation}
\tag{4}
$$
where $\|\,{\cdot}\,\|$ denotes the spectral norm. The following theorem was proved in [16]. Theorem 1. Let $W\in SU(2)$ be a single qubit quantum gate. If $[W,H_0+f_0V]\,{\neq}\,0$, then for any $T>0$ traps do not exist. If $[W,H_0+f_0V]=0$, then any control, except possibly $f\equiv f_0$, is not a trap for any $T>0$ and the control $f_0$ is not a trap for $T>T_0$. The case of whether control $f_0$ can be a trap for $T\leqslant T_0$ or not was partially studied in [21]. Without loss of generality, it is sufficient to consider the case $H_0=\sigma_z$ and $V=v_x\sigma_x+v_y\sigma_y$, where $\upsilon_x,\upsilon_y\in \mathbb{R}$ ($\upsilon_x^2+\upsilon_y^2>0$) and $\sigma_x$, $\sigma_y$, and $\sigma_z$ are the Pauli matrices:
$$
\begin{equation}
\sigma_x=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}, \qquad \sigma_y=\begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}, \qquad \sigma_z=\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}.
\end{equation}
\tag{5}
$$
In this case, the special time is $T_0=\pi/2$ and the special control is $f_0=0$. By Theorem 1, if $[W,\sigma_z]\neq 0$, then, for any $T>0$, there are no traps for $J_W$. If $[W,\sigma_z]=0$, then $W=e^{i\varphi_W \sigma_z+i\beta}$, where $\varphi_W\in (0,\pi]$ and $\beta\in [0,2\pi)$. The phase can be neglected, so without loss of generality, we set $\beta=0$. Below we consider only such gates. The following result was proved in [21]. Theorem 2. Let $W=e^{i\varphi_W \sigma_z}$. If $\varphi_W\in (0,\pi/2)$, then, for any $T>0$, there are no traps. If $\varphi_W\in [\pi/2,\pi]$, then, for any $T>\pi-\varphi_W$, there are no traps. For fixed $\varphi_W$ and $T$, the value of the objective evaluated at $f_0$ is If $\varphi_W+T=\pi$, then $J_W[f_0]=1$ and $f_0$ is a point of global maximum. If $\varphi_W+T=\pi/2$ and $\varphi_W+T=3\pi/2$, then $J_W[f_0]=0$ and $f_0$ is a point of global minimum.The Taylor expansion of the functional $J_W$ at $f$ up to the second order has the form (for the theory of calculus of variations in infinite dimensional spaces see [41]):
$$
\begin{equation}
\begin{aligned} \, J_W[f+\delta f] &= J_W[f]+J_W^{(1)}[f](\delta f) \nonumber \\ &\qquad+\frac 12J_W^{(2)}[f](\delta f,\delta f) +o(\|\delta f\|^{2})\quad\text{for }\|\delta f\|\rightarrow 0. \end{aligned}
\end{equation}
\tag{7}
$$
The first Fréchet derivative is
$$
\begin{equation*}
J_W^{(1)}[f](\delta f)=\int_0^T\frac{\delta J_W}{\delta f(t)}\,\delta f(t)\, dt,
\end{equation*}
\notag
$$
where the integral kernel, which determines the gradient of the objective, is
$$
\begin{equation*}
\frac{\delta J_W}{\delta f(t)}=\frac 12 \Im\bigl(\operatorname{Tr} Y^\ast \operatorname{Tr}(YV_t)\bigr).
\end{equation*}
\notag
$$
Here, as in [21] we use the notations $Y=W^\unicode{8224} U^f_T$ and $V_t=U^{f\unicode{8224}}_t VU^f_t$. The second order term is
$$
\begin{equation}
\frac 12J_W^{(2)}[f](\delta f,\delta f)=\frac12(\mathbf{Hess}\, \delta f, \delta f)_{L^2} =\frac12\int_0^T\int_0^T\operatorname{Hess} (t,s)f(t)f(s)\, dt\, ds,
\end{equation}
\tag{8}
$$
where the Hessian $\mathbf{Hess} \colon L^2([0,T],\mathbb{R})\to L^2([0,T],\mathbb{R})$ is an integral operator:
$$
\begin{equation}
(\mathbf{Hess} f)(t)=\int_0^T\operatorname{Hess} (t,s)f(s)\, ds.
\end{equation}
\tag{9}
$$
The integral kernel of the Hessian has the form
$$
\begin{equation*}
\operatorname{Hess}(t,s)= \begin{cases} \dfrac 12 \operatorname{Re}\bigl(\operatorname{Tr}(YV_{t})\operatorname{Tr}(Y^\ast V_{s})-\operatorname{Tr}(YV_{s}V_{t})\operatorname{Tr} Y^\ast\bigr) &\text{if } s\geqslant t, \\ \dfrac 12 \operatorname{Re}\bigl(\operatorname{Tr}(YV_{s})\operatorname{Tr}(Y^\ast V_{t})-\operatorname{Tr}(YV_{t}V_{s})\operatorname{Tr} Y^\ast\bigr)& \text{if } s<t. \end{cases}
\end{equation*}
\notag
$$
The control $f_0=0$ is a critical point, that is, the gradient of the objective evaluated at this control is zero. The integral kernel of the Hessian at $f_0=0$ has the form (see [21])
$$
\begin{equation}
\operatorname{Hess}(s,t)=-2\upsilon^2\cos\varphi\cos(2|t-s|+\varphi),
\end{equation}
\tag{10}
$$
where $\varphi=-\varphi_W-T$ and $\upsilon=\sqrt{\upsilon^2_x+\upsilon^2_y}$. We consider for the values of the parameters $(\varphi_W,T)$ the following cases (see Figure 2, where the set $\mathcal{D}_2$ in addition is divided into three subsets described in § 4):
Remark 1. Note that these notations for the domains $\mathcal{D}_2$, $\mathcal{D}_3$, and $\mathcal{D}_4$ are different from those used in [22]. The present notations seem to be more convenient. The following theorem was obtained in [22]. Theorem 3. If $(\varphi_W,T)\in \mathcal{D}_1\cup \mathcal{D}_2\cup \mathcal{D}_3\cup \mathcal{D}_4$, then the Hessian of the objective functional $J_W$ at $f_0=0$ is an injective compact operator on $L^2([0,T];\mathbb{R})$. Moreover, the following holds. 1. If $(\varphi_W,T)\in \mathcal{D}_1$, then the Hessian at $f_0$ has only negative eigenvalues. 2. If $(\varphi_W,T)\in \mathcal{D}_2\cup \mathcal{D}_3\cup \mathcal{D}_4$, then the Hessian at $f_0$ has both negative and positive eigenvalues. In this case, the special control $f_0=0$ is a saddle point for the objective functional. Numerical experiments suggest that if $(\varphi_W,T)\in\mathcal{D}_1$, then $f_0$ is a point of global maximum [40]. Note that the numbers of positive and negative eigenvalues mentioned in item 2 of this theorem, as well as their magnitudes, were not computed in [22]. However, these numbers and magnitudes are important since they determine the numbers of directions towards increasing or decreasing of $J_W$, and the magnitudes of the eigenvalues determine the speed of increasing and decreasing the objective along these directions. All of that affects the relative level of ease or difficulty of practical optimization in a vicinity of $f_0$. § 3. Main theoremOur main result in the present paper is the following theorem. Theorem 4. 1. If $(\varphi_W,T)\in \mathcal{D}_1$, then the Hessian at $f_0$ has only negative eigenvalues. 2. If $(\varphi_W,T)\in \mathcal{D}_2$, then the Hessian at $f_0$ has two positive and infinitely many negative eigenvalues. 3. If $(\varphi_W,T)\in\mathcal{D}_3$ and $\varphi_W+T<\pi/2$, then the Hessian at $f_0$ has one positive and infinitely many negative eigenvalues. If $(\varphi_W,T)\in\mathcal{D}_3$ and $\varphi_W+T>\pi/2$, then the Hessian at $f_0$ has one negative and infinitely many positive eigenvalues. 4. If $(\varphi_W,T)\in\mathcal{D}_4$, then the Hessian at $f_0$ has one negative and infinitely many positive eigenvalues. § 4. Proof of the main theoremIn this section, we will investigate the spectrum of the Hessian $\mathbf{Hess}$ and prove Theorem 4. If $(\varphi_W,T)\in \mathcal{D}_1\cup \mathcal{D}_2\cup \mathcal{D}_3\cup \mathcal{D}_4$, then $\sin2\varphi=-\sin2(\varphi_W+T)\neq 0$. Instead of the Hessian, we can consider the operator
$$
\begin{equation*}
\mathbf{K}=\frac1{\upsilon^2\sin2\varphi}\, \mathbf{Hess},
\end{equation*}
\notag
$$
which differs from $\mathbf{Hess}$ by a scalar factor that makes the calculations a bit simpler. Here, $\mathbf{K}\colon L^2([0,T],\mathbb{R}) \to L^2([0,T],\mathbb{R})$ is an integral operator: with the integral kernel The operators $\mathbf{K}$ and $\mathbf{Hess}$ differ by a scalar factor, and hence their spectra are related:
$$
\begin{equation*}
\sigma(\mathbf{K})=\frac 1{\upsilon^2\sin2\varphi}\, \sigma(\mathbf{Hess}).
\end{equation*}
\notag
$$
Let $g\in L^2([0,T],\mathbb{R})$ and $h=\mathbf{K} g$. Then
$$
\begin{equation}
h(t)=-\frac 1{\sin\varphi}\int_0^t\cos(2t-2s+\varphi)g(s)\, ds-\frac 1{\sin\varphi}\int_t^T\cos(2s-2t+\varphi)g(s)\, ds.
\end{equation}
\tag{12}
$$
Differentiating twice in a generalized sense expression (12), we obtain (see [22] for details) Moreover, for any continuous $g$, we can find $h=\mathbf{K} g$ as a unique solution of ODE (13), which satisfies the initial conditions
$$
\begin{equation}
h(0) =-\frac 1{\sin\varphi}\int_0^T\cos(2s+\varphi)g(s)\, ds,
\end{equation}
\tag{14}
$$
$$
\begin{equation}
h'(0) =-\frac {2}{\sin\varphi}\int_0^T\sin(2s+\varphi)g(s)\, ds.
\end{equation}
\tag{15}
$$
These initial conditions are obtained by substituting into the right-hand side of expression (12) and into its derivative $t=0$. Equality (13) implies that if $h\equiv 0$, then $g\equiv 0$. Hence $\mathbf{K}$ is an injective operator. Let $\mu\neq 0$ be an eigenvalue of the operator $\mathbf{K}$ and $g$ be a corresponding eigenfunction, then $h=\mathbf{K} g=\mu g$. Let $\lambda=1/\mu$. Now (13), (14), and (15) together imply that the search for eigenvalues of the operator $\mathbf{K}$ reduces to the problem of finding such $h\in C^\infty([0,T],\mathbb{R})$ and non-zero $\lambda \in\mathbb{R}$ that
$$
\begin{equation}
\begin{cases} h''(t)=4(\lambda-1)h(t), \\ \lambda h(0)=-\dfrac 1{\sin\varphi}{\displaystyle\int_0^T\cos(2s+\varphi)h(s)\, ds}, \\ \lambda h'(0)=-\dfrac {2}{\sin\varphi}{\displaystyle\int_0^T\sin(2s+\varphi)h(s)\, ds}. \end{cases}
\end{equation}
\tag{16}
$$
This problem is similar to the Sturm–Liouville problem (see [42]). 4.1. Case $\lambda<1$Consider the case of non-zero $\lambda<1$. Let $a^2=(1-\lambda)$ and $a>0$. If $h$ satisfies (16), then $h$ has the form If we substitute $h$ in the boundary conditions of (16), then we get a system of two linear algebraic equations on $(b,c)$. It has a non-zero solution if the determinant of the coefficients of this system is not equal to zero. This determinant has the form (see [22])
$$
\begin{equation*}
\begin{aligned} \, F^1_{\varphi_W,T}(a) &=-2a-a^2\sin(2aT)\sin(2\varphi_W)-\sin(2aT)\sin(2\varphi_W) \\ &\qquad +2a\cos(2aT)\cos(2\varphi_W). \end{aligned}
\end{equation*}
\notag
$$
So the function $h$ and $\lambda<1$ are a solution of problem (16) if and only if the function $F^1_{\varphi_W,T}$ has a positive root. It is easy to see that for $(\varphi_W,T)\in\mathcal{D}_4$ the roots of the function $F^1_{\varphi_W,T}$ are $a_n=\pi n/T$. If $(\varphi_W,T)\in\mathcal{D}_1$ and $\varphi_W=\pi/2$, then the roots of the function $F^1_{\varphi_W,T}$ are $a_n=(2n-1)\pi/(2T)$. Hence, in these cases, $\mu_n=1/(1-a_n^2)$ belong to the spectrum of the operator $\mathbf{K}$. We will show below that there is also only one positive eigenvalue for $(\varphi_W,T)\in\mathcal{D}_4$ and there are not negative eigenvalues for $(\varphi_W,T)\in\mathcal{D}_1$ such that $\varphi_W=\pi/2$. Lemma 1. Let $T\in(0,\pi/2)$. The equation has only one root on $(0,1)$ if $T< \alpha <\tan T$ and has no roots on $(0,1)$ if $\alpha\in(-\infty,T]$ and $\alpha \in[\tan T,+\infty)$. The proof of this lemma is illustrated in Figure 1 (left subplot). Lemma 2. Let $T\in(0,\pi/2)$. The equation has only one root on $(0,1)$ if $\cot T< \alpha$ and has no roots on $(0,1)$ if $\alpha\,{\in}\,(-\infty,\cot T]$. The proof of this lemma is illustrated in Figure 1 (right subplot).  Figure 1.Illustration for Lemma 1 (a) and for Lemma 2 (b). (a) Two inclined straight lines have angles of inclination $T$ and $\tan(T)$, that is, they are defined by equations $y=Tx$ and $y=\tan(T)x$, respectively. The line $y=\tan(Tx)$ on the interval $(0,1)$ lies between these two strait lines. Hence on the interval $(0,1)$ the line $y=\alpha x$ intersects the line $y=\tan(Tx)$ if and only if $T<\alpha<\tan(T)$. (b) The inclined straight line $y=\cot(T)x$ intersects with the line $y=\cot(Tx)$ at $x=1$. Hence equation $\alpha x=\cot(Tx)$ on $(0,1)$ has one solution if $\alpha> \cot T$ and has no solutions if $\alpha\leqslant\cot T$ In addition, we divide the domain $\mathcal{D}_2$ into the following three subdomains (see Figure 2):
 Figure 2.The domains of the rectangle $(\varphi_W,T)\in[0,\pi]\times [0,\pi/2]$. $\mathcal{D}_3$ is the left square except of the dashed diagonal and borders. $\mathcal{D}_1$ is the bottom triangle in the right square with left vertical border and without the solid diagonal and bottom horizontal border. $\mathcal{D}_4$ is the vertical line marked with crosses. $\mathcal{D}''_2$ is the dash-dotted line. $\mathcal{D}_2'$ is the area between $\mathcal{D}_1$ and $\mathcal{D}_2''$. $\mathcal{D}_2'''$ is the area between $\mathcal{D}_2''$ and $\mathcal{D}_4$. Note that these notations for the domains $\mathcal{D}_2$, $\mathcal{D}_3$, and $\mathcal{D}_4$ are different from those used in [22]. In the domain $\mathcal{D}_1$, the Hessian at the critical point $f_0=0$ has only negative eigenvalues. On the dashed diagonal (in the left square), the critical point $f_0=0$ is a point of global minimum. On the solid diagonal (in the right square), the critical point $f_0=0$ is a point of global maximum. On the domain $\mathcal{D}_2=\mathcal{D}_2'\cup \mathcal{D}_2''\cup \mathcal{D}_2'''$, the critical point $f_0=0$ is a saddle point with two positive and infinitely many negative eigenvalues of the Hessian. On the bottom triangle of $\mathcal{D}_3$, the critical point $f_0=0$ is a saddle point with one positive and infinitely many negatives eigenvalues of the Hessian. On the top triangle of $\mathcal{D}_3$ and on the domain $\mathcal{D}_4$, the critical point $f_0=0$ is a saddle point with one negative and infinitely many positive eigenvalues of the Hessian Proposition 1. Positive roots of the function $F^1_{\varphi_W,T}$ are positive solutions $\{a'_m\}$ and $\{a_n\}$ of the equation and the equation respectively. Here, $\{a'_m\}$ and $\{a_n\}$ are two countable sets whose elements are distributed in ascending order. Then the numbers $\mu'_m=1/(1-{a'}^2_m)$ and $\mu_n=1/(1-a_n^2)$ belong to the spectrum of the operator $\mathbf{K}$. Moreover, the following holds. 1. If $(\varphi_W,T)\in \mathcal{D}_1$ and $\varphi_W\neq \pi/2$, then $a'_m\in ((m-1)\pi/T,(2m-1)\pi/(2T))$ and $a_n\in ((n-1)\pi/T,(2n-1)\pi/(2T))$, where $n,m\in \mathbb{N}$. In this case, the numbers $\{\mu'_m\}$ and $\{\mu_n\}$ are negative for all $n,m\in\mathbb{N}$. 2. If $(\varphi_W,T)\in \mathcal{D}''_2\cup \mathcal{D}'''_2$, then $a'_m\in ((m-1)\pi/T,(2m-1)\pi/(2T))$ and $a_n\in ((n-1)\pi/T,(2n-1)\pi/(2T))$, where $m\in \{2,3,\ldots\}$ and $n\in \mathbb{N}$. In this case, $\mu_1=1/(1-a_1^2)>1$ is positive and the numbers $\{\mu'_m\}$ and $\{\mu_n\}$ are negative for $n>1$ and $m>1$. 3. If $(\varphi_W,T)\in \mathcal{D}'_2$, then $a'_m\in ((m-1)\pi/T,(2m-1)\pi/(2T))$ and $a_n\in ((n- 1)\pi/T, (2n-1)\pi/(2T))$, where $n,m\in \mathbb{N}$. In this case, $\mu'_1=1/(1-{a'_1}^2)>1$ and $\mu_1=1/(1-a_1^2)>1$ are positive. The numbers $\{\mu'_m\}$ and $\{\mu_n\}$ are negative for $n>1$ and $m>1$. 4. If $(\varphi_W,T)\,{\in}\, \mathcal{D}_3$, then $a'_m\,{\in}\, ((2m-1)\pi/(2T),m\pi/T)$ and $a_n\,{\in}\, ((2n- 1)\pi/(2T), n\pi/T)$, where $n,m\in \mathbb{N}$. In this case, the numbers $\{\mu'_m\}$ and $\{\mu_n\}$ are negative for all $n,m\in\mathbb{N}$. Proof. Let us analyze positive roots of the function $F^1_{\varphi_W,T}$. For this purpose we consider quadratic (with respect to $x$) equation:
$$
\begin{equation*}
x^2\sin(2aT)\sin(2\varphi_W)+2x(1-\cos(2aT)\cos(2\varphi_W))+\sin(2aT)\sin(2\varphi_W)=0.
\end{equation*}
\notag
$$
The roots of this quadratic equation are as follows: Hence $a$ is a root of the function $F^1_{\varphi_W,T}$ if and only if $x=a$ is a solution of either equation (17) or equation (18). If $(\varphi_W,T)\in \mathcal{D}_3$, then both equations have a single root on each interval $((2n-1)\pi/(2T),n\pi/T)$ for $n\in\mathbb{N}$. If $(\varphi_W,T)\in \mathcal{D}_1\cup \mathcal{D}_2$ and $\varphi_W\neq \pi/2$, then equation (18) has a single root on each interval $((n-1)\pi/T, (2n- 1)\pi/(2T))$ for $n\in\mathbb{N}$. If $(\varphi_W,T)\in \mathcal{D}_1\cup\mathcal{D}'_2$ and $\varphi_W\neq \pi/2$, then equation (17) has a single root on each interval $((n-1)\pi/T,(2n-1)\pi/(2T))$ for $n\in \mathbb{N}$. Lemma 1 implies that, in the case $(\varphi_W,T)\in \mathcal{D}''_2\cup\mathcal{D}'''_2$, equation (17) has no roots on $(0,\pi/(2T))$ and has a single root on each interval $((n-1)\pi/T,(2n- 1)\pi/(2T))$ for $n>1$.
The number $\mu=1/(1-a^2)$ is a positive eigenvalue of the operator $\bf K$ if and only if $a\in (0,1)$. If $(\varphi_W,T)\in \mathcal{D}_1$ and $\varphi_W\neq \pi/2$, then Due to Lemma 2 equation (18) has no roots on $(0,1)$. Due to Lemma 1 equation (17) has no roots on $(0,1)$.If $(\varphi_W,T)\in \mathcal{D}_2$, then Hence by Lemma 2 equation (18) has one root on $(0,1)$. If $T\geqslant -\tan\varphi_W$, then (17) has no solution on $(0,1)$ by Lemma 1. So if $(\varphi_W,T)\in \mathcal{D}''_2\cup \mathcal{D}'''_2$, then the function $F^1_{\varphi_W,T}$ has only one root $a_1$ on the interval $(0,1)$. If $(\varphi_W,T)\in \mathcal{D}'_2$, then the function $F^1_{\varphi_W,T}$ has two roots $a_1$ and $a'_1$ on the interval $(0,1)$.If $(\varphi_W,T)\in \mathcal{D}_3$, then $-\tan\varphi_W<0$ and Lemmas 1 and 2 imply that both equations (17) and (18) have no solutions on $(0,\pi/(2T))$. Hence, the numbers $\{\mu'_m\}$ and $\{\mu_n\}$ are negative for all $n,m\in\mathbb{N}$. 4.2. Case $\lambda=1$If $\mu=1$ is an eigenvalue of the operator $\mathbf{K}$, then the corresponding eigenfunctions should have the form If we substitute $g$ in (14) and (15), then we get a system of two linear algebraic equations on $(b,c)$. This system has a non-zero solution if and only if the determinant of the coefficients of this system is not equal to zero. This determinant has the form (see [22])
$$
\begin{equation}
\Delta=-2\sin\varphi_W(\sin\varphi_W+T\cos\varphi_W).
\end{equation}
\tag{21}
$$
Proposition 2. The number $\mu'_1=1$ is an eigenvalue of $\mathbf{K}$ only in the following cases: 1) $(\varphi_W,T)\in \mathcal{D}_{4}$; 2) $(\varphi_W,T)\in \mathcal{D}_2''$. 4.3. Case $\lambda>1$Consider the case $\lambda>1$. Let $a^2=(\lambda-1)$ and $a>0$. If $h$ satisfies (16), then $h$ has the form If we substitute $h$ in boundary conditions of (16), then we get a system of two linear algebraic equations on $b$ and $c$. This system has a non-zero solution if and only if the determinant of the coefficients of this system is not equal to zero. This determinant has the form [22]
$$
\begin{equation}
\begin{aligned} \, F^2_{\varphi_W,T}(a) &= -a^2\sinh(2aT)\sin(2\varphi_W) \nonumber \\ &\qquad+2a(1-\cosh(2aT)\cos(2\varphi_W))+\sinh(2aT)\sin(2\varphi_W). \end{aligned}
\end{equation}
\tag{22}
$$
So, the function $h$ and $\lambda>1$ are a solution of problem (16) if and only if the function $F^2_{\varphi_W,T}$ has a positive root. It is easy to see that for $(\varphi_W,T)\in\mathcal{D}_4$ and $(\varphi_W,T)\in\mathcal{D}_1$ such that $\varphi_W=\pi/2$ the function $F^2_{\varphi_W,T}$ has no positive roots. Proposition 3. 1. If $(\varphi_W,T)\in \mathcal{D}_3$, then $F^2_{\varphi_W,T}$ has only one positive root $a'_0>0$, where $a'_0$ is a solution of the equation Then $\mu'_0=1/(1+a_0^2)<1$ is a positive eigenvalue of the operator $\mathbf{K}$. 2. If $(\varphi_W,T)\in \mathcal{D}_1\cup \mathcal{D}'_2\cup \mathcal{D}''_2$ and $\varphi_W\neq \pi/2$, then the function $F^2_{\varphi_W,T}$ has no positive roots. 3. If $(\varphi_W,T)\in \mathcal{D}'''_2$, then $F^2_{\varphi_W,T}$ has only one positive root $a'_1>0$, where $a'_1$ is a solution of the equation Then $\mu'_1=1/(1+{a'_1}^2)<1$ is a positive eigenvalue of the operator $\mathbf{K}$. Proof. Let us analyze the positive roots of the function $F^2_{\varphi_W,T}$. For this purpose we consider quadratic (with respect to $x$) equation: The roots of this equation are as follows:
Hence $a$ is a root of the function $F^2_{\varphi_W,T}$ if and only if $x=a$ is a solution of either equation (23) or equation (24). If $(\varphi_W,T)\in \mathcal{D}_3$ then due to $\tan\varphi_W>0$ equation (24) has no positive roots. Equation (23) has only one positive root. If $(\varphi_W,T)\in \mathcal{D}_1\cup \mathcal{D}_2$ and $\varphi_W\neq \pi/2$, then due to $\cot\varphi_W<0$ equation (23) has no positive roots. If $(\varphi_W,T)\in \mathcal{D}_1\cup \mathcal{D}'_2\cup \mathcal{D}''_2$ and $\varphi_W\neq \pi/2$, then $T\leqslant-\tan\varphi_W$ and $\tanh(Tx)<-\tan(\varphi_W)x$ for positive $x$ and equation (24) has no positive roots. If $(\varphi_W,T)\in \mathcal{D}'''_2$, then equation (24) has one positive root. The conclusion of Theorem 4 follows directly from Propositions 1, 2, and 3. It is also worth pointing out that these propositions give, in addition, estimates for the magnitudes of the eigenvalues. § 5. ConclusionsAnalysis of either existence or absence of traps (which are points of local, but not global, extrema of the objective quantum functional) is important for quantum control. It was known that in the problem of single-qubit phase shift quantum gate generation all controls, except maybe the special control $f_0=0$ at small times, cannot be traps. In the previous work [22], we studied the spectrum of the Hessian at this control $f_0$ and investigated under what conditions this control is a saddle point of the quantum objective functional. In this work, we have calculated the numbers of negative and positive eigenvalues of the Hessian at this control point and obtained estimates for the magnitudes of these eigenvalues. At the same time, we significantly simplified the proof of Theorem 3 of the paper [22]. The authors thank A. I. Mikhailov for helpful discussions and advice.
Citation in format AMSBIB
\Bibitem{VolPec23} |
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