# Time scaling and quantum speed limit in non-Hermitian Hamiltonians

F. Impens1, F. M. D’Angelis1, F. A. Pinheiro1 and D. Guéry-Odelin2 1 Instituto
de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ
21941-972, Brazil
2 Laboratoire Collisions, Agrégats, Réactivité, IRSAMC, Université de
Toulouse, CNRS, UPS, France

###### Abstract

We report on a time scaling technique to enhance the performances of quantum
protocols in non-Hermitian systems. The considered time scaling involves no
extra-couplings and yields a significant enhancement of the quantum fidelity
for a comparable amount of resources. We discuss the application of this
technique to quantum state transfers in 2 and 3-level open quantum systems. We
derive the quantum speed limit in a system governed by a non-Hermitian
Hamiltonian. Interestingly, we show that, with an appropriate driving, the
time-scaling technique preserves the optimality of the quantum speed with
respect to the quantum speed limit while reducing significantly the damping of
the quantum state norm.

Fast quantum control protocols have a promising future in quantum platforms as
they mitigate the deleterious effects of disorder or/and dissipation.
Introduced about a decade ago, shortcuts-to-adiabaticity (STA) RMPdavid  have
already a strong track record of improving quantum procedures in a wide
variety of quantum platforms including ultra-cold atom setups Schaff10 ;
Martinez13 ; Amri19 ; Impens20 , NMR Zhou20  and solid-state Zhou17  systems,
superconducting qubits Wang19  and topological spin chains FMD2020 ;
Theocharis21 . There are several well-established methods to build STA
protocols, such as the optimal control Chasseur15 , the counterdiabatic
driving Rice03 ; Rice05 ; Rice08 ; Berry09 , the use of Lewis-Riesenfeld
invariants Lewis69 ; Chen12 ; Ruschhaupt12 ; Impens17 , or the use of properly
scaled dynamical variables Deffner14 , to name a few. Those different methods
provide different strategies to hamper, compensate or mitigate the effects of
non-adiabatic transitions.

As a matter of fact, STA protocols require more resources than adiabatic
methods and may involve a larger number of dynamical couplings. For instance,
the Fast-Forward technique as originally introduced by Masuda Masuda08 ;
Masuda14 ; Takahashi14 ; Zhu21 , introduces extra couplings to be regularized
in the limit of strong acceleration Setiawan17 . The same conclusion holds for
most counterdiabatic protocols. However, the dynamical control of additional
interactions may constitute a limit for their practical implementation.

In the presence of dissipation, the concept of adiabaticity breaks down. The
question is rather how to mitigate the effect of dissipation for a given
protocol duration, and to approach the ultimate limit provided by the quantum
speed limit (QSL) QSL45 ; QSL73 ; QSL90 ; Vaidmann92 ; QSL03 ; QSL05 ; QSL09 ;
QSL12 ; QSLOpen13a ; QSLOpen13b  in non-Hermitian systems. In this article, we
investigate a time scaling method for finite Hilbert spaces that tackles those
two issues: it does not introduce extra couplings, enables a minimization of
the resources and provides a strategy to keep the quantum speed optimality
while reducing the deleterious effect of the dissipation on the state norm.

As a starting point, we consider a given quantum trajectory
$|\psi_{0}(t)\rangle$ solution of the time-dependent Schrödinger equation for
the Hamiltonian $\hat{H}(t)$:

$i\hbar\frac{\partial|\psi_{0}(t)\rangle}{\partial
t}=H(t)|\psi_{0}(t)\rangle,$ (1)

where the time-dependence of the Hamiltonian is encapsulated in a set of
parameters: $\hat{H}(t)=\hat{H}[\lambda_{1}(t),...,\lambda_{N}(t)]$. The
quantum trajectory $|\psi_{0}(\Lambda(t))\rangle$ is then a solution of the
time-dependent Schrödinger equation for the rescaled Hamiltonian
$\hat{H}_{\Lambda}(t)$:

$\hat{H}_{\Lambda}(t)=\dot{\Lambda}\hat{H}[\lambda_{1}(\Lambda(t)),...,\lambda_{N}(\Lambda(t))]\,.$
(2)

where $\Lambda(t)$ is assumed to be a monotone, differentiable function such
that $\Lambda(0)=0$ and $\dot{\Lambda}(t)\geq 0$ at any time.

The Hamiltonian (2) simply provides a time rescaling of the original solution.
If $T$ denotes the final time at which the systems reaches the desired quantum
state target under the driving $\hat{H}(t)$, the evolution under the rescaled
Hamiltonian $\hat{H}_{\Lambda}(t)$ reaches the very same target in a time
$T_{\Lambda}=\Lambda^{-1}(T)$, that can be much shorter. As a result, time
scaling provides a priori the simplest way to realize a shortcut to
adiabaticity protocol.

The time-scaling method provides an enhancement of the protocol performance
while maintaining the original quantum trajectory. In the following, we
explain how to design the time scaling $\Lambda(t)$ in a wide variety of
contexts. To work out quantitatively a strategy that minimizes the effect of
dissipation, we define in Sec. I a “quasi-unitary” driving that ensures a
constant damping rate during the whole parametrized evolution. This systematic
approach provides a clear improvement over the original driving and is
illustrated in 2 and 3-level systems. This strategy can be applied jointly
with geometric corrections on the driving field mitigating the effects of
dissipation Impens19 . We explain how a suitable choice for the time-scaling
function enables one to minimize the energetic cost of STA protocols for both
closed and open systems while achieving the same quantum fidelity. In Sec. II,
we discuss the relation between the time-scaling transform and the quantum
speed limit. Generalizations of the QSL to open systems have been obtained
within the density matrix formalism QSLOpen13a , and in connection with the
concept of Fisher information QSLOpen13b . Here, we put forward a simple
derivation of the QSL for quantum systems driven by non-Hermitian Hamiltonians
in the spirit of the Vaidmann bound Vaidmann92 . We show that the time-scaling
transform preserves the ratio of the quantum speed to the QSL in 2 and 3-level
dissipative systems with appropriate corrections to the quantum driving.

##  I Time scaling for dissipative 2 and 3-level systems

We discuss here the application of the time scaling method to open quantum
systems described by non-Hermitian Hamiltonian Plenio98 ; Moiseyev98 . First,
we address the commonly-called FAQUAD (Fast quasiadiabatic) protocol FAQUAD1 ;
FAQUAD2  for a dissipative two-level system CohenBookPhotonAtoms . We then
investigate the application of time scaling to the STIRAP (Stimulated Raman
adiabatic passage) protocol in a 3-level systems.

###  I.1 FAQUAD driving in a 2-level dissipative system

The FAQUAD protocol has been originally proposed for dissipationless quantum
systems to perform a state to state transformation as quickly as possible
while remaining as adiabatic as possible at all times. Let’s remind the main
features of this protocol for a two-level quantum system described by the
control Hamiltonian:

$\hat{H}_{0}(t)=\frac{\hbar}{2}\left(\begin{array}[]{cc}\delta(t)&\Omega(t)\\\
\Omega^{*}(t)&-\delta(t)\end{array}\right)$ (3)

expressed here in the decoupled $\\{|e\rangle,|g\rangle\\}$ basis. Its
instantaneous eigenvalues are
$E_{\pm}(t)=\pm\hbar\sqrt{\delta(t)^{2}+\Omega(t)^{2}}$. They are associated
to the instantaneous eigenvectors of $\hat{H}_{0}(t)$

$|\phi_{+}(\theta)\rangle=\left(\begin{array}[]{c}\cos(\frac{\theta}{2})\\\
\sin(\frac{\theta}{2})\end{array}\right),\;|\phi_{-}(\theta)\rangle=\left(\begin{array}[]{c}\sin(\frac{\theta}{2})\\\
-\cos(\frac{\theta}{2})\end{array}\right),$ (4)

where $\theta(t)=\acos[\delta(t)/\sqrt{\delta(t)^{2}+\Omega^{2}(t)}]$. In the
FAQUAD approach, the time evolution of $\theta(t)$ is obtained by keeping
constant the adiabatic criterium (equal to a constant $c$). One finds

$\dot{\theta}=\frac{c}{\hbar}\frac{|E_{+}(t)-E_{-}(t)|}{|\langle\phi_{+}(\theta)|\partial_{\theta}\phi_{-}(\theta)\rangle|}$
(5)

where $0<c<1$. The $c\rightarrow 0$ limit is nothing but the adiabatic limit.
In this protocol, non-adiabatic transitions are controlled through the
“driving speed”.

In view of the implementation of the time-scaling method, it is worth noticing
that the equality (5) remains unchanged by the following scaling:
$t\to\Lambda(t)$, $\Omega(t)\to\dot{\Lambda}(t)\Omega(\Lambda(t))$ and
$\delta(t)\to\dot{\Lambda}(t)\delta(\Lambda(t))$. As a result, the scaling
function $\Lambda(t)$ can be engineered to fulfill extra requirements. As an
example, we propose hereafter to set this scaling function by defining the
acceptable dissipation rate for the desired transformation.

We model the dissipation for this two level problem with the non-Hermitian
Hamiltonian $\hat{H}=\hat{H}_{0}-i\hbar\hat{\gamma}$ with
$\hat{\gamma}=\gamma_{e}|e\rangle\langle e|+\gamma_{g}|g\rangle\langle g|$,
where $|e\rangle$ refers to the excited state and $|g\rangle$ to the ground
state. In the following, we consider an original FAQUAD passage based on the
quasi-adiabatic evolution of the eigenvector $|\phi_{+}(\theta)\rangle$. As a
result of the dissipation, the norm $\mathcal{N}$ of the quantum state
decreases as a function of time. For a quasi-adiabatic evolution one gets

$\frac{1}{{\mathcal{N}}}\frac{d\mathcal{N}}{dt}=-\langle\phi_{+}(\theta(t))|\hat{\gamma}|\phi_{+}(\theta(t))\rangle.$
(6)

F The instantaneous damping rate thus depends only on the parameter $\theta$
and on the dissipation operator $\hat{\gamma}$. Taking advantage of the extra
freedom provided by the time scaling function, we can also impose a fixed
“geometric” damping rate along the trajectory:

$\frac{1}{{\mathcal{N}_{\Lambda}}}\frac{d\mathcal{N}_{\Lambda}}{d\theta}=-c^{\prime}.$
(7)

As a result, $\mathcal{N}_{\Lambda}(T)=\mathcal{N}_{\Lambda}(0)\exp[-\pi
c^{\prime}]$. The condition (7) prescribes a driving speed proportional to the
instantaneous dissipation rate
$\dot{\theta}^{\Lambda}(t)=\langle\phi_{+}(\theta^{\Lambda}(t))|\hat{\gamma}|\phi_{+}(\theta^{\Lambda}(t))\rangle/c^{\prime}$.
It encodes mathematically the intuitive idea according to which one should
increase the driving speed in region of strong dissipation. Interestingly, the
driving speed depends here only on geometric features of the trajectory
$|\phi_{+}(\theta)\rangle$, i.e. on the orientation of the associated Bloch
vector, and not on its norm. Finally, the time-scaling $\Lambda(t)$ connecting
the prescribed driving speed $\dot{\theta}^{\Lambda}(t)$ to the original
driving $\theta(t)$ is obtained by taking the ratio of Eqs. (6) and (7):

$\dot{\Lambda}=\frac{1}{c^{\prime}\dot{\theta}(\Lambda)}\langle\phi_{+}(\theta(\Lambda))|\hat{\gamma}|\phi_{+}(\theta(\Lambda)\rangle$
(8)

The coefficient $c^{\prime}$ is fixed self-consistently by the total duration
of the time scaling.

As a concrete example, we consider a time-rescaled FAQUAD protocol keeping the
same protocol duration, $\Lambda(T)=T$, and with a constant Rabi frequency
$\Omega(t)=\Omega_{0}$. The single control parameter is therefore the time-
dependent detuning $\delta(t)$ FAQUAD1 ; FAQUAD2 . Equation (5) yields
$\cos\theta(t)-\cos\theta_{0}=-4c\Omega_{0}t.$ To ensure a high fidelity
transfer, the angle $\theta(t)$ must fulfill the boundary conditions
$\theta_{0}\simeq 0$ and $\theta_{T}\simeq\pi$. We choose
$\cos\theta_{0}=1-\epsilon$ and $\cos\theta_{T}=-1+\epsilon$ with
$0<\epsilon\ll 1$ as a null parameter $\epsilon=0$ generates unrealistic
infinite detuning at the time boundaries FAQUAD1 ; FAQUAD2 . The adiabaticity
constant reads $c=(1-\epsilon)/(2\Omega_{0}T)$. One then readily finds the
angle $\theta(t)={\rm arccos}[f_{\epsilon}(t)]$ with
$f_{\epsilon}(t)=(1-\epsilon)(1-2t/T)$, and the corresponding detuning
$\delta(t)=\Omega/\tan\theta(t)$. We find
$\langle\phi_{+}(\theta(t))|\hat{\gamma}|\phi_{+}(\theta(t))\rangle=\frac{1}{2}(\gamma_{e}+\gamma_{g})+\frac{1}{2}(\gamma_{e}-\gamma_{g})f_{\epsilon}(t)$
and $\dot{\theta}(t)=2(1-\epsilon)/[T\sqrt{1-f_{\epsilon}(t)^{2}}]$. The
scaling $\Lambda(t)$ is subsequently obtained by solving (8).

In Fig. 1, we summarize the results of the original FAQUAD and of its time-
rescaled version for a specific example in the presence of dissipation. We
have chosen the parameters $\Omega_{0}T=10$ and $\epsilon=0.01$, yielding an
adiabaticity constant $c\simeq 0.05$. We use $\gamma_{e}T=0.1$ and
$\gamma_{g}=0.01\gamma_{e}$. For our parameters, the condition $\Lambda(T)=T$
dictates the value of the constant $c^{\prime}\simeq 5.3\times 10^{-3}$. The
final purity $p=p_{g}/(p_{g}+p_{e})$, defined as the fraction of the target
state population, is $p\simeq 0.998$ for both protocols. However, our time-
rescaled FAQUAD protocol yields a norm reduction $\mathcal{N}_{\Lambda}\simeq
0.97$ to be compared to $\mathcal{N}\simeq 0.90$ for the initial protocol. The
time-scaling thus significantly enhances the performance of the FAQUAD driving
in the presence of dissipation, by keeping a high purity while reducing the
norm reduction rate by at least a factor of $3$ in this specific case.

Figure 1: Time-scaled FAQUAD protocol for a 2-level system: (a): Time-scaling
function $\Lambda(t)/T$ (red solid line) as a function of the renormalized
time $t/T$. The black solid line corresponds to the original protocol
($\Lambda(t)=t$). (b) Time-dependent occupation probability of the ground
state $|g\rangle$ for the original (black solid line) FAQUAD protocol and for
its time-scaled version (red solid line) as a function of the renormalized
time $t/T$. Parameters: $\Omega_{0}T=10$, $\epsilon=0.01$, $\gamma_{e}T=0.1$
and $\gamma_{g}=0.01\gamma_{e}$ .

###  I.2 Time scaling in a STIRAP transfer

####  I.2.1 The dissipationless STIRAP solution

In this section, we investigate the interest of time scaling for an
accelerated population transfer in a dissipative 3-level system. More
precisely, we consider a 3-level system in a $\Lambda$-configuration. In the
absence of dissipation, the quantum state,
$|\psi(t)\rangle=C_{1}(t)|1\rangle+C_{2}(t)|2\rangle+C_{3}(t)|3\rangle$, obeys
the Schrödinger equation associated to the control Hamiltonian

$\hat{H}_{0}(t)=\frac{\hbar}{2}\left(\begin{array}[]{ccc}0&\Omega_{p}^{0}(t)&0\\\
\Omega_{p}^{0}(t)&0&\Omega_{s}^{0}(t)\\\ 0&\Omega_{s}^{0}(t)&0\\\
\end{array}\right).$ (9)

The transfer of the population from the ground state $|1\rangle$ to the ground
state $|3\rangle$ can be realized by following adiabatically the dark state,
which amounts to applying Stokes ($\Omega_{s}$) and pump ($\Omega_{p}$) field
pulses with a slight delay whilst maintaining a significant temporal overlap
between the two pulses ReviewSTIRAP1 ; ReviewSTIRAP2 . Using an invariant-
based inverse engineering technique, such a transfer can be accelerated at the
expense of a transient population in the excited state $|2\rangle$ Chen12 . In
this latter protocol, the dissipation-free quantum trajectory can be
parametrized as

$|\psi_{0}(t)\rangle=\left(\begin{array}[]{c}\cos\gamma_{0}(t)\cos\beta_{0}(t)\\\
-i\sin\gamma_{0}(t)\\\ -\cos\gamma_{0}(t)\sin\beta_{0}(t)\end{array}\right)$
(10)

Interestingly, the quantum state (10) can be mapped onto a real-valued vector
$\mathbf{S}_{0}(t)=\cos\gamma_{0}(t)\sin\beta_{0}(t)\hat{\mathbf{x}}-\sin\gamma_{0}(t)\hat{\mathbf{y}}+\cos\gamma_{0}(t)\cos\beta_{0}(t)\hat{\mathbf{z}}$
which behaves as an effective spin that obeys a precession equation:

$\frac{d\mathbf{S}_{0}}{dt}=\gamma\>\mathbf{B}_{0}\times\mathbf{S}_{0}.$ (11)

where the effective magnetic field
$\mathbf{B}_{0}(t)=\frac{1}{2}[\Omega_{p}^{0}(t)\hat{\mathbf{x}}+\Omega_{s}^{0}(t)\hat{\mathbf{z}}]$
is determined by the pump and Stokes fields.

In the absence of dissipation, we introduce a reference trajectory,
$\mathbf{S}_{0}(t)$, associated to a prescribed evolution of the angles
$\beta_{0}(t)$ and $\gamma_{0}(t)$ that fulfills the required boundary
conditions to ensure the transfer of the population from state 1 to state 3.
The pump and Stokes fields $\Omega_{p0}(t),\Omega_{s0}(t)$ are subsequently
inferred from the chosen trajectory (see Appendix A).

####  I.2.2 The STIRAP solution in the presence of dissipation

We now take into account dissipation. We assume that the intermediate level
$|2\rangle$ suffers a finite damping,modelled by the anti-Hermitian
Hamiltonian $\hat{H}_{\Gamma}=-i\hbar\Gamma_{2}|2\rangle\langle 2|$. The
effective spin now obeys the differential equation

$\frac{d\mathbf{S}}{dt}=\gamma\>\mathbf{B}\times\mathbf{S}-\overline{\overline{\Gamma}}\>\mathbf{S},$
(12)

where the dissipation tensor is
$\overline{\overline{\Gamma}}=\Gamma_{2}\hat{y}\hat{y}$. By superimposing to
the original field $\mathbf{B}_{0}(t)$ the following geometric correction

$\delta\mathbf{B}_{0}(t)=\gamma^{-1}\mathbf{S}_{0}(t)\times\overline{\overline{\Gamma}}\mathbf{S}_{0}(t),$
(13)

the effective spin $\mathbf{S}$ follows the same trajectory as its
dissipationless counterpart $\mathbf{S}_{0}$ despite the damping, or otherwise
stated the renormalized state
$|\widetilde{\psi}(t)\rangle=|\psi(t)\rangle/|||\psi(t)\rangle||$ coincides
with its dissipationless counterpart Impens19 . The corresponding pulse
corrections reads:

$\displaystyle\delta\Omega_{p}^{0}(t)$ $\displaystyle=$
$\displaystyle-{\Gamma_{2}}\>\sin 2\gamma_{0}(t)\>\cos\beta_{0}(t),$
$\displaystyle\delta\Omega_{s}^{0}(t)$ $\displaystyle=$
$\displaystyle{\Gamma_{2}}\>\sin 2\gamma_{0}(t)\>\sin\beta_{0}(t).$ (14)

The effective spin $\mathbf{S}$ evolves in the magnetic field
$\mathbf{B}(t)=\mathbf{B}_{0}(t)+\delta\mathbf{B}_{0}(t)$. Interestingly, this
approach restores the dissipation-free purity
$p=p_{|3\rangle}/(p_{|1\rangle}+p_{|2\rangle}+p_{|3\rangle})\simeq 99,8\%$ of
the final population in the target state. However, the quantum state norm
$\mathcal{N}(t)=|||\psi(t)\rangle||$ may suffer a significant damping. The
interest of the time-rescaling is also here to mitigate this latter effect.
For the STIRAP problem, one readily derives the rescaled pulse fields:
$\Omega_{p,s}^{0\>\Lambda}(t)=\dot{\Lambda}(t)\Omega_{p,s}^{0}(\Lambda(t))$
and $\delta\Omega_{p,s}^{0\>\Lambda}(t)=\delta\Omega_{p,s}^{0}(\Lambda(t))$.
With the considered $\mathbf{S}_{0}$ trajectory, the population in the damped
intermediate state $p(t)=|\langle
2|\psi(t)\rangle|^{2}=\sin^{2}(\gamma_{0}(t))$ reaches its maximum value at
the half time $t=T/2$.

####  I.2.3 Comparing different time-scaled STIRAP solutions

We propose hereafter two different time scalings that accelerate about this
half time to reduce the norm decrease. First, we consider a polynomial scaling
that fulfils this latter requirement $\Lambda_{1}(t)=T_{1}P(t/T_{1})$ with
$P(x)=3x^{2}-2x^{3}$. Alternatively, we will consider a quasi-unitary time-
scaling $\Lambda_{2}(t)$ (see Fig. 2b) associated to a uniform damping of the
quantum state norm in the sense of (7) and with respect to the geometric angle
$\beta_{0}$. The scaling $\Lambda_{2}(t)$ is obtained by solving a
differential equation analogous to (8):

$\dot{\Lambda}_{2}=\frac{\Gamma_{2}}{\dot{\beta}_{0}(\Lambda_{2})c^{\prime}}\sin^{2}\gamma_{0}(\Lambda_{2}).$
(15)

Figure 2 compares the performance of the three protocols: the pulse sequence
$\Omega_{p,s}(t)=\Omega_{p,s}^{0}(t)+\delta\Omega_{p,s}^{0}(t)$ and their
time-scaled versions based on $\Lambda_{1}(t)$ and $\Lambda_{2}(t)$.

For numerical applications, we use the angular trajectories detailed in
Appendix A parametrized with $\epsilon=0.05,\delta=\pi/4$ and for a damping
rate equal to $\Gamma_{2}T=0.1$. First, we consider time-scalings
$\Lambda_{1,2}(t)$ such that $\Lambda_{1,2}(T)=T$. This condition amounts to
setting $T_{1}=T$ and $c^{\prime}\simeq 4.94\times 10^{-3}$. One obtains the
respective quantum fidelities $\mathcal{F}_{0}=0.954,$
$\mathcal{F}_{\Lambda_{1},T}=0.966$ and $\mathcal{F}_{\Lambda_{2},T}=0.982$
for respectively the initial protocol, for the polynomial scaling
$\Lambda_{1}(t)$ and for the quasi-unitary scaling $\Lambda_{2}(t)$. For the
three protocols the final purity remains equal to the dissipation-free value
$p\simeq 99,8\%.$ The enhancement of the quantum fidelity results from a
reduction of the norm $\mathcal{N}(t)$ damping.

####  I.2.4 Energetic cost and optimization

Alternatively, one can choose the total duration $T_{1,2}$ of the time
scalings $\Lambda_{1,2}(t)$ as to yield a protocol with the same energy as the
original STIRAP protocol. The energy, taken as
$E_{\Lambda_{k}}=\int_{0}^{T_{k}}dt^{\prime}||\mathbf{B}_{\Lambda_{k}}(t^{\prime})||^{2}$,
is proportional to the integrated Stokes and pump field intensities and
inversely proportional to the total duration $T_{k}$. One finds the durations
$T_{1}\simeq 1.10T$ and $T_{2}\simeq 1.53T,$ giving the quantum fidelities
$\mathcal{F}_{\Lambda_{1},T_{1}}\simeq 0.963$ and
$\mathcal{F}_{\Lambda_{2},T_{2}}\simeq 0.974$. At constant energy, the quasi-
unitary time scaling $\Lambda_{2}(t)$ thus enables a reduction of the
discrepancy $\tilde{\epsilon}=1-\mathcal{F}$ with the perfect transfer in the
absence of dissipation by nearly $45\%$ with respect to the original STIRAP
protocol. This improvement at constant resources confirms the viability of the
time-scaling technique.

Figure 2: Application of the time-scaling to the STIRAP protocol: (a): The
scaling functions $\Lambda_{1}(t)/T$ (blue dashed line) and $\Lambda_{2}(t)/T$
(red solid line) as a function of the normalized time $t/T$ (with the choice
$\Lambda_{1,2}(T)=T$). The black dotted line represents the trivial time
scaling $\Lambda(t)=t$. (b): Rabi frequencies $\Omega_{p}$ (in units of
$T^{-1}$) for the pump field for the dissipation-corrected STIRAP protocol
(APPENDIX A,I.2.2) (black line), and its time-scaled versions for
$\Lambda_{1}(t)$ (blue dashed line) and $\Lambda_{2}(t)$ (red dot-dashed line)
as a function of the normalized time $t/T$. The original protocol is
associated to the angular trajectories (18) with the parameters
$\epsilon=0.05,\delta=\pi/4$, and the dissipation rate is equal to
$\Gamma_{2}=0.1/T$. (c) Time-scaled resource optimization of a STIRAP
transfer: Optimal time scaling $\Lambda(t)/T$ as a function of the normalized
time $t/T$ (dotted black line) and the corresponding pump $\Omega_{p}$ (dashed
blue line) and Stokes $\Omega_{s}$ (solid red line) Rabi frequencies
$\Omega_{\Lambda s,p}(t)=\dot{\Lambda}(t)\Omega^{0}_{s,p}(t)$ renormalized by
the constant value $\Omega_{0}=\sqrt{\Omega_{\Lambda
p}^{2}(t)+{\Omega_{\Lambda s}^{2}(t)}}$.

The previous formalism provides a strategy to determine a protocol that
minimizes the energy for the STIRAP transfer. First, we notice that even for
strong dissipation rates such that $\Gamma_{\perp}T=1$ and
$\Gamma_{/\\!/\\!}T=0.1,$ with the chosen angle $\epsilon=0.05$, the energy
overhead associated to the correction remains small $\delta E_{\rm
corr}/E\simeq 0.6\%.$ Regarding the optimization of the protocol through the
time scaling, we can thus ignore the contribution associated with the
correction and take
$E[\Lambda,\dot{\Lambda}]=\hbar\int_{0}^{T}dt\dot{\Lambda}(t)^{2}\left(\Omega_{p}^{0}(\Lambda(t))^{2}+\Omega_{s}^{0}(\Lambda(t))^{2}\right)$,
with the pump and Stokes fields (APPENDIX A). By minimizing this functional of
$\Lambda$ and $\dot{\Lambda}$, we get the differential equation that $\Lambda$
obeys
$\dot{\Lambda}(t)=c\left(\dot{\beta}_{0}(\Lambda(t))^{2}\cot^{2}\gamma_{0}(\Lambda(t))+\dot{\gamma}_{0}(\Lambda(t))^{2}\right)^{-1/2}$.
The constant $c$ is determined self-consistently by imposing the boundary
value $\Lambda(T)$. The solution of the differential equation obeyed by
$\Lambda$ imposes a constant norm for the effective field vector,
$\Omega_{\Lambda}=\sqrt{\Omega_{\Lambda p}^{2}(t)+{\Omega_{\Lambda
s}^{2}(t)}}=cte$. This optimal solution is reminiscent of the $\pi$-pulse
optimal solution for the 2-level problem. Figure 2(c) represents this optimal
time scaling $\Lambda(t)$ (for $\Lambda(T)=T$) as well as the pump and Stokes
pulses obtained for such an optimization. The time-scaling accelerates when
the effective magnetic field is minimal, as for instance at the initial and
final times. With these optimal pulses, one obtains an energy $E_{\rm
opt}\simeq 64.9\hbar/T,$ which is is roughly $10\%$ lower than the original
pulse $E_{\rm 0}=72.1\hbar/T$ for the same purity $p=99.8$ %.

##  II Time Scaling and Quantum Speed Limit for non-Hermitian Hamiltonians

The time optimality of a quantum state transfer is measured through the
concept of QSL. Quantum systems evolve at a fraction of the QSL. This fraction
constitutes a measure of the driving efficiency, and for an optimal driving it
reaches unity. One can readily show that this driving efficiency is invariant
under a time-scaling transform for a unitary evolution. Indeed, for closed
quantum system the time-scaling changes equally the time and energy scales
respectively related to the quantum speed and to the QSL. In contrast, the
dissipation is unaffected by the time-scaling transform and one would thus
expect this invariance to break down in dissipative systems. In the following,
we derive the expression of the QSL for dissipative systems modelled by a non-
Hermitian Hamiltonian. Interestingly, with appropriate corrective terms in the
quantum driving, the dissipation-free efficiency can be restored in a
dissipative 3-level system. In this example, the driving efficiency remains
invariant through time-scaling transforms even in the presence of dissipation.

###  II.1 Quantum Speed Limit for non-Hermitian Hamiltonian

The quantum speed limit (QSL) amounts to measuring the minimal time -
associated to the maximal evolution velocity - from a given initial state
$|\tilde{\psi}(0)\rangle$ to a state orthogonal $|\tilde{\psi}(t)\rangle$ to
the initial one (we denote
$|\tilde{\psi}(t)\rangle=|\psi(t)\rangle/|||\psi(t)\rangle||$.) For a system
evolving under the action of a time-independent Hamiltonian $\hat{H}_{0}$, it
translates as an upper bound on the rate of variation of the angle
$\cos\phi=|\langle\psi(0)|\psi(t)\rangle|$:

$\frac{d\phi}{dt}\leq\frac{\Delta\hat{H}_{0}}{\hbar}.$ (16)

In appendix B, we show how Vaidman’s derivation of the QSL in dissipationless
system Vaidmann92  can be readily adapted to non-Hermitian time-dependent
Hamiltonians $\hat{H}=\hat{H}_{0}-i\hat{\Gamma}$. The new bound reads

$\dot{\phi}=\frac{d\phi}{dt}\leq\frac{\sqrt{(\Delta\hat{H}_{0})^{2}+(\Delta\hat{\Gamma})^{2}-i\langle[\hat{H}_{0},\hat{\Gamma}]\rangle}}{\hbar}.$
(17)

As a consistency check, we have performed numerical simulations in 2-level
systems that confirm the validity of this upper bound.

###  II.2 Quantum Speed Limit in a 2-level system

Consider a 2-level quantum system where $|e\rangle$ denotes the excited state
and $|g\rangle$ the ground state. The QSL is saturated when
$\langle\widetilde{\psi}(t)|\widetilde{\psi}(0)\rangle\langle\widetilde{\psi}(0)|\dot{\widetilde{\psi}}(t)\rangle$
is a real quantity where $|\dot{\widetilde{\psi}}(t)\rangle$ denotes the time
derivative of the quantum state. For a generic parametrization of the state
$|\widetilde{\psi}(\theta,\varphi)\rangle=a_{t}|e\rangle+b_{t}|g\rangle,$ and
the initial state $|\psi(0)\rangle=|e\rangle$, the saturation of the QSL bound
is reached for $\dot{\varphi_{e}}=0$ with $a_{t}=|a_{t}|e^{i\varphi_{e}}$. We
define the control Hamiltonian as
$\hat{H}_{0}(t)=\frac{1}{2}\Omega_{0}\left(|e\rangle\langle
g|+|g\rangle\langle e|\right)$ with a constant Rabi frequency
$\Omega_{0}=\pi/T$. We take into account the dissipation for both states
thanks to the operator $\hat{\Gamma}=\Gamma_{1}|e\rangle\langle
e|+\Gamma_{2}|g\rangle\langle g|$. Under the Hamiltonian
$\hat{H}=\hat{H}_{0}-i\hat{\Gamma}$, the quantum state $|\psi_{t}\rangle$
varies as a function of time with a coefficient $a_{t}$ real (i.e.
$\varphi_{e}(t)=0$ at all times). As a result, the quantum speed is equal to
the QSL with $(\Delta H_{0})^{2}(t)=\frac{1}{4}\Omega_{0}^{2}(t)$,
$(\Delta\hat{\Gamma})^{2}(t)=\Gamma_{1}^{2}|a_{t}|^{2}+\Gamma_{2}^{2}|b_{t}|^{2}-(\Gamma_{1}|a_{t}|^{2}+\Gamma_{2}|b_{t}|^{2})^{2},$
and
$i\langle[\hat{H}_{0}(t),\hat{\Gamma}]\rangle=\frac{i}{4}(\Gamma_{1}-\Gamma_{2})\Omega_{0}(t)(a_{t}b_{t}^{*}-a_{t}^{*}b_{t}).$

To clarify how dissipation affects the quantum speed, we consider in the
following two opposite cases: $\Gamma_{e}>\Gamma_{g}$ and
$\Gamma_{e}<\Gamma_{g}$. In the first configuration, the faster decay of the
excited state contributes to flip down the Bloch vector. One thus expects a
quantum velocity faster than in the dissipation-free case. In the opposite
sitaution ($\Gamma_{e}<\Gamma_{g}$), the ground state is less stable and one
expects dissipation to slow down the quantum state transfer. Our expression
for the QSL (22) captures this physics through the commutator
$i\langle[\hat{H}_{0}(t),\hat{\Gamma}]\rangle:$ depending on the relative
strength of the excited/ground state dissipation rates, this contribution
increases or decreases the QSL.

As an example, with the dissipation rates $\Gamma_{e}T=0.2$ and
$\Gamma_{g}T=0.01$, the quantum state evolves faster than in the dissipation-
free system for $\Gamma_{e}>\Gamma_{g}$, and the $\pi$ pulse is achieved for
$T^{*}\simeq 0.964T$ while $T^{*}\simeq 1.0405T$ when the values of the
dissipation rates are exchanged. In both cases the damping seriously
deteriorates the quality of the transfer and the final quantum fidelity. In
these examples, the quantum speed reaches the QSL at all times. Such a
saturation of the QSL persists after a time-scaling transform. More generally,
we show below that the time-scaling transform can also preserve the ratio of
the quantum speed to the QSL in a dissipative 3-dimensional system.

###  II.3 Quantum Speed Limit in a dissipative STIRAP system

We now consider the dissipative 3-level system of Section I.2. We use the
pulse sequence
$\Omega_{p,s}(t)=\Omega_{p,s}^{0}(t)+\delta\Omega_{p,s}^{0}$corresponding to
the sum of the dissipation-free pulses $\Omega_{p,s}^{0}(t)$ (APPENDIX A) and
the associated dissipative corrections $\delta\Omega_{p,s}^{0}(t)$ (I.2.2).
Thanks to the pulse correction, the renormalized quantum state
$|\widetilde{\psi}(t)\rangle$ follows exactly the dissipation-free trajectory,
i.e. $|\widetilde{\psi}(t)\rangle=|\psi_{0}(t)\rangle$ at all times. Thus, the
angle
$\phi(t)=\arccos\left(|\langle\widetilde{\psi}(t)|\widetilde{\psi}(0)\rangle|\right)$
fulfills $\phi(t)=\phi_{0}(t)$ at all times, where
$\phi_{0}(t)=\arccos\left(|\langle\psi_{0}(t)|\psi(0)\rangle|\right)$ is the
angle associated to the dissipation-free trajectory. The effective quantum
speed $\dot{\phi}(t)$ is thus given by the dissipation-free trajectory.

In the corrected protocol, the non-Hermitian QSL $\chi(t)$ depends a priori on
the dissipation-free control Hamiltonian $\hat{H}_{0}$, the geometric
correction $\delta\hat{H}_{0}$ and the dissipation operator $\hat{\Gamma}$. We
find
$\chi(t)^{2}=(\Delta\hat{H}_{0})^{2}+\langle\\{\delta\hat{H}_{0},\hat{H}_{0}\\}\rangle+(\Delta\delta\hat{H}_{0})^{2}+(\Delta\hat{\Gamma})^{2}-i\langle[\hat{H}_{0},\hat{\Gamma}]\rangle-i\langle[\delta\hat{H}_{0},\hat{\Gamma}]\rangle.$
Remarkably, by using the explicit form of the geometric correction (I.2.2),
the dissipative QSL $\chi(t)$ boils down to $\chi(t)=\Delta\hat{H}_{0}(t)$.
Thanks to the geometric correction, the non-Hermitian QSL (25) is exactly
equal to the dissipation-free QSL (26) of the original protocol. The
preservation of the dissipation-free quantum speed $\dot{\phi}_{0}(t)$ and QSL
$\chi_{0}(t)$ despite dissipation comes at the price of an energy overhead
associated to the extra term added to the Hamiltonian, $\delta\hat{H}_{0}$.

As a corollary, when the time-scaling technique is applied, both the quantum
speed and the dissipative QSL undergo similar transformations as
$\dot{\phi}_{\Lambda}(t)=\dot{\Lambda}(t)\dot{\phi}_{0}(\Lambda(t))$ and
$\chi_{\Lambda}(t)=\dot{\Lambda}(t)\chi_{0}(\Lambda(t)).$ That is to say, the
time-scaling preserves the ratio $r_{0}(t)$ of the quantum speed to the QSL as
$r_{\Lambda}(t)=r_{0}(\Lambda(t))$ with
$r_{0}(t)=\hbar\dot{\phi}_{0}(t)/\dot{\chi}_{0}(t)$. Remarkably and thanks to
the geometric correction, the dissipative dynamics keeps the same quantum
speed and quantum speed limit as for the original dissipation-free protocol.

##  III Conclusion

In conclusion, we have demonstrated the applicability and relevance of time
scaling for quantum state transfer to optimize the resources and/or mitigate
the effect of dissipation in non-Hermitian quantum systems. Actually, the
scaling function provides an extra freedom in the system that can be used to
minimize the energy or the norm reduction. The quantum speed limit has been
here generalized to non-Hermitian Hamiltonians, and we have shown that time
scaling does not affect the speed limit and thus the optimality when the
system is driven at the quantum speed limit persists.

###### Acknowledgements.

This work was supported by the Agence Nationale de la Recherche research
funding Grant No. ANR-18-CE30-0013. F.I., F.M.D.A.., and F. A. P. thank CNPq,
CAPES, and FAPERJ for financial support.

## APPENDIX A

We consider a polynomial form for the angular functions
$\gamma(t)=\sum_{j=0}^{4}a_{j}t^{j}$ and $\beta(t)=\sum_{j=0}^{3}b_{j}t^{j}$.
They fulfill the boundary conditions of Protocol 2 of Ref. Chen12

$\displaystyle\gamma_{0}(0)=\epsilon,\,\dot{\gamma}_{0}(0)=0,\,\gamma_{0}(T/2)=\delta$
(18) $\displaystyle\gamma_{0}(T)=\epsilon,\,\dot{\gamma}_{0}(T)=0$
$\displaystyle\beta_{0}(0)=0,\,\beta_{0}(T)=\pi/2$
$\displaystyle\dot{\beta}_{0}(0)=0,\,\dot{\beta}_{0}(T)=0,\,\gamma_{0}(T/2)=\delta$

Such shortcut-to-adiabaticity solutions give rise to a trade-off between the
amplitudes of Rabi frequencies and the transient population of the
intermediate state $|2\rangle$ Chen12 ; FastSTIRAP21 .

The pump and Stokes fields $\Omega_{p0}(t),\Omega_{s0}(t)$ are subsequently
inferred from the chosen trajectory:

$\displaystyle\Omega_{p}^{0}(t)=2\left(\dot{\beta}_{0}(t)\frac{\sin\beta_{0}(t)}{\tan\gamma_{0}(t)}+\dot{\gamma}_{0}(t)\cos\beta_{0}(t)\right),$
$\displaystyle\Omega_{s}^{0}(t)=2\left(\dot{\beta}_{0}(t)\frac{\cos\beta_{0}(t)}{\tan\gamma_{0}(t)}-\dot{\gamma}_{0}(t)\sin\beta_{0}(t)\right).$
(19)

A small angle initial angle $\epsilon$ is used, which yields an error
$1-\mathcal{F}=O(\epsilon^{2})$ for the protocol defined by Eq. (APPENDIX
A,18) alone. For sake of simplicity, in our discussion on the quantum fidelity
and of the quantum speed limit, we consider this protocol as such. However, a
perfect transfer may be restored by adding an initial and a final stage to the
STIRAP protocol (18), namely by using an initial/final small pulse of angle
$\epsilon$ with the pump field $\Omega_{p}(t)$/Stokes field $\Omega_{s}(t)$
used separately. The full protocol would then corresponds to a sequence
$|1\rangle\rightarrow\left(\cos\epsilon|1\rangle-i\sin\epsilon|2\rangle\right)\rightarrow\left(-i\sin\epsilon|2\rangle+\cos\epsilon|3\rangle\right)\rightarrow|3\rangle$.

## APPENDIX B: QUANTUM SPEED LIMIT FOR NON-HERMITIAN HAMILTONIAN

As a starting point, we remind that for any Hermitian operator $\hat{A}$ and
any quantum state $|\psi\rangle$ Vaidmann92

$A|\psi\rangle=\langle A\rangle|\psi\rangle+\Delta A|\psi_{\perp}\rangle$ (20)

where $|\psi_{\perp}\rangle$ is orthogonal to $|\psi\rangle$ and
$\Delta\hat{A}$ the variance of the operator $\hat{A}$.

Interestingly, this relation can be generalized to non-Hermitian Hamiltonian
$\hat{H}=\hat{H}_{0}-i\hat{\Gamma}$ (with the Hermitian operators
$\hat{H}_{0}^{\dagger}=\hat{H}_{0}$ and $\hat{\Gamma}^{\dagger}=\hat{\Gamma}$)
on a given normalized quantum state $|\psi\rangle$. With the same notations as
previously, and for any quantum state $|\psi\rangle$, we write

$\chi|\psi_{\perp}\rangle=\hat{H}|\psi\rangle-\langle\hat{H}\rangle|\psi\rangle$
(21)

where $\chi$ is a positive real scalar (see below). To get an explicit
expression for the coefficient $\chi$, we write
$\langle\hat{H}^{\dagger}\hat{H}\rangle=|\langle\hat{H}\rangle|^{2}+\chi^{2}$
and use
$\hat{H}^{\dagger}\hat{H}=\hat{H}_{0}^{2}+\hat{\Gamma}^{2}-i[\hat{H}_{0},\hat{\Gamma}].$
As a result, we find

$\chi=\left((\Delta\hat{H}_{0})^{2}+(\Delta\hat{\Gamma})^{2}-i\langle[\hat{H}_{0},\hat{\Gamma}]\rangle\right)^{1/2}\,.$
(22)

The anti-hermiticity of the commutator $[\hat{H}_{0},\hat{\Gamma}]$ guarantees
that the quantity $i\langle[\hat{H}_{0},\hat{\Gamma}]\rangle$ is real-valued.

In closed quantum systems, the usual definition of the quantum velocity rests
on the fidelity with respect to the initial state
$\mathcal{F}(t)=|\langle\psi(t)|\psi(0)\rangle|^{2}$ – the quantum velocity is
inversely proportional to the time for which this fidelity goes to zero. By
using the decomposition (21) in the Schrödinger equation, one obtains the time
derivative of the fidelity for non-unitary dynamics

$\displaystyle\dot{\mathcal{F}}(t)$ $\displaystyle=$
$\displaystyle-2\langle\hat{\Gamma}\rangle(t)|\langle\psi(t)|\psi(0)\rangle|^{2}$
$\displaystyle-\frac{2\chi(t)}{\hbar}\mbox{Re}\left[i\langle\psi(t)|\psi(0)\rangle\langle\psi(0)|\psi_{\perp}(t)\rangle\right]$
$\displaystyle=\dot{\mathcal{F}}_{r}+\dot{\mathcal{F}}_{\theta}$

The right hand side has two contributions with distinct physical
interpretations. The first component
$\dot{\mathcal{F}}_{r}=-2\langle\hat{\Gamma}\rangle(t)|\langle\psi(t)|\psi(0)\rangle|^{2}$
corresponds to a pure quantum state damping. In contrast, the contribution
$\dot{\mathcal{F}}_{\theta}$ accounts for a genuine rotation of the quantum
state. $\dot{\mathcal{F}}_{\theta}$ is thus the only relevant contribution to
the quantum velocity.

We now propose a definition of the quantum velocity unaffected by the trivial
quantum state damping. For this purpose, we introduce the renormalized quantum
state
$|\widetilde{\psi}(t)\rangle=|\psi(t)\rangle/\sqrt{\langle\psi(t)|\psi(t)\rangle}$,
and consider the corresponding quantum fidelity
$\widetilde{\mathcal{F}}(t)=|\langle\widetilde{\psi}(t)|\widetilde{\psi}(0)\rangle|^{2}$.
By construction, only the relevant angular velocity
$\dot{\mathcal{F}}_{\theta}$ contributes to the variation of this quantum
fidelity, i.e.
$\dot{\widetilde{\mathcal{F}}}(t)=\dot{\widetilde{\mathcal{F}}}_{\theta}$.

To determine an upper bound on $\dot{\mathcal{F}}_{\theta}$, we we apply the
concepts introduced in Ref. Vaidmann92 . The initial state can always be
expanded over at most three orthogonal states as
$|\psi(0)\rangle=\langle\widetilde{\psi}(t)|\widetilde{\psi}(0)\rangle|\widetilde{\psi}(t)\rangle+\langle\widetilde{\psi}_{\perp}(t)|\widetilde{\psi}(0)\rangle|\widetilde{\psi}_{\perp}(t)\rangle+\alpha|\widetilde{\psi}_{\perp\perp}(t)\rangle.$
This guarantees that
$|\langle\widetilde{\psi}(0)|\widetilde{\psi}_{\perp}(t)\rangle|\leq\sqrt{1-|\langle\widetilde{\psi}(0)|\widetilde{\psi}(t)\rangle|^{2}}$.
As a result,

$|\dot{\widetilde{\mathcal{F}}}|=|\dot{\widetilde{\mathcal{F}}}_{\theta}|\leq\frac{2\chi(t)}{\hbar}|\langle\widetilde{\psi}(0)|\widetilde{\psi}(t)\rangle|\sqrt{1-|\langle\widetilde{\psi}(0)|\widetilde{\psi}(t)\rangle|^{2}}$

By introducing the usual definition
$\cos\phi=\widetilde{\mathcal{F}}^{1/2}=|\langle\widetilde{\psi}(t)|\widetilde{\psi}(0)\rangle|$,
we obtain the following upper bound for the quantum velocity

$\dot{\phi}(t)\leq\frac{\sqrt{(\Delta\hat{H}_{0})^{2}+(\Delta\hat{\Gamma})^{2}-i\langle[\hat{H}_{0},\hat{\Gamma}]\rangle}}{\hbar}.$
(25)

Other derivations of the QSL for dissipative systems, based on a matrix
density formalism, can be found in Refs. QSLOpen13a ; QSLOpen13b . Our non-
unitary QSL has a similar form as the QSL obtained for closed quantum systems
Vaidmann92

$\dot{\phi}(t)\leq\frac{\Delta\hat{H}_{0}}{\hbar}$ (26)

up to a replacement of the energy variance $\Delta\hat{H}_{0}$ by the quantity
$\chi(t)$ (22). By the Ehrenfest’s theorem, the variance $\Delta\hat{H}_{0}$
is time-independent for a unitary evolution in a constant Hamiltonian, leading
to a constant QSL in this context. Nevertheless, for the time-dependent and
non-Hermitian Hamiltonians considered here, the QSL $\chi(t)$ will generally
vary with time.

Our expression highlights the role of the dissipation operator in the
evolution of the quantum state. By construction the quantity
$\chi^{2}=(\Delta\hat{H}_{0})^{2}+(\Delta\hat{\Gamma})^{2}-i\langle[\hat{H}_{0},\hat{\Gamma}]\rangle$
is real-valued and positive, and is indeed bounded below by
$\chi(t)^{2}\geq(\Delta H_{0}-\Delta\Gamma)^{2}\geq 0$. This inequality shows
that a strictly positive quantum speed limit $\chi(t)>0$ exists for an
eigenstate of the Hermitian Hamiltonian ($\Delta\hat{H}_{0}=0$) as long as
dissipation has a strictly positive variance $\Delta\hat{\Gamma}>0.$

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