Control landscapes for high-fidelity generation of C-NOT and C-PHASE gates with coherent and environmental driving

Abstract

High-fidelity generation of two-qubit gates is important for quantum computation, since such gates are components of popular universal sets of gates. Here, we consider the problem of high-fidelity generation of two-qubit C-NOT and C-PHASE (with a detailed study of C–Z) gates in presence of the environment. We consider the general situation when qubits are manipulated by coherent and incoherent controls; the latter is used to induce generally time-dependent decoherence rates. For estimating efficiency of optimization methods for high-fidelity generation of these gates, we study quantum control landscapes which describe the behavior of the fidelity as a function of the controls. For this, we generate and analyze the statistical distributions of best objective values obtained by incoherent GRadient Ascent Pulse Engineering (inGRAPE) approach. We also apply inGRAPE and a dual annealing algorithm (DAA) based on a stochastic zeroth-order method to numerically estimate minimal infidelity values. The results are different from the case of single-qubit gates and indicate a smooth trap-free behavior of the fidelity.

Appendices

Appendices

A: Goerz–Reich–Koch approach

The following theorem was proved by  Goerz et al. in [44, 45].

Theorem 1

Let $\rho _1=\sum _{n=1}^N \lambda _n|\phi _n\rangle \langle \phi _n|$, where $\{|\phi _n\rangle \}_{n=1}^N$ is an orthogonal basis in $\mathbb {C}^N$, be a density matrix with N nonzero nondegenerate eigenvalues. Let a density matrix $\rho _2$ be a one dimensional orthogonal projector on $\mathbb {C}^N$ such that $\rho _2|\phi _n\rangle \ne 0$ for $n = 1,\ldots ,N$. Let $\rho _3=\frac{1}{N}\mathbb {I}_N$. Let for some quantum channel $\Phi$ and for some unitary gate U the three equalities $\Phi \rho _m=U\rho _mU^\dagger$ ($m = 1,2,3$) be satisfied. Then $\Phi \rho =U \rho U^\dagger$ for any density matrix $\rho$, i.e., $\Phi =\Phi _U$.

Theorem 1 implies that achieving zero value for the objective functional $F_U^{\textrm{GRK,sd}}$, implies successful generation of the target unitary gate U. As the dimension N increases, optimization for functionals $F_U^{\textrm{GRK,sd}}$ and $F_U^{\textrm{GRK,sp}}$ is significantly faster than for $F_U^{\textrm{sd}}$, because the last one depends on $N^2$ operators from $\mathcal {M}_N$ instead of the three operators. But as it is shown in Sect. 6, small values of the objective functionals $F_U^{\textrm{GRK,sd}}$ and $F_U^{\textrm{GRK,sp}}$ do not necessarily mean small values of the objective functional $F_U^{\textrm{sd}}$. The function $J_U^{\textrm{sd}}$ defines the square of the metric on the space of superoperators. It can be shown that the functionals  $J_U^{\textrm{GRK,sd}}$ and $J_U^{\textrm{GRK,sp}}$ are continuous in the topology given by this metric. It implies that when the functional $F_U^{\textrm{sd}}$ reaches sufficiently small values, the functionals $F_U^{\textrm{GRK,sd}}$ and $F_U^{\textrm{GRK,sp}}$ will also take on small values.

In our paper, following [44, 45] we consider the following three density matrices which satisfy the conditions of Theorem 1 for the case of two qubits:

$$\begin{aligned} \rho _1 = \textrm{diag}\left( \frac{2}{5}, \frac{3}{10}, \frac{1}{5}, \frac{1}{10} \right) , \quad \rho _2 = \frac{1}{4}J_4, \quad \rho _3=\frac{1}{4}\mathbb {I}_4. \end{aligned}$$

(15)

Here $J_4$ means the $4 \times 4$ matrix whose all elements are equal to 1. Note that the action of C-NOT and C-PHASE gates does not change two density matrices of the three density matrices (15). Namely, if $U=\text {C-NOT}$, then $\Phi _U\rho _1= \textrm{diag}\left( \frac{2}{5}, \frac{3}{10}, \frac{1}{10}, \frac{1}{5} \right) \ne \rho _1$, $\Phi _U\rho _2 = \rho _2$ and $\Phi _U\rho _3= \rho _3$. If $U=\text {C-PHASE} (\lambda )$, then $\Phi _U\rho _1= \rho _1$, $\Phi _U\rho _3= \rho _3$ and

$$\begin{aligned} \Phi _U\rho _2= \frac{1}{4}\begin{pmatrix} 1 &{}\quad 1 &{}\quad 1 &{}\quad e^{-i\lambda } \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad e^{-i\lambda } \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad e^{-i\lambda } \\ e^{i\lambda } &{}\quad e^{i\lambda } &{}\quad e^{i\lambda } &{}\quad 1 \end{pmatrix} \ne \rho _2. \end{aligned}$$

B: Gradient and Hessian of the objective functionals

Practical application of gradient and Hessian-based algorithms relies on analytical formulas for the gradient and the Hessian of the objective functionals, as in inGRAPE [35] of BGFS [52, 53]. Moreover, explicit formulas for the Hessian evaluated at critical points of the objective functional are necessary for the analysis of presence or absence of the trapping behavior of objective functional [83, 86, 89, 95]. In this Appendix, we derive the formulas for the gradient and the Hessian of the objective functionals $F_U^{\textrm{sd}}$, $F_U^{\textrm{GRK,sd}}$, and $F_U^{\textrm{GRK,sp}}$ for any N. Then we substitute piecewise constant controls into the formulas for the gradient and obtain expressions used in this work when applying inGRAPE in Appendix C.

For generality, we provide general formulas for functional variation for a system whose controlled dynamics is determined by a linear evolutionary equation. Consider as a control space a normed space ${\mathcal {U}}$ continuously embedded into the space $L_1=L_1([0,T];{\mathbb {R}}^d)$. For example, someone can choose $L_1$ itself, $L_2=L_2([0,T];{\mathbb {R}}^d)$, $L_\infty =L_\infty ([0,T];{\mathbb {R}}^d)$ or the space of piecewise constant ${\mathbb {R}}^d$-valued functions with $L_\infty$-norm as ${\mathcal {U}}$. The Hilbert space $L_2$ is convenient for studying the Hessian [89, 93]; the space $L_\infty$ is used when there are restrictions on the value of the control function. Equation (2) is a particular case of the general evolutionary equation with control $f \in {\mathcal {U}}$:

$$\begin{aligned} \dfrac{\textrm{d} \Phi _t^f}{\textrm{d} t} = \mathcal {L}^{f}_t\Phi _t^f, \quad \Phi ^f_0 = \mathbb {I}, \end{aligned}$$

(16)

where $\mathcal {L}_t^f:=\mathcal {K}+f_\mu (t)\mathcal {N}^\mu$. Here $\mathcal {K}$, $\mathcal {N}^\mu$ ($\mu =1,\ldots ,d)$ are linear operators on a finite dimensional space $\mathcal {V}$. In this section, we summarize by repeating Greek indices. Carathéodory’s theorem implies that for $f\in L_1$ Eq. (16) has an unique absolutely continuous solution [120]. The solution of (16) is determined through a chronological exponent:

$$\begin{aligned} \Phi _t^f =\;\hat{T} \exp \!\left( {\displaystyle \int _0^t \textrm{d} s\, \mathcal {L}^{f}_s}\right) = \mathbb {I} + \displaystyle \sum _{n = 1}^{\infty } \dfrac{1}{n!} \int _0^t \dots \int _0^{t} \hat{T} \left\{ \mathcal {L}^{f}_{\tau _n}\cdots \mathcal {L}^{f}_{\tau _1}\right\} \textrm{d} \tau _1 \cdots \textrm{d}\tau _n \nonumber \\ =\mathbb {I} + \int _0^t \mathcal {L}_s^f \,\textrm{d}s + \frac{1}{2} \int _0^t \textrm{d}\tau \!\int _0^{\tau } \!\mathcal {L}^{f}_\tau \mathcal {L}^{f}_s \, \textrm{d}s + \dots , \end{aligned}$$

(17)

where $\hat{T}\{\,\cdot \,\}$ is the chronological ordering operator, which sets the multipliers in the chronological order of their application in the composition of operators. The series (17) converges for finite-dimensional operators. About the notion of chronological exponent, also see [121, Ch. 2].

Let $\mathcal {J}$ be a  $C^2$-smooth function on $GL(\mathcal {V})$, the general linear group on $\mathcal {V}$. Our goal is to calculate the gradient and the Hessian of an arbitrary functional of the form

$$\begin{aligned} \mathcal {F}(f) = \mathcal {J}(\Phi _T^f), \end{aligned}$$

(18)

that is, to calculate the first and second derivatives in the sense of Fréchet in the functional space of controls.

Let us introduce the notation ${(\mathcal {N}_t^{f})}^\mu = {\Phi _t^f}^{-1}\!\mathcal {N}^\mu \Phi _t^f.$ This operator exists due to invertibility of the evolution operator

$$\begin{aligned} {\Phi _t^f}^{-1} = \hat{T}_a \exp \!\left( - {\displaystyle \int _0^t \mathcal {L}^{f}_s \,\textrm{d}s} \right) , \end{aligned}$$

where $\hat{T}_a$ is the antichronological ordering operator, i.e., unlike the $\hat{T}$ operator, this operator rearranges the factors in the reverse chronological order of their appearance.

Proposition 1

The first and the second Fréchet derivatives of the functional (18) in the function space ${\mathcal {U}}$ are

$$\begin{aligned} \dfrac{\delta \mathcal {F}(f)}{\delta f_\mu (t)} =&\dfrac{\delta \mathcal {J}\left( \Phi _T^f\right) }{\delta \Phi } \Phi _T^f{(\mathcal {N}_t^{f})}^\mu ,\nonumber \\ \frac{\delta ^2 \mathcal {F}(f)}{\delta f_\mu (t_1)\,\delta f_\nu (t_2)} =&\dfrac{\delta \mathcal {J}\left( \Phi _T^f\right) }{\delta \Phi } \Phi _T^f \hat{T}\{{(\mathcal {N}^{f}_{t_1})}^\mu {(\mathcal {N}^{f}_{t_2})}^\nu \} + \dfrac{\delta ^2\mathcal {J}\left( \Phi _T^f\right) }{\delta \Phi ^2} \left( \Phi _T^f{(\mathcal {N}^{f}_{t_1})}^\mu , \Phi _T^f{(\mathcal {N}^{f}_{t_2})}^\nu \right) . \end{aligned}$$

(19)

Proof

To calculate the derivatives of the objective functional, consider the increment $\delta f$ in the neighborhood of the control f. Introduce the operator $W^{f,\delta f}_t$, where

$$\begin{aligned} \Phi ^{f + \delta f}_t = \Phi _t^f W^{f,\delta f}_t. \end{aligned}$$

This operator exists due to invertibility of the evolution operator. The operator $\Phi ^{f+\delta f}_t$ satisfies Eq. (16) with control $f+\delta f$. This allows us to obtain the following equation for $W^{f,\delta f}_t$:

$$\begin{aligned} \dfrac{\textrm{d} }{\textrm{d} t}W^{f,\delta f}_t= \delta f_\mu (t) \, {\Phi _t^f}^{-1}\!\mathcal {N}^\mu \Phi _t^f W^{f,\delta f}_t, \quad W^{f,\delta f}_0 = \mathbb {I}. \end{aligned}$$

Its solution is

$$\begin{aligned} W^{f,\delta f}_T = \hat{T} \exp \!\left( \displaystyle \int _0^T \delta f_\mu (t) {(\mathcal {N}_t^{f})}^\mu \,\textrm{d}t \right) . \end{aligned}$$

Using (17), we obtain the Taylor expansion

$$\Phi _T^{f + \delta f} = \Phi _T^f W_T^{f,\delta f} = \Phi _T^f + \int_0^T \delta f_\mu (t)\Phi _T^f ({{\mathcal{N}}}_t^f )^\mu dt\quad + \frac{1}{2}\int_0^T {\int_0^T {\Phi _T^f } } \hat{T}\{ ({{\mathcal{N}}}_{t_1 }^f )^\mu ({\rm{\mathcal{N}}}_{t_2 }^f )^\nu \} \delta f_\mu (t_1 )\delta f_\nu (t_2 ){\mkern 1mu} dt_1 dt_2 + \ldots$$

(20)

Then the first Fréchet derivative of the control evolution operator equals to

$$\begin{aligned} \dfrac{\delta \Phi _T^f}{\delta f_\mu (t)} = \Phi _T^f{(\mathcal {N}_t^{f})}^\mu . \end{aligned}$$

Hence, the derivative of the functional (18) equals to

$$\begin{aligned} \frac{\delta \mathcal {F}^f}{\delta f_\mu (t)} = \frac{\delta \mathcal {J}\left( \Phi _T^f\right) }{\delta \Phi } \frac{\delta \Phi _T^f}{\delta f_\mu (t)}= \frac{\delta \mathcal {J}\left( \Phi _T^f\right) }{\delta \Phi } \Phi _T^f{(\mathcal {N}_t^{f})}^\mu . \end{aligned}$$

Moreover, decomposition (20) gives an expression for the second-order derivative of the evolution operator:

$$\begin{aligned} \dfrac{\delta ^2\Phi _T^f}{\delta f_\mu (t_1) \delta f_\nu (t_2)} = \Phi _T^f \hat{T}\{{(\mathcal {N}_{t_1}^{f})}^\mu {(\mathcal {N}_{t_2}^{f})}^\nu \}. \end{aligned}$$

The formula for the second derivative of an arbitrary functional (18) is obtained by differentiating (19) as follows:

$$\begin{aligned} \frac{\delta ^2 \mathcal {F}(f)}{\delta f_\mu (t_1) \delta f_\nu (t_2)} = \dfrac{\delta \mathcal {J}\left( \Phi _T^f\right) }{\delta \Phi } \Phi _T^f \hat{T}\{{(\mathcal {N}_{t_1}^{f})}^\mu {(\mathcal {N}_{t_2}^{f})}^\nu \} + \dfrac{\delta ^2\mathcal {J}\left( \Phi _T^f\right) }{\delta \Phi ^2} \left( \Phi _T^f{(\mathcal {N}_{t_1}^{f})}^\mu , \Phi _T^f{(\mathcal {N}_{t_2}^{f})}^\nu \right) , \end{aligned}$$

where $\dfrac{\delta ^2\mathcal {J}}{\delta \Phi ^2} (\,\cdot \,,\,\cdot \,)$, being a second-order derivative, is a bilinear map. $\hfill \square$

Let the control $f=(f_1,f_2,f_3)=(u,n_1,n_2)$ belong to the interior of the set of admissible controls $\{f\in L_\infty ([0,T],\mathbb {R}^3):f_2\ge 0,f_3\ge 0\}$. Proposition 1 implies that the first and the second Fréchet derivatives of the objective functional $F_U$ at the control point f have the form

$$\begin{aligned} (F_{U})'[f](\delta f)&=\int _0^T\frac{\delta F_{U}[f]}{\delta f_\mu (t)}\delta f_\mu (t)dt,\\ (F_{U})''[f](\delta f_1,\delta f_2)&= \int _0^T\int _0^T\frac{\delta ^2{F_{U}}[f]}{\delta f_\mu (t_1) \delta f_\nu (t_2)} (\delta f_1)_\mu (t_1)(\delta f_2)_\nu (t_2)dt_1dt_2, \end{aligned}$$

where $F_U$ means $F_U^{\textrm{sd}}$ or $F_U^{\textrm{GRK,sd}}$ or $F_U^{\textrm{GRK,sp}}$.

Proposition 2

Gradient and Hessian of the functional $F_{U}^{\textrm{sd}}$ have the forms

$$\begin{aligned} \frac{\delta F_{U}^{\textrm{sd}}[f]}{\delta f_\mu (t)}&=\frac{1}{N^2}\textrm{Tr}\left( {(\Phi _T^f-\Phi _U)}^\dagger \Phi _T^f{(\mathcal {N}^{f}_t)}^\mu \right) .\\ \frac{\delta ^2{F_{U}^{\textrm{sd}}[f]}}{\delta f_\mu (t_1) \delta f_\nu (t_2)}&=\frac{1}{N^2}\textrm{Tr}\left( (\Phi _T^f-\Phi _U)^\dagger \Phi _T^f \hat{T}\{{(\mathcal {N}_{t_1}^{f})}^\mu {(\mathcal {N}_{t_2}^{f})}^\nu \}\right) +\frac{1}{N^2} \textrm{Tr}\left( (\Phi _T^f {(\mathcal {N}_{t_1}^{f})}^\mu )^\dagger \Phi _T^f {(\mathcal {N}_{t_2}^{f})}^\nu \right) . \end{aligned}$$

Proposition 3

Gradient and Hessian of the functional $F^{\textrm{GRK,sd}}_{U}$ have the forms

$$\begin{aligned} \frac{\delta F^{\textrm{GRK,sd}}_{U}[f]}{\delta f_\mu (t)}=&\sum _{m=1}^3\frac{1}{3}\textrm{Tr}\left( (\Phi _T^f\rho _{m}-\Phi _U\rho _{m})\Phi _T^f{(\mathcal {N}^{f}_t)}^\mu \rho _{m}\right) .\\ \frac{\delta ^2{F^{\textrm{GRK,sd}}_{U}[f]}}{\delta f_\mu (t_1) \delta f_\nu (t_2)}=&\sum _{m=1}^3\frac{1}{3}\textrm{Tr}\left( (\Phi _T^f\rho _{m}-\Phi _U\rho _{m}) \Phi _T^f\hat{T}\{{(\mathcal {N}_{t_1}^{f})}^\mu {(\mathcal {N}_{t_2}^{f})}^\nu \}\rho _{m}\right) \\&+\sum _{m=1}^3\frac{1}{3}\textrm{Tr}\left( \Phi _T^f {(\mathcal {N}_{t_1}^{f})}^\mu \rho _{m}\Phi _T^f {(\mathcal {N}_{t_2}^{f})}^\nu \rho _{m}\right) . \end{aligned}$$

Proposition 4

Gradient and Hessian of the functional $F_U^{\textrm{GRK,sp}}$ have the forms

$$\begin{aligned} \frac{\delta F_U^{\textrm{GRK,sp}}[f]}{\delta f_\mu (t)}&= -\sum _{m=1}^3\frac{1}{3\textrm{Tr} {\rho }^2_m} \left( \textrm{Tr}\left[ \Phi _U{\rho }_m\Phi _T^f{(\mathcal {N}^{f}_t)}^\mu \rho _m\right] \right) .\\ \frac{\delta ^2 F_U^{\textrm{GRK,sp}}[f]}{\delta f_\mu (t_1) \delta f_\nu (t_2)}&=-\sum _{m=1}^3 \frac{1}{3\textrm{Tr} {\rho }^2_m}\left( \textrm{Tr}\left[ \Phi _U{\rho }_m\Phi _T^f\hat{T}\{{(\mathcal {N}_{t_1}^{f})}^\mu {(\mathcal {N}_{t_2}^{f})}^\nu \}\rho _m\right] \right) \end{aligned}$$

Propositions 2, 3, 4 are direct consequences of Proposition 1.

C: Realification of the quantum system and of the objective functionals

Realification of the set of density matrices and the set of operators on this set is important in the context of optimization in order to reduce computational space and time cost. It is done by performing only real-value calculations instead of complex-value calculations. As it will be shown further, this enables reduction by half the dimension of spaces carrying states and operators on states.

1.1 C.1: Parametrization of the density matrix

Taking into account the hermiticity of the density matrix, we consider a parameterization of density matrix $\rho$ by a real vector $x=x_\rho =(x_1,\ldots ,x_{16}) \in \mathbb {R}^{16}$. More specifically, we consider the following expansion in the special Hermitian basis $\{M_k\}$:

$$\begin{aligned} \rho = \sum _{j = 1}^{16} x_j M_j = \begin{pmatrix} x_1 &{} x_2 + i x_3 &{} x_4 + i x_5 &{} x_6 + i x_7 \\ x_2 - i x_3 &{} x_8 &{} x_9 + i x_{10} &{} x_{11} + i x_{12} \\ x_4 - i x_5 &{} x_9 - i x_{10} &{} x_{13} &{} x_{14} + i x_{15} \\ x_6 - i x_7 &{} x_{11} - i x_{12} &{} x_{14} - i x_{15} &{} x_{16} \end{pmatrix}. \end{aligned}$$

(21)

The condition $\textrm{Tr}\rho =1$ implies linear dependence $x_1+x_8+x_{13}+x_{16}=1$.

Rewritten for the coordinate vector x, the dynamical systems 1, 2, and 3 have the following form:

$$\begin{aligned} \dfrac{\textrm{d} x^{u,n}}{\textrm{d} t} = L^{u,n}_t x, \quad x^{u,n}(0) = x_{\rho _0}, \end{aligned}$$

(22)

where $L_t^{u,n}$ is the matrix of the generator ${\mathcal L } _t^{u,n}$ in the basis $M = \{M_j\}_{j = 1}^{16}$ (21):

$$\begin{aligned} L_t^{u,n} = \left( A + B_u u(t) + B_{n_1} n_1(t) + B_{n_2} n_2(t) \right) , \end{aligned}$$

the $16 \times 16$ matrices A, $B_u$, $B_{n_1}$, $B_{n_2}$ are found by substituting the expansion (21) into the master Eq. (1) with the Hamiltonian (5) and the dissipator (9) for each system 1, 2, and 3 defined in Sect. 4; $x_{\rho _0}$ is the coordinate vector of $\rho _0$. For the interaction Hamiltonians $V_1$ and $V_2$, these dynamical systems and initial conditions were explicitly written in [71]. For brevity, here we do not reproduce these equations and the corresponding matrices.

Introduce an evolution operator $\Psi ^{u,n}_t$ which is a matrix of the evolution operator $\Phi ^{u,n}_t$ in the basis $M = \{M_j\}_{j = 1}^{16}$ (21), gives evolution of a vector x: $x^{u,n}(t) = \Psi ^{u, n}_t x_{\rho _0}$, and satisfies equation:

$$\begin{aligned} \dfrac{\textrm{d} \Psi }{\textrm{d} t} = L_t^{u,n} \Psi , \quad \Psi _0 = \mathbb {I}. \end{aligned}$$

(23)

Note that any quantum channel $\Phi$ maps density matrices to density matrices, therefore a matrix $\Psi$ of any quantum channel is real-valued in the Hermitian basis:

$$\begin{aligned} \Psi \in \mathbb {R}^{N\times N} \subset \mathbb {C}^{N\times N}. \end{aligned}$$

1.2 C.2: Objective functionals and their Gradients in terms of real-valued states

Here, we provide expressions for the objective kinematic functionals and their gradient in the real-valued parameterization proposed in (21). Firstly, we rewrite expressions for the objective kinematic functionals $J_U^{\textrm{sd}}$, $J_U^{\textrm{GRK,sd}}$, and $J_U^{\textrm{GRK,sp}}$ as functionals of a matrix $\Psi$ of an operator $\Phi$ in the basis $M = \{M_j\}_{j = 1}^{16}$. Then we rewrite expressions for the gradient of the objective dynamic functionals in a convenient and effective for optimization form.

The first functional $J_U^{\textrm{sd}}$ needs calculation of the squared Hilbert–Schmidt norm $\Vert \Phi \Vert ^2 = \textrm{Tr}(\Phi ^\dagger \Phi )$. For that we prove the following proposition.

Proposition 5

Let $\Psi$ and $\Psi '$ be matrices of operators $\Phi$ and $\Phi '$ on $\mathbb {C}^{N\times N}$, respectively, in an orthogonal basis $M = \{M_j\}_{j = 1}^{N^2}$, $M_j \in \mathbb {C}^{N\times N}$, $\langle M_i, M_j\rangle = \textrm{Tr}(M_i^\dagger M_j) = \beta _j\delta _{ij}$; $i,j = 1, \ldots , N^2$. Then the Hilbert–Schmidt scalar product of $\Phi , \Phi '$ equals

$$\begin{aligned} \langle \Phi , \Phi '\rangle = \sum _{i,j = 1}^{N^2}\frac{\beta _i}{\beta _j}\overline{\Psi }_{ij}\Psi '_{ij}. \end{aligned}$$

(24)

Proof

Let $\rho$ and $\sigma$ be matrices in $\mathbb {C}^{N\times N}$ and have coordinates x and y, respectively, in the orthogonal basis M. Then their scalar product equals

$$\begin{aligned} \langle \rho , \sigma \rangle = \overline{(\beta \circ x)}^Ty, \end{aligned}$$

(25)

where “$\circ$“ denotes the Hadamard product that returns the vector which components equal

$$\begin{aligned} (\beta \circ x)_j = \beta _j x_j,\quad j = 1, \ldots , N^2. \end{aligned}$$

Denote by ${\Psi }^\dagger$ a matrix of ${\Phi }^\dagger$ in the basis M. From the equality $\langle \rho , \Phi \sigma \rangle = \langle {\Phi }^\dagger \rho , \sigma \rangle$ we have that the components of ${\Psi }^\dagger$ equal

$$\begin{aligned} {\Psi }^\dagger _{ij} = \frac{\beta _i}{\beta _j}\overline{\Psi }_{ji}. \end{aligned}$$

Then the Hilbert–Schmidt scalar product equals

$$\begin{aligned} \langle \Phi , \Phi '\rangle = \textrm{Tr}{\Psi }^\dagger \Psi ' = \sum _{i,j = 1}^N\frac{\beta _i}{\beta _j}\overline{\Psi }_{ij}\Psi '_{ij}. \end{aligned}$$

$\square$

The considered parameterization basis $M = \{M_j\}_{j = 1}^{16}$ (21) is orthogonal, the vector of squared norms $\beta _j = \textrm{Tr}M_j^\dagger M_j$ equals

$$\begin{aligned} \beta = (1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 1). \end{aligned}$$

Equations (24) and (25) justify introducing scalar products and norms in $\mathbb {R}^{16}$ and in $\mathbb {R}^{16\times 16}$:

$$\begin{aligned} \langle x, y\rangle _M&= \langle \beta \circ x, y\rangle = \sum _{j = 1}^{16} \beta _j x_j y_j,\quad \Vert x\Vert ^2_M = \langle x, x\rangle _M,\quad x,y \in \mathbb {R}^{16};\\ \langle \Psi , \Psi '\rangle _M&= \sum _{i,j = 1}^{16}\frac{\beta _i}{\beta _j}\Psi ^1_{ij}\Psi '_{ij},\quad \Vert \Psi \Vert ^2_M = \langle \Psi , \Psi \rangle _M,\quad \Psi , \Psi '\in \mathbb {R}^{16\times 16}. \end{aligned}$$

Thus, we can write the objective kinematic functionals as a function of $\Psi$ as follows:

Proposition 6

The objective kinematic functionals $J_U^{\textrm{sd}}$, $J_U^{\textrm{GRK,sd}}$, and $J_U^{\textrm{GRK,sp}}$ as functionals of a matrix $\Psi$ of an operator $\Phi$ in the basis $\{M_j\}_{j = 1}^{16}$ used in (21) are equal to

$$\begin{aligned} J_U^{\textrm{sd}}(\Phi )&= \frac{1}{32}\Vert \Psi - \Psi _U\Vert ^2_M, \\ J_U^{\textrm{GRK,sd}}(\Phi )&= \frac{1}{6}\sum _{m = 1}^3 \Vert \Psi x_{\rho _m} - \Psi _Ux_{\rho _m}\Vert ^2_M, \\ J_U^{\textrm{GRK,sp}}(\Phi )&= 1 - \frac{1}{3}\sum _{m = 1}^3 \dfrac{\langle \Psi x_{\rho _m}, \Psi _U x_{\rho _m}\rangle _M}{\Vert x_{\rho _m}\Vert ^2_M}. \end{aligned}$$

where $\Psi _U$ is a matrix of $\Phi _U$ and $x_{\rho _m}$ is the coordinate vectors of $\rho _m$, $m = 1,2,3$, in the parameterization basis (21).

Let control $f = (u, n_1, n_2)$ be a piecewise constant control given by (10). In addition, consider control $g = (u, w_1, w_2)$, where $w_1$ and $w_2$ are given by the change of variable (11). Denote by $x_{\rho _m}^f(t)$ the solution of the system (22) with the initial condition $x_{\rho _m}^f(0) = x_{\rho _m}$. When considering piecewise constant controls, the generator matrix $L^f_t$ is constant on the intervals: $L_t^f = L^f_{t_{k-1}} = A + B_uu_k + B_{n_1}n_{k,1} + B_{n_2}n_{k,2}$ for any $t \in [t_{k-1}, t_k)$. The evolution matrix values $\Psi _{t_k}^f$ on the borders $t_k$ are equal to $\Psi ^f_{t_k} = \exp (\Delta t L^f_{t_{k-1}}) \cdots \exp (\Delta t L^f_0)$. For the considered piecewise constant control, the objective functional $F_U$ become function of 3K variables $g_{k,\mu }$.

For the optimization process described in Sect. 5 we use the following expressions for gradient of the objective functionals $F_U^{\textrm{sd}}$, $F_U^{\textrm{GRK,sd}}$, and $F_U^{\textrm{GRK,sp}}$ which are derived as direct corollary of Propositions 2, 3, and 4 in Appendix B.

Proposition 7

Gradient of the functional $F_U^{\textrm{sd}}$ has the form

$$\begin{aligned} \dfrac{\partial F_U^{\textrm{sd}}}{\partial g_{k,\mu }} = \frac{1}{16}\left\langle \Psi _T^f - \Psi _U, \dfrac{\partial \Psi ^f_T}{\partial g_{k,\mu }}\right\rangle _M. \end{aligned}$$

(26)

Proposition 8

Gradient of the functional $F_U^{\textrm{GRK,sd}}$ has the form

$$\begin{aligned} \dfrac{\partial F_U^{\textrm{GRK,sd}}}{\partial g_{k,\mu }} = \frac{1}{3}\sum _{m = 1}^3\left\langle \Psi _T^f x_{\rho _m} - \Psi _Ux_{\rho _m}, \dfrac{\partial x^f_{\rho _m}(T)}{\partial g_{k,\mu }}\right\rangle _M. \end{aligned}$$

(27)

Proposition 9

Gradient of the functional $F_U^{\textrm{GRK,sp}}$ has the form

$$\begin{aligned} \dfrac{\partial F_U^{\textrm{GRK,sp}}}{\partial g_{k,\mu }} = -\frac{1}{3}\sum _{m = 1}^3\frac{1}{\Vert x_{\rho _m}\Vert ^2_M}\left\langle \Psi _U x_{\rho _m}, \dfrac{\partial x^f_{\rho _m}(T)}{\partial g_{k,\mu }}\right\rangle _M. \end{aligned}$$

(28)

Proposition 10

Gradient of the evolution matrix $\Psi _T^f$ and the final states $x_{\rho _m}(T,f)$ equals

$$\begin{gathered} \frac{{\partial \Psi _T^f }}{{\partial u_k }} = \Delta t\Psi _{t_k ,T}^f \int_0^1 {\exp } \left( {(1 - \tau )\Delta tL_{t_{k - 1} }^f } \right)B_u \exp \left( {\tau \Delta tL_{t_{k - 1} }^f } \right)d\tau \Psi _{t_{k - 1} }^f , \hfill \\ \frac{{\partial \Psi _T ^f }}{{\partial w_{k,l} }} = 2w_{k,l} \Delta t\Psi _{t_k ,T}^f \int_0^1 {\exp } \left( {(1 - \tau )\Delta tL_{t_{k - 1} }^f } \right)B_{n_l } \exp \left( {\tau \Delta tL_{t_{k - 1} }^f } \right)d\tau \Psi _{t_{k - 1} }^f \hfill \\ \frac{{\partial x_{\rho _m }^f (T)}}{{\partial u_k }} = \Delta t\Psi _{t_k ,T}^f \int_0^1 {\exp } \left( {(1 - \tau )\Delta tL_{t_{k - 1} }^f } \right)B_u \exp \left( {\tau \Delta tL_{t_{k - 1} }^f } \right)d\tau x_{\rho _m }^f (t_{k - 1} ), \hfill \\ \frac{{\partial x_{\rho _m }^f (T)}}{{\partial w_{k,l} }} = 2w_{k,l} \Delta t\Psi _{t_k ,T}^f \int_0^1 {\exp } \left( {(1 - \tau )\Delta tL_{t_{k - 1} }^f } \right)B_{n_l } \exp \left( {\tau \Delta tL_{t_{k - 1} }^f } \right)d\tau x_{\rho _m }^f (t_{k - 1} ), \hfill \\ \end{gathered}$$

(29)

$k = 1, \ldots , K$; $l = 1,2$; $m = 1, 2, 3$.

Here, the evolution matrix from time s to time t is defined as $\Psi _{s,t}^f = \Psi _t^f{\Psi _s^f}^{-1}$. Its values for $s = t_k$ and $t = T$ are equal to $\Psi ^f_{t_k,T} = \exp (\Delta t L^f_{t_{K-1}}) \cdots \exp (\Delta t L^f_{t_{k}}).$