Optimal shortcut-to-adiabaticity quantum control

Control of quantum systems is at the core of quantum applications and quantum technologies [1, 2, 3, 4, 5, 6]. A particularly well-known and effective method for designing control protocols is the Shortcut-To-Adiabaticity approach (STA) [7, 8, 9, 10]. STA is a generic term for various techniques that aim to ensure that the quantum system of interest follows a given adiabatic trajectory at an arbitrary speed via a Hamiltonian transformation. A practical approach is to consider an additional term in the Hamiltonian system, called counterdiabatic driving, to cancel out the non-adiabatic losses despite the finite duration of the process. This technique, which was first introduced in [11], later in [12, 13, 14, 15] and then in a quantum thermodynamic context [16, 17, 18, 19, 20, 21, 22] to avoid quantum friction [23, 24, 25], has gained much importance in adiabatic quantum computing [26], experimental state engineering [27], and quantum information processing [28], to name a few. STA techniques do not provide a definite scheme for accelerating the dynamics, since there are in principle infinitely many ways of defining an adiabatic trajectory, and require an ansatz typically based on physical considerations. On the other hand, optimal control theory (OCT) is a general mathematical procedure whose goal is to find time-dependent control parameters while minimizing or maximizing a functional that can be the control time-length or the energy used by the control, to name a few [29, 30, 2, 31]. The mathematical construction of OCT is based on the Pontryagin’s Maximum Principle (PMP) which was established in the late 1950s [32, 33, 34, 35, 36]. Today, OCT has become a powerful tool to optimize a variety of operations in quantum technologies [1, 2, 37].

STA provides a simple answer to perform a Hamiltonian transformation from a given Hamiltonian $H_{i}$ to a final Hamiltonian $H_{f}$ while following the adiabatic trajectory defined by a known protocol $H_{0}(t)$ where $H_{0}(0)=H_{i}$ and $H(t_{f})=H_{f}$. Such a protocol may be motivated, for example, by some thermodynamic protocols such as quantum Otto or Carnot cycles [38], or by other requirements in quantum annealing processes, qubits resets, or even in adiabatic Grover search algorithm [39]. In the situation in which the initial and the final Hamiltonians do not commute with each other, the protocol $H_{0}(t)$ does not commute with itself at different times, and it may induce some transitions between different energy levels, leading in particular to quantum friction [23, 24, 25] and larger work expenditure. However, there are two natural situations that avoid such extra work: (i) slow driving so that the induced dynamics satisfies the quantum adiabatic theorem [40, 41, 42], and (ii) some adequately chosen initial non-passive states [43]. STA techniques offer an alternative as they allow fast transformations while still preserving the adiabatic trajectory defined by $H_{0}(t)$, but at the cost of adding an extra driving term $V(t)$. In this framework, one of the widely used STA technique is the counter-diabatic drive [12, 13, 14, 15], which can be expressed as

$$V_{\text{CD}}(t)=i\hbar\sum_{n}\big{[}|\dot{n}(t)\rangle\langle n(t)|-\langle n(t)|\dot{n}(t)\rangle|n(t)\rangle\langle n(t)|\big{]},$$

(1)

where $|n(t)\rangle$ denotes the instantaneous eigenstates of the Hamiltonian $H_{0}(t)$, $H_{0}(t)=\sum_{n}e_{n}(t)|n(t)\rangle\langle n(t)|$. The effect of $V_{\text{CD}}(t)$ is actually to cancel the transitions between the energy levels of $H_{0}(t)$. However, the additional drive $V_{\text{CD}}(t)$ has to come with an additional energy cost [44, 45]. Several arguments have been put forward such as the quantum speed limit [46], qualitative estimate of the power needed to generate the control fields [47], or a connection with the classical entropy production generated during the generation of the control signal [48].

In this work, we define a new class of STA, where the additional driving is uniquely designed from the optimization of its energy cost. All the above propositions suggest a figure of merit of the form $W_{\text{cost}}\propto\omega_{i}^{2-\alpha}\int_{0}^{t_{f}}du||V_{\text{CD}}(u)||^{\alpha}$, with $\alpha=1$ or 2 and $\omega_{i}$ a typical frequency of the Hamiltonian $H_{0}$. Note that this additional energy cost is related to the power consumption of the devices used to control the quantum system, and is therefore usually much higher than the work cost associated with the Hamiltonian transformation (which is a cost at the quantum level). In addition, unnecessarily large controls applied to the quantum system can lead to extra dissipation and heating of the quantum system, which in turn leads to additional energy costs to cool the experimental setup [49].

Then, a natural question, of growing importance due to the intense debate on the energy cost of quantum technologies [50, 51], concerns the design of a STA protocol with minimal energy consumption. In other words, the goal is to find the minimum amount of energy required to implement an STA protocol. In this paper, we propose to solve this problem by using OCT. In the case of qubits, we highlight some processes where the energetically optimized STA is much less energetically expensive than the counter-diabatic drive. As expected, in the situation of very slow initial drive $H_{0}(t)$ (in a sense that will be made explicit later), both optimal and counter-diabatic drive become equal.

The paper is organized as follows. The optimization problem is formulated in Sec. II. In Sec. III, we show how the Pontryagin Maximum Principle can be applied and we derive the optimal equations. Formal solutions of these equations are given. Sections IV and V are dedicated to two examples, namely the case of a two-level quantum system with a constant energy gap and the Landau-Zener model. We discuss the similarities and differences of the energetically-optimized protocol and of the counter-adiabatic driving. A systematic analysis of the robustness with respect to the different Hamiltonian parameters is carried out in the two examples. A conclusion and prospective views are given in Sec. VI. Additional results are provided in the Appendix A.

In this work we focus on qubit systems. We consider arbitrary initial and final Hamiltonians, denoted by $H_{i}$ and $H_{f}$, respectively, for the desired transformation. The Hamiltonians are parameterized in units where $\hbar=1$ as

$$H_{i}:=\omega_{i}\sum_{k=x,y,z}u_{i,k}\sigma_{k}=\omega_{i}(|e_{i}\rangle\langle e_{i}|-|g_{i}\rangle\langle g_{i}|),$$

(2)

and

$$H_{f}:=\omega_{f}\sum_{k=x,y,z}u_{f,k}\sigma_{k}=\omega_{f}(|e_{f}\rangle\langle e_{f}|-|g_{f}\rangle\langle g_{f}|),$$

(3)

with $\sigma_{k}$, $k=x,y,z$, are the Pauli matrices and $\vec{u}_{i}=(u_{i,x},u_{i,y},u_{i,z})$, $\vec{u}_{f}=(u_{f,x},u_{f,y},u_{f,z})$ are unit vectors characterizing the initial and final Hamitonians. We also denote by $|e_{i}\rangle$ and $|g_{i}\rangle$ ($|e_{f}\rangle$ and $|g_{f}\rangle$) the initial (final) excited and ground states, respectively. We consider a protocol $H_{0}(t)$ that satisfies $H_{0}(0)=H_{i}$ and $H_{0}(t_{f})=H_{f}$, i.e. $H_{0}(t)$ realizes the above Hamiltonian transformation and is given by the physics of the problem (imposed, e.g., by a thermodynamic or algorithmic protocol on a certain platform).

In most of quantum control applications, one is only interested in the final state and not in all intermediate states. On this basis, we derive an optimal driving $V_{\text{opt}}(t)$ allowing the exact connection between the initial and target Hamiltonians as the adiabatic or STA processes, but with no constraint on the instantaneous trajectory followed by the system, other than minimizing the energetic cost. In particular, this new class of STA does not suffer from the high energy expenditure of standard STA protocols based on a counterdiabatic driving that have to compensate the adiabatic losses at all times.

More specifically, the goal is to find a control protocol of the form

$$V_{\text{opt}}(t)=\omega_{i}\vec{v}(t).\vec{\sigma}=\omega_{i}\sum_{k=x,y,z}v_{k}(t)\sigma_{k},$$

(4)

such that the eigenstates of $H_{i}$, $|e_{i}\rangle$ and $|g_{i}\rangle$ are brought to the corresponding eigenstates of $H_{f}$, $|e_{f}\rangle$ and $|g_{f}\rangle$, the dynamics being governed by $H(t)=H_{0}(t)+V_{\text{opt}}(t)$. In optimal control terminology, the original Hamiltonian, $H_{0}(t)$, can be interpreted as a time-dependent drift term that cannot be modified. Note that the final eigenstates can be reached up to a global phase factor. Then, starting from the state $|\psi(0)\rangle=|e_{i}\rangle$, the target state is

$$|\psi_{\text{traget}}\rangle=e^{i\xi_{f}}|e_{f}\rangle,$$

(5)

where $\xi_{f}$ is an unspecified global phase. Finding controls such that the generated dynamic $U(t_{f})$ brings $|\psi(0)\rangle=|e_{i}\rangle$ to $|\psi_{\text{target}}\rangle$ automatically implies that $U(t_{f})$ brings $|g_{i}\rangle$ to $|g_{f}\rangle$ up to a global phase. This is because $U(t_{f})|e_{i}\rangle=e^{i\xi_{f}}|e_{f}\rangle$ leads to $\langle e_{f}|U(t_{f})|g_{i}\rangle=0$.

In addition, for any state $\rho_{i}$ commuting with the initial Hamiltonian, and for any state $\rho_{f}$ commuting with the final Hamiltonian $H_{f}$, it is also interesting to require that

$${\rm Tr}[\rho_{i}H_{i}]={\rm Tr}[\rho_{i}H(0)]={\rm Tr}[\rho_{i}H_{i}]+\omega_{i}{\rm Tr}[\rho_{i}\vec{v}(0).\vec{\sigma}]$$

(6)

and

$${\rm Tr}[\rho_{f}H_{f}]={\rm Tr}[\rho_{f}H(t_{f})]={\rm Tr}[\rho_{f}H_{f}]+\omega_{i}{\rm Tr}[\rho_{f}\vec{v}(t_{f}).\vec{\sigma}],$$

(7)

which means that the additional drive $\omega_{i}\vec{v}(t).\vec{\sigma}$ does not affect the initial (final) energy of such states. In other words, this additional constraint guarantees that if the initial and final states are diagonal in the initial and final Hamiltonian bases, the protocol $H(t)$ does not contribute to the initial and final energy of the qubit. This property can also be found in some standard Shortcut-to-Adiabaticity protocols [46]. The two conditions can be expressed in terms of the control functions as

	$\displaystyle\vec{v}(0).\vec{u}_{i}=0,$
	$\displaystyle\vec{v}(t_{f}).\vec{u}_{f}=0.$

In the reminder of the paper, we will use $\{|e_{i}\rangle,|g_{i}\rangle\}$, the eigenbasis of $H_{i}$ as the reference basis.

An optimal solution satisfies the boundary conditions described in Sec. II while minimizing a cost functional. Since the goal here is to design a low-energy STA-like protocol, it is natural to consider an energy cost in the optimization process. As mentioned in the introduction, there have been several proposals to evaluate such a cost associated with a quantum control scheme. In this paper, following [48], we choose an energy cost defined by $\alpha=2$, which leads to the following cost functional

$${\cal C}=\frac{\omega_{i}}{2}\int_{0}^{t_{f}}du\vec{v}^{2}(u).$$

(8)

The corresponding optimal control can be designed by applying the PMP. In this approach, the optimal control problem is transformed into a generalized Hamiltonian system with specific boundary conditions. The time evolution of the control parameters is found by maximizing this Hamiltonian over the allowed controls. We refer the interested reader to recent reviews on the subject for details [2, 1, 35, 34].

In our control problem, the Pontryagin Hamiltonian can be written in the normal case as [34]

$$H_{p}=\Im[\langle\chi(t)|H(t)|\psi(t)\rangle]-\frac{\omega_{i}}{2}\vec{v}^{2}(t),$$

(9)

where $\langle\chi(t)|$ is the adjoint state of $|\psi(t)\rangle$. The PMP states that the state and the adjoint state are solutions of the Schrödinger equation given by

	$\displaystyle\frac{d|\psi(t)\rangle}{dt}$	$\displaystyle=$	$\displaystyle-iH(t)|\psi(t)\rangle,$		(10)
	$\displaystyle\frac{d|\chi(t)\rangle}{dt}$	$\displaystyle=$	$\displaystyle-iH(t)|\chi(t)\rangle.$		(11)

Since there is no additional constraint on the controls $\vec{v}$, the maximization condition on $H_{p}$ gives $\partial H_{p}/\partial\vec{v}=0$. The optimal controls denoted by $\vec{v}^{*}$ can then be expressed as

$$v_{k}^{*}(t)=\Im[\langle\chi(t)|\sigma_{k}|\psi(t)\rangle].$$

(12)

Plugging them into the Hamiltonian $H(t)$, we obtain (see details in Appendix A),

	$\displaystyle H^{*}(t)$	$\displaystyle=$	$\displaystyle H_{0}(t)-i\omega_{i}\Big{(}|\psi(t)\rangle\langle\chi(t)|-|\chi(t)\rangle\langle\psi(t)|\Big{)}$		(13)
			$\displaystyle-\omega_{i}\Im[\langle\chi(t)|\psi(t)\rangle].$

Using Eq. (13), the dynamical system satisfied by $|\psi(t)\rangle$ and $|\chi(t)\rangle$ can be written as

	$\displaystyle\frac{d|\psi(t)\rangle}{dt}$	$\displaystyle=$	$\displaystyle-iH_{0}(t)|\psi(t)\rangle-\omega_{i}\Big{(}d|\psi(t)\rangle-|\chi(t)\rangle\Big{)},$
	$\displaystyle\frac{d|\chi(t)\rangle}{dt}$	$\displaystyle=$	$\displaystyle-iH_{0}(t)|\chi(t)\rangle-\omega_{i}\Big{(}c|\psi(t)\rangle-d|\chi(t)\rangle\Big{)},$

where $d:=\Re[\langle\chi(t)|\psi(t)\rangle]$ and $c:=\langle\chi(t)|\chi(t)\rangle\geq 0$. Since $|\psi(t)\rangle$ and $|\chi(t)\rangle$ follow unitary evolutions, we deduce that $\langle\psi(t)|\chi(t)\rangle$ and $\langle\chi(t)|\chi(t)\rangle$, and thus $c$, $d$ and $s=\Im[\langle\chi(t)|\psi(t)\rangle]$ are constants of motion. Note that $c$ is a real positive number which can be different from 1 ($|\chi(t)\rangle$ is not necessarily normalized), while $d$ and $s$ can be any real number. Then, introducing, $|\tilde{\psi}(t)\rangle=U_{0}^{\dagger}(t)|\psi(t)\rangle$ and $|\tilde{\chi}(t)\rangle=U_{0}^{\dagger}(t)|\chi(t)\rangle$, with $U_{0}(t)$ the time evolution generated by $H_{0}(t)$, we can show that

	$\displaystyle\frac{d|\tilde{\psi}(t)\rangle}{dt}$	$\displaystyle=$	$\displaystyle-\omega_{i}\Big{(}d|\tilde{\psi}(t)\rangle-|\tilde{\chi}(t)\rangle\Big{)},$		(14)
	$\displaystyle\frac{d|\tilde{\chi}(t)\rangle}{dt}$	$\displaystyle=$	$\displaystyle-\omega_{i}\Big{(}c|\tilde{\psi}(t)\rangle-d|\tilde{\chi}(t)\rangle\Big{)}.$		(15)

We can parameterize $|\tilde{\chi}(0)\rangle=|\chi(0)\rangle$ in the eigenbasis $\{|e_{i}\rangle,|g_{i}\rangle\}$ of $H_{i}$ as

$$|\chi(0)\rangle=(d+is)|e_{i}\rangle+r|g_{i}\rangle.$$

(16)

This yields the relation $c=d^{2}+s^{2}+|r|^{2}$. In general, $|\tilde{\psi}(t)\rangle$ and $|\tilde{\chi}(t)\rangle$ are not orthogonal, which can lead to some additional difficulties in solving the dynamics. Therefore, we introduce the normalized vector $|\tilde{\phi}(t)\rangle$ orthogonal to $|\tilde{\psi}(t)\rangle$ as

	$\displaystyle|\tilde{\phi}(t)\rangle$	$\displaystyle:=$	$\displaystyle\frac{\Big{(}|\tilde{\chi}(t)\rangle-\langle\tilde{\psi}(t)|\tilde{\chi}(t)\rangle|\tilde{\psi}(t)\rangle\Big{)}}{|||\tilde{\chi}(t)\rangle-\langle\tilde{\psi}(t)|\tilde{\chi}(t)\rangle|\tilde{\psi}(t)\rangle||}$		(17)
		$\displaystyle=$	$\displaystyle\frac{1}{|r|}\Big{(}|\tilde{\chi}(t)\rangle-(d+is)|\tilde{\psi}(t)\rangle\Big{)}.$

Note that $|\tilde{\phi}(0)\rangle=|\phi(0)\rangle=\frac{r}{|r|}|g_{i}\rangle$. Using this new vector $|\tilde{\phi}(t)\rangle$, the dynamical system becomes

	$\displaystyle\frac{d|\tilde{\psi}(t)\rangle}{dt}$	$\displaystyle=$	$\displaystyle-\omega_{i}\Big{(}-is|\tilde{\psi}(t)\rangle-|r||\tilde{\phi}(t)\rangle\Big{)},$		(18)
	$\displaystyle\frac{d|\tilde{\phi}(t)\rangle}{dt}$	$\displaystyle=$	$\displaystyle-\omega_{i}\Big{(}|r||\tilde{\psi}(t)\rangle+is|\tilde{\phi}(t)\rangle\Big{)}.$		(19)

Now, we introduce the kets

$$|X_{\pm}(t)\rangle:=x_{\pm}|\tilde{\psi}(t)\rangle+y_{\pm}|\tilde{\phi}(t)\rangle,$$

(20)

such that

$\displaystyle\frac{d}{dt}|X_{\pm}(t)\rangle$

$\displaystyle=\lambda_{\pm}|X_{\pm}(t)\rangle.$

(21)

Using Eqs. (18) and (19), one can show that we have to choose

$$\frac{x_{\pm}}{y_{\pm}}=\frac{i}{|r|}(s\pm\sqrt{s^{2}+|r|^{2}}),$$

(22)

with

$$\lambda_{\pm}=\pm i\omega_{i}\sqrt{s^{2}+|r|^{2}}.$$

(23)

Note that only the ratio $x_{\pm}/y_{\pm}$ matters, or, in other words, we can choose $y_{\pm}=1$. Then, we have

$$|X_{\pm}(t)\rangle=e^{\lambda_{\pm}t}|X_{\pm}(0)\rangle,$$

(24)

from which we deduce the time evolution of $|\tilde{\psi}(t)\rangle$ and $|\tilde{\phi}(t)\rangle$ by inverting the relation (20),

	$\displaystyle|\tilde{\psi}(t)\rangle$	$\displaystyle=$	$\displaystyle\frac{1}{x_{+}y_{-}-x_{-}y_{+}}$		(25)
			$\displaystyle\times\left(y_{-}e^{\lambda_{+}t}|X_{+}(0)\rangle-y_{+}e^{\lambda_{-}t}|X_{-}(0)\rangle\right)$
		$\displaystyle=$	$\displaystyle\frac{1}{x_{+}/y_{+}-x_{-}/y_{-}}$
			$\displaystyle\times\Big{[}\left(e^{\lambda_{+}t}x_{+}/y_{+}-e^{\lambda_{-}t}x_{-}/y_{-}\right)|e_{i}\rangle$
			$\displaystyle\hskip 42.67912pt+\left(e^{\lambda_{+}t}-e^{\lambda_{-}t}\right)\frac{r}{|r|}|g_{i}\rangle\Big{]},$
	$\displaystyle|\tilde{\phi}(t)\rangle$	$\displaystyle=$	$\displaystyle\frac{1}{x_{+}y_{-}-x_{-}y_{+}}$
			$\displaystyle\times\left(-x_{-}e^{\lambda_{+}t}|X_{+}(0)\rangle+x_{+}e^{\lambda_{-}t}|X_{-}(0)\rangle\right)$
		$\displaystyle=$	$\displaystyle\frac{1}{x_{+}/y_{+}-x_{-}/y_{-}}$
			$\displaystyle\times\Big{[}-\frac{x_{+}x_{-}}{y_{+}y_{-}}\left(e^{\lambda_{+}t}-e^{\lambda_{-}t}\right)|e_{i}\rangle$
			$\displaystyle+\left(-e^{\lambda_{+}t}x_{-}/y_{-}+e^{\lambda_{-}t}x_{+}/y_{+}\right)\frac{r}{|r|}|g_{i}\rangle\Big{]}.$

From this result, we find the expression of $|\tilde{\chi}(t)\rangle=|r||\tilde{\phi}(t)\rangle+(d+is)|\tilde{\psi}(t)\rangle$, and we can deduce also $H(t)$ using expression Eq. (13) as well as the optimal control,

	$\displaystyle v_{k}^{*}(t)$	$\displaystyle=$	$\displaystyle\Im[\langle\chi(t)|\sigma_{k}|\psi(t)\rangle]$		(27)
		$\displaystyle=$	$\displaystyle\Im[\langle\tilde{\chi}(t)|\sigma_{k}^{0}(t)|\tilde{\psi}(t)\rangle],$

where $\sigma_{k}^{0}(t):=U_{0}^{\dagger}(t)\sigma_{k}U_{0}(t)$. Note that in order to have an explicit expression of the controls $v_{k}(t)$, we compute the evolution generated by the original protocol $H_{0}(t)$, which is easily done numerically.

The solutions we have just derived are candidates for optimality. The last step of the procedure consists in finding the solutions that also satisfy the boundary conditions. To this aim, we choose adequately the free parameters ($r$, $d$ and $s$, or equivalently $|\chi(0)\rangle$) such that $|\psi(t_{f})\rangle=e^{i\xi_{f}}|e_{f}\rangle$, as well as ${\rm Tr}[\rho(H_{i}+\omega_{i}\vec{v}(0).\vec{\sigma})]={\rm Tr}[\rho H_{i}]$ for all state $\rho=p_{e}|e_{i}\rangle\langle e_{i}|+p_{g}|g_{i}\rangle\langle g_{i}|$ that commutes with the Hamiltonian $H_{i}$, and a similar condition for the final Hamiltonian. This implies ${\rm Tr}[\rho V_{{\rm opt}}(0)]=\omega_{i}{\rm Tr}[\rho\vec{v}(0).\vec{\sigma}]=0$ from which we obtain

	$\displaystyle 0$	$\displaystyle=$	$\displaystyle{\rm Tr}[\rho(|\psi(0)\rangle\langle\chi(0)|-|\chi(0)\rangle\langle\psi(0)|)]+is$
		$\displaystyle=$	$\displaystyle p_{e}[\langle e_{i}|\psi(0)\rangle\langle\chi(0)|e_{i}\rangle-\langle e_{i}|\chi(0)\rangle\langle\psi(0)|e_{i}\rangle]+is$
			$\displaystyle+p_{g}[\langle g_{i}|\psi(0)\rangle\langle\chi(0)|g_{i}\rangle-\langle g_{i}|\chi(0)\rangle\langle\psi(0)|g_{i}\rangle]$
		$\displaystyle=$	$\displaystyle p_{e}[d-is-(d+is)]+is=-(2p_{e}-1)is$

where in the last line we use $|\psi(0)\rangle=|e_{i}\rangle$. Then, the initial condition gives $s=0$. Similarly, for the condition at final time $t_{f}$,we arrive at

	$\displaystyle 0$	$\displaystyle=$	$\displaystyle p_{e}[\langle e_{f}|\psi(t_{f})\rangle\langle\chi(t_{f})|e_{f}\rangle-\langle e_{f}|\chi(t_{f})\rangle\langle\psi(t_{f})|e_{f}\rangle]+is$
			$\displaystyle+p_{g}[\langle g_{f}|\psi(t_{f})\rangle\langle\chi(t_{f})|g_{f}\rangle-\langle g_{f}|\chi(t_{f})\rangle\langle\psi(t_{f})|g_{f}\rangle]$
		$\displaystyle=$	$\displaystyle p_{e}[d-is-(d+is)]+is=-(2p_{e}-1)is,$

where the last line assumes that the drive has successfully implemented the expected dynamics, namely $|\psi(t_{f})\rangle=e^{i\xi_{f}}|e_{f}\rangle$, which also implies that $\langle e_{f}|\chi(t_{f})\rangle=e^{-i\xi_{f}}(d+is)$. Then, we obtain that the condition $s=0$ guarantees that the additional drive does not affect the initial and final energies of the system (as long as the initial and the final states commute respectively with the initial and final Hamiltonians).

The last condition to take into account is $|\psi(t_{f})\rangle=e^{i\xi_{f}}|e_{f}\rangle$. Before that, we analyze geometrically the consequence of the condition $s=0$ that simplifies the equations of motion. Indeed, $s=0$ leads to

$$\frac{x_{\pm}}{y_{\pm}}=\pm i,\ \lambda=\pm i\omega_{i}|r|,$$

and

	$\displaystyle|\tilde{\psi}(t)\rangle$	$\displaystyle=$	$\displaystyle\cos{(\omega_{i}|r|t)}|e_{i}\rangle+e^{i\phi_{r}}\sin{(\omega_{i}|r|t)}|g_{i}\rangle,$		(28)
	$\displaystyle|\tilde{\phi}(t)\rangle$	$\displaystyle=$	$\displaystyle-\sin{(\omega_{i}|r|t)}|e_{i}\rangle+e^{i\phi_{r}}\cos{(\omega_{i}|r|t)}|g_{i}\rangle,$		(29)

with $\phi_{r}:=\arg(r)$, which corresponds to a rotation on the Bloch sphere about the axis $\vec{u}_{r}=(-\cos\phi_{r},\sin\phi_{r},0)$ at angular velocity $2\omega_{i}|r|$.

Equivalently, the resulting Hamiltonian in the rotating picture with respect to $H_{0}(t)$ is

	$\displaystyle\tilde{H}(t)$	$\displaystyle=$	$\displaystyle\tilde{V}_{\text{opt}}(t):=U_{0}(t)^{\dagger}V_{\text{opt}}(t)U_{0}(t)$
		$\displaystyle=$	$\displaystyle-i\omega_{i}\Big{(}|\tilde{\psi}(t)\rangle\langle\tilde{\chi}(t)|-|\tilde{\chi}(t)\rangle\langle\tilde{\psi}(t)|\Big{)}$
		$\displaystyle=$	$\displaystyle-i\omega_{i}\Big{(}|\tilde{\psi}(t)\rangle\langle\tilde{\phi}(t)|-|\tilde{\phi}(t)\rangle\langle\tilde{\psi}(t)|\Big{)}.$

This gives, using the above expressions, the following time-independent Hamiltonian,

$$\tilde{V}_{\text{opt}}=\omega_{i}|r|\big{[}-\sin(\phi_{r})\sigma_{x}^{(i)}+\cos(\phi_{r})\sigma_{y}^{(i)}\big{]},$$

(30)

which also corresponds to the aforementioned rotation in the Bloch sphere, where

	$\displaystyle\sigma_{x}^{(i)}$	$\displaystyle:=$	$\displaystyle|e_{i}\rangle\langle g_{i}|+|g_{i}\rangle\langle e_{i}|,$
	$\displaystyle\sigma_{y}^{(i)}$	$\displaystyle:=$	$\displaystyle-i|e_{i}\rangle\langle g_{i}|+i|g_{i}\rangle\langle e_{i}|,$
	$\displaystyle\sigma_{z}^{(i)}$	$\displaystyle:=$	$\displaystyle|e_{i}\rangle\langle e_{i}|-|g_{i}\rangle\langle g_{i}|,$

are the Pauli matrices in the eigenbasis of $H_{i}$, $\{|e_{i}\rangle,|g_{i}\rangle\}$. Note that the parameter $d$, which characterizes the real part of the overlap between $|\tilde{\psi}(t)\rangle$ and $|\tilde{\chi}(t)\rangle$, has disappeared from the dynamics, and is therefore an irrelevant parameter (which can actually already be seen by combining Eqs. (14), (15), and (16)).

Then, we deduce that for a fixed final time $t_{f}$, $|\tilde{\psi}(t)\rangle$ can reach any state on the Bloch sphere by choosing adequately $|r|$ and $\phi_{r}$, and in particular we can fulfill our final condition $|\psi(t_{f})\rangle=e^{i\xi_{f}}|e_{f}\rangle$ (which is equivalent to $|\tilde{\psi}(t_{f})\rangle=e^{i\xi_{f}}U_{0}^{\dagger}(t_{f})|e_{f}\rangle$). Denoting by $k_{f}:=\langle e_{f}|U_{0}(t_{f})\sigma_{k}^{(i)}U_{0}^{\dagger}(t_{f})|e_{f}\rangle$, for $k=x,y,z$, the Bloch coordinates of $e^{i\xi_{f}}U_{0}^{\dagger}(t_{f})|e_{f}\rangle$ in the eigenbasis of $H_{i}$, a direct geometrical analysis shows that the final condition $|\psi(t_{f})\rangle=e^{i\xi_{f}}|e_{f}\rangle$ is satisfied if

	$\displaystyle\cos(\phi_{r})\sin(2\omega_{i}|r|t_{f})=x_{f},$
	$\displaystyle\sin(\phi_{r})\sin(2\omega_{i}|r|t_{f})=y_{f},$
	$\displaystyle\cos{(2\omega_{i}|r|t_{f})}=z_{f},$

which is equivalent to

	$\displaystyle|r|=\frac{\arccos(z_{f})}{2\omega_{i}t_{f}},$
	$\displaystyle\frac{\Re(r)}{|r|}=\frac{x_{f}}{\sqrt{1-z_{f}^{2}}},$
	$\displaystyle\frac{\Im(r)}{|r|}=\frac{y_{f}}{\sqrt{1-z_{f}^{2}}},$

and finally to

	$\displaystyle\Re(r)$	$\displaystyle=$	$\displaystyle\frac{x_{f}\arccos(z_{f})}{2\omega_{i}t_{f}\sqrt{1-z_{f}^{2}}},$		(31)
	$\displaystyle\Im(r)$	$\displaystyle=$	$\displaystyle\frac{y_{f}\arccos(z_{f})}{2\omega_{i}t_{f}\sqrt{1-z_{f}^{2}}}.$		(32)

We have shown that, for any initial and final Hamiltonian and any protocol $H_{0}(t)$, we can solve our problem completely and exactly, as long as we can compute, at least numerically, the transformation $U_{0}(t)$ generated by $H_{0}(t)$. The explicit expression of the optimal controls is

$$v_{k}(t)=\frac{|r|}{2}{\rm Tr}\Big{(}\big{[}-\sin(\phi_{r})\sigma_{x}^{(i)}+\cos(\phi_{r})\sigma_{y}^{(i)}\big{]}U_{0}^{\dagger}(t)\sigma_{k}U_{0}(t)\Big{)},$$

(33)

for $k=x,y,z$.

Finally, we conclude this derivation by a comment on the global phase. When using the Bloch representation, this phase is automatically discarded. However, it can be seen that Eqs. (28) and (29) do not allow any control of the global phase. This is actually due to the choice $s=0$. It can be shown that the global phase can indeed be controlled by adjusting adequately $s$. However, this would imply the loss of the conservation of initial and final energy with respect to $H_{0}(0)$ and $H_{0}(t_{f})$.