# Quantum Optimal Control:
Practical Aspects and Diverse Methods

T S Mahesh<EMAIL_ADDRESS>Priya Batra
<EMAIL_ADDRESS>M. Harshanth Ram
<EMAIL_ADDRESS>Department of Physics and NMR Research
Center,
Indian Institute of Science Education and Research, Pune 411008, India

###### Abstract

Quantum controls realize the unitary or nonunitary operations employed in
quantum computers, quantum simulators, quantum communications, and other
quantum information devices. They implement the desired quantum dynamics with
the help of electric, magnetic, or electromagnetic control fields. Quantum
optimal control (QOC) deals with designing an optimal control field modulation
that most precisely implements a desired quantum operation with minimum energy
consumption and maximum robustness against hardware imperfections as well as
external noise. Over the last two decades, numerous QOC methods have been
proposed. They include asymptotic methods, direct search, gradient methods,
variational methods, machine learning methods, etc. In this review, we shall
introduce the basic ideas of QOC, discuss practical challenges, and then take
an overview of the diverse QOC methods.

quantum optimal control, quantum gate, state preparation, quantum dynamics

##  I Introduction

The quest for control over quantum dynamics has a long history of several
decades (eg. Peirce  _et al._ (1988); Kosloff  _et al._ (1989); Zare (1998);
Rabitz  _et al._ (2000)). Although quantum control methods have been employed
in several fields from chemical kinetics to imaging, here we focus on the
methods employed for quantum information related areas. Thanks to the ongoing
revolution of quantum technologies, quantum control methods have received a
major impetus in recent years Dowling and Milburn (2003).

In this review, we mostly use examples from nuclear magnetic resonance (NMR),
although the basic principles are applicable in other architectures as well.
The early methods of quantum control for NMR quantum information tasks
involved bandwidth selective fields, called shaped pulses Cavanagh  _et al._
(1996). Several quantum information tasks were realized using such band-width
selective pulses Dorai  _et al._ (2000). Although the shaped pulses are
convenient to realize, they suffered from long pulse-durations as well as
undesired evolutions. The need for precise control over quantum dynamics led
to the development of advanced control methods based on optimal control
theory. Classical optimal control theory uses Pontryagin’s maximum principle
to design the best controls for a given task within certain constraints Kopp
(1962); Pontryagin (1987); Kirk (2004); Boscain  _et al._ (2021a, b).
Extending the control problem to the framework of quantum theory yields
quantum optimal control (QOC) theory Werschnik and Gross (2007); Cong (2014);
Glaser  _et al._ (2015); d’Alessandro (2021).

This review is an attempt to provide a broad overview of various methods,
without getting into finer details. While the review is surely non exhaustive,
we intend to capture several strategies and convey their gist, which may
benefit a reader novice in the field. The review is arranged as follows. The
next section outlines the quantum control problem formulated for quantum
processors. Section 3 discusses various practical aspects which must be taken
into consideration while designing the quantum control. Section 4 contains the
review of a diverse collection of QOC methods. Finally, section 5 provides a
summary and outlook.

##  II Quantum Control Problem

###  II.1 The system, control fields, and environment

We consider an overall Hamiltonian of the form

$\displaystyle H(t)=H^{S}+H^{\Omega}(t)+H^{E}(t).$ (1)

Here, $H^{S}$ is the static system Hamiltonian. In quantum information
processing (QIP), $H^{S}$ includes both self Hamiltonians of various qubits as
well as their mutual interactions. The time-dependent control Hamiltonian
$H^{\Omega}(t)$ is used to realize the desired dynamics. The remaining
undesirable interactions are collected in the effective environmental
Hamiltonian $H^{E}(t)$, which may not be completely known. We ignore this term
unless we want to see its effects or design control sequences that minimize
them. The instantaneous state $\rho(t)$ of the processor satisfies the
Liouville–von Neumann equation

$\displaystyle\dot{\rho}(t)=-i\left[{H(t)},{\rho(t)}\right],$ (2)

whose solution is of the form

$\displaystyle\rho(t)$
$\displaystyle=U(0,t)\rho(0)U^{\dagger}(0,t),~{}~{}\mbox{with unitary
operator}$ $\displaystyle U(0,t)$
$\displaystyle=D\exp\left[-i\int_{t^{\prime}=0}^{t}dt^{\prime}H(t^{\prime})\right].$
(3)

Here $D$ is the Dyson’s time-ordering operator. Throughout the review, we have
set $\hbar=1$.

###  II.2 Control Sequence via Discretized Control Field

Table 1: Examples of control fields used in various quantum processor architectures.  Quantum Processor |  System interactions |  Control field(s) |  Additional control(s) |  Eg. references
---|---|---|---|---
Trapped atoms/ions |  Electric or Magnetic Dipole interactions / Coulomb interaction between ions |  Optical / Microwave |  Electric / Magnetic fields |  Schäfer  _et al._ (2020); Häffner  _et al._ (2008)
Quantum dots |  Zeeman interaction, spin-spin interactions between electrons in adjacent dots |  Microwave |  Electric / Magnetic fields |  Veldhorst  _et al._ (2015); Watson  _et al._ (2018)
Super-conducting circuits |  Qubit capacitance, qubit-resonator coupling capacitance, qubit-qubit coupling capacitance |  Microwave |  Magnetic field |  Krantz  _et al._ (2019)
NV centers in diamond |  Zero-field splitting, hyperfine interaction, nuclear quadrupole interaction |  Microwave for electronic spin and RF for nuclear spin |  Laser, Magnetic field, Electric field |  Bucher  _et al._ (2019)
NMR |  Zeeman interaction, spin-spin interaction |  RF |  Magnetic field gradients |  Fortunato  _et al._ (2002); Khaneja  _et al._ (2005); Levitt (2013)

The control field may be an electric field (eg. Bayer  _et al._ (2001)), or
magnetic field (eg. Tiwari and Poonia (2021)), or electromagnetic field (eg.
Jelezko  _et al._ (2004)), or a combination of these (eg. Pfender  _et al._
(2017)). Table 1 lists control fields in various QIP architectures. To bypass
the complicated integral in Eq. 3 and facilitate the numerical evaluation of
the propagator, we shall discretize the entire control field $\Omega(t)$ of
duration $T$ into $N$ piecewise constant segments
$\\{\Omega_{1},\Omega_{2},\cdots,\Omega_{n},\cdots,\Omega_{N}\\}$,
collectively termed as the control sequence $\Omega$. Discretization by itself
need not be an approximation, since the modern digital signal generators can
produce the discretized control sequence directly (more on this in Sec.
III.1). In general, each control segment $\Omega_{n}$ is a multi-component
array with $M$ distinct channels each with one or more distinct controls (see
Fig. 1). For example, a typical NMR spectrometer is equipped with two to five
radio-frequency (RF) channels that allow simultaneous control of different
nuclear spin isotopes Levitt (2013). In the case of NV centers, one channel
controls the electronic spin and the remaining channels control nuclear spin
isotopes 15N and 13C Doherty  _et al._ (2013). Typically, each channel has
several control parameters such as amplitude, frequency, and initial phase.
Collectively, we shall represent the controls of segment $n\in[1,N]$ in
channel $m\in[1,M]$ as a control vector $\vec{\omega}_{mn}$. Amplitude is the
most common control parameter since it is easier to execute in theory as well
as in the experiment. In the following, for the case of a single control
parameter, we shall drop the vector symbol and denote it by $\omega_{mn}$ (see
Fig. 1). In general, the control Hamiltonian for the $n$th segment is of the
form

$\displaystyle H^{\Omega_{n}}=\sum_{m=1}^{M}{\cal H}_{m}(\vec{\omega}_{mn}),$
(4)

with the $m$th channel control Hamiltonian ${\cal H}_{m}(\vec{\omega}_{mn})$.

Figure 1: An example structure of the control sequence.

The discretized Hamiltonian for the $n$th segment of duration $\tau_{n}$ and
the corresponding segment propagator are

$\displaystyle H_{n}=H^{S}+H^{\Omega_{n}},~{}~{}U_{n}$
$\displaystyle=e^{-iH_{n}\tau_{n}}.$ (5)

Thus the overall propagator is simply the ordered product

$\displaystyle U(0,T)=U_{1:N}$ $\displaystyle=U_{N}U_{N-1}\cdots U_{n}\cdots
U_{2}U_{1}.$ (6)

If segments are all of the equal duration, $\tau_{n}=\tau=T/N$.

The control need not be entirely unitary. For example, we can insert
projective measurements or controlled dephasing (twirl operations Anwar  _et
al._ (2005); Bhole  _et al._ (2016)) in between unitary segments, and realize
a net nonunitary propagator. If $\rho_{n-1}$ represents the state after $n-1$
segments,

$\displaystyle\rho_{n}=\hat{{\cal
E}}_{n}\left[U_{n}\rho_{n-1}U_{n}^{\dagger}\right],$ (7)

where $\hat{{\cal E}}_{n}$ is the superoperator implementing the $n$th non-
unitary operation. We now discuss two types of quantum control problems.

###  II.3 Quantum Gate Synthesis Problem

Given a target unitary operator $U_{F}$ corresponding to a quantum gate, the
goal here is to find a control sequence $\Omega$ that maximizes the gate
fidelity Nielsen and Chuang (2002)

$\displaystyle
F_{G}(U_{F},U_{1:N})=\left|\frac{\langle{U_{F}}|{U_{1:N}}\rangle}{\langle{U_{F}}|{U_{F}}\rangle}\right|^{2}.$
(8)

Here and throughout the rest of the article we have used

$\displaystyle\langle{A}|{B}\rangle=\mathrm{Tr}\left(A^{\dagger}B\right)$ (9)

to denote the overlap between the operators $A$ and $B$. The control sequence
so obtained will be applicable independent of the initial state.

###  II.4 State to State Transfer or State Preparation Problem

Here the control sequence $\Omega$ needs to transfer a specific initial state
$\rho_{I}$ to a specific final state $\rho_{F}$. If the two states are of the
same purity, we seek a unitary sequence $U_{1:N}$ that prepares the state

$\displaystyle\rho_{N}$ $\displaystyle=U_{1:N}\rho_{I}U_{1:N}^{\dagger}$ (10)

as close to $\rho_{F}$ as possible. Examples include adiabatic
inversion/excitation Cavanagh  _et al._ (1996); Tannús and Garwood (1997),
preparing singlet-triplet order Manu and Kumar (2012); Khurana and Mahesh
(2017a), etc.

Often, the initial and the final desired states need not have the same purity.
For instance, preparing pseudopure state in NMR from thermal state Cory  _et
al._ (1997); Suter and Mahesh (2008); Bhole  _et al._ (2016). In such a case
we seek a nonunitary sequence as described in Eq. 7.

In state preparation, the control sequence $\Omega$ is obtained by maximizing
one of the following measures.

  * (i)

Uhlmann state-fidelity (eg. Wu and Rabitz (2012)):

$\displaystyle
F_{S}(\rho_{F},\rho_{N})=\left(\mathrm{Tr}\sqrt{\sqrt{\rho_{F}}\rho_{N}\sqrt{\rho_{F}}}\right)^{2}.$
(11)

  * (ii)

Trace fidelity (eg. Khaneja  _et al._ (2005)):

$\displaystyle F_{T}(\rho_{F},\rho_{N})=\langle{\rho_{F}}|{\rho_{N}}\rangle.$
(12)

  * (iii)

For traceless parts of the density matrices, it is particularly convenient to
use the following measures Fortunato  _et al._ (2002):

    * –

Correlation:

$\displaystyle F_{C}(\rho_{F},\rho_{N})$
$\displaystyle=\frac{\langle{\rho_{F}}|{\rho_{N}}\rangle}{\sqrt{\mathrm{Tr}(\rho_{F}^{2})}\cdot\sqrt{\mathrm{Tr}(\rho_{N}^{2})}}.$

    * –

Attenuated correlation:

$\displaystyle F_{A}(\rho_{I},\rho_{F},\rho_{N})$
$\displaystyle=\frac{\langle{\rho_{F}}|{\rho_{N}}\rangle}{\sqrt{\mathrm{Tr}(\rho_{I}^{2})}\cdot\sqrt{\mathrm{Tr}(\rho_{N}^{2})}},$
(14)

which is also sensitive to the change in the purity of $\rho_{F}$ w.r.t. the
initial state $\rho_{I}$, in addition to the overlap between $\rho_{N}$ and
$\rho_{F}$.

While it is relatively easier to generate a state-specific control sequence
compared to the the general quantum gate discussed in Sec. II.3, the state to
state control sequence generated for a particular pair of initial and final
states is not applicable for other states.

##  III Practical Aspects

###  III.1 Control field limitations

####  III.1.1 Distribution of a control parameter

Suppose a control segment with a nominal amplitude value $\omega_{mn}$
actually has a distribution over the range
$[\omega_{mn}^{\mathrm{min}},\omega_{mn}^{\mathrm{max}}]$. We may discretize
the distribution into $L$ bins, and assign probability $p_{l}$ for the $l$th
bin (see Fig. 2).

Figure 2: (a) Illustrating ideal (top) and typical (bottom) spatial
distribution of RF amplitude about a nominal value of $\omega_{mn}$. Middle
row: the Bloch-sphere trajectories on applying Hadamard gate
$H=R_{x}(\pi)R_{y}(\pi/2)$ on $|{0}\rangle$ with 10% weaker (left), nominal
(center), and 10% stronger (right) field-components. Note that, only for the
nominal component, the qubit reaches the desired state
$(|{0}\rangle+|{1}\rangle)/\sqrt{2}$. (b) The fidelity profiles of simple and
robust control sequences.

It is important to have a quantitative understanding of the distribution. For
example, Pravia et al. Pravia  _et al._ (2003) described the experimental
estimation of RF distribution via Torrey oscillation. A microscopic system,
like an NV center, may sample one particular bin of the distribution, while a
macroscopic object, such as an NMR spin ensemble, experiences the entire
distribution Cavanagh  _et al._ (1996). To realize a control sequence
${\Omega}$ that is robust against the distribution, we optimize the weighted
average by defining the performance function as Fortunato  _et al._ (2002);
Khaneja  _et al._ (2005),

$\displaystyle\Phi(\Omega)=\bar{F}=\sum_{l=1}^{L}p_{l}F_{l},$ (15)

where $F_{l}$ is the fidelity of the $l$th bin (see Fig. 2(b)).

####  III.1.2 Power restriction

The hardware limitation imposes a maximum nominal value of
$\omega_{m}^{\mathrm{MAX}}$ for each control parameter $\omega_{mn}$. To
restrict the control sequence ${\Omega}$ within the allowed range, we maximize
the improved performance function

$\displaystyle\Phi(\Omega)=\bar{F}-{\cal P}(\\{\omega_{m}^{\mathrm{MAX}}\\}),$
(16)

where the penalty function ${\cal P}$ is normally zero, but rapidly grows as
any control parameter reaches close to or exceeds the limits Fortunato  _et
al._ (2002); Fletcher (1983).

####  III.1.3 Frequency bandwidth and nonlinear response

Electronic circuits producing a control field are designed for a certain
maximum frequency bandwidth, beyond which the signal profiles are distorted.
This limits how fast the control parameters can be changed from one segment to
another. To this end, we may seek a control sequence in terms of low-bandwidth
and smoothly varying basis functions, such as Slepian functions Lucarelli
(2018); Feng  _et al._ (2018). A related issue is the finite raise-time &
fall-time, as well as other nonlinearities in a practical pulse. Boulant et
al. Boulant  _et al._ (2003) used an interesting approach that involved
reading the actual nonlinear pulse profile as seen by a spy detector coil.
This enabled them to model the nonlinear transfer function, find its inverse,
and iteratively obtain a compensated pulse.

###  III.2 Noise and decoherence

As mentioned in Eq. 1, a quantum system being controlled is often prone to
certain unavoidable environmental fields or interactions, leading to
decoherence. The challenge here is to design a protected quantum control, that
is robust against these external influences. One often addresses this problem
with a time-optimal control sequence (eg. Möttönen  _et al._ (2006); Zhang
_et al._ (2011); Tibbetts  _et al._ (2012)), that performs the desired task
before noise becomes significant. The other powerful approach is by
interlacing dynamical decoupling pulses with the control sequence to realize
protected quantum gates Xu  _et al._ (2012); Zhang  _et al._ (2014); Viola
_et al._ (1999); Ram  _et al._ (2021). If additional qubits are available, one
can also encode information onto logical qubits in a decoherence free subspace
(DFS) Lidar  _et al._ (1998). An extensive review for controlling the open
quantum system has been covered by Koch Koch (2016).

###  III.3 Computational resource for optimization

The computational resource needed for generating a control sequence is an
important consideration in choosing the optimization method for a given
control problem. For systems with large Hilbert-space dimensions, the
bottleneck is in the matrix exponentiation in Eq. 5. Some ideas are discussed
below.

####  III.3.1 Limiting off-diagonal operators

Certain quantum control tasks, especially nonlocal quantum gates, require long
delays (evolution under system Hamiltonian $H^{S}$) in between control pulses.
The delay propagator $U_{d}(\tau)$ can be efficiently evaluated in the
eigenbasis $\\{|{s}\rangle\\}$ of the system Hamiltonian $H_{S}$, i.e.,

$\displaystyle
U_{d}(\tau)=e^{-iH^{S}\tau}=\sum_{s}e^{-is\tau}|{s}\rangle\langle{s}|.$ (17)

Therefore, a due emphasis on delays in such control sequences greatly improves
the efficiency of optimization Mahesh and Suter (2006).

####  III.3.2 Diagonalizing Control Hamiltonians

In certain scenarios, it is possible to find a clever choice of basis
operators that helps avoiding iterative matrix exponentiation Bhole and Mahesh
(2017); Bhole and Jones (2018). The trick is to use the Trotter decomposition
and rewrite Eq. 5 as

$\displaystyle U_{n}=e^{-i(H^{S}+H^{\Omega_{n}})\tau}$ $\displaystyle\approx
e^{-iH^{S}\tau/2}e^{-iH^{\Omega_{n}}\tau}e^{-iH^{S}\tau/2}$
$\displaystyle~{}~{}=U_{d}(\tau/2)~{}e^{-iH^{\Omega_{n}}\tau}~{}U_{d}(\tau/2).$
(18)

The nontrivial central exponential is then diagonalized using a pair of fixed
unitary operators $W_{1}$ and $W_{2}$ so that

$\displaystyle e^{-iH^{\Omega_{n}}\tau}$
$\displaystyle=W_{1}\left(\sum_{d}e^{-id\tau_{n}}|{d}\rangle\langle{d}|\right)W_{2},$
(19)

using which $U_{n}$ can be efficiently computed Bhole and Jones (2018).

####  III.3.3 Use of fixed controls or bang-bang controls

While designing quantum controls for large quantum systems, often it might be
sufficient to use only the maximum power pulses, also known as the bang-bang
controls, separated by delays. In this case, the matrix exponentiation in Eq.
5 reduces to

$\displaystyle U_{n}=\begin{cases}&U_{d}(\tau_{n})~{}~{}\mbox{if delay
segment, else}\\\
&\exp\left[-i\tau\left(H_{S}+\sum_{m=1}^{M}\omega_{m}^{\mathrm{max}}H_{m}\right)\right].\end{cases}$

Both cases are effectively one-time calculations and accordingly, the bang-
bang controls can be generated efficiently even for larger quantum systems
Bhole  _et al._ (2016); Khurana and Mahesh (2017a).

##  IV Diverse Methods

We now discuss several of the diverse methods which have been employed for QOC
(see Fig. 3).

Figure 3: An overview of the diverse QOC methods discussed in this review.

###  IV.1 Asymptotic Evolution Methods

The following methods assume slow variation of some parameter, and approach
exact solutions in the asymptotic limit.

####  IV.1.1 Adiabatic pulse method

Adiabatic pulses are considered a method of choice for state transfers in the
presence of highly inhomogeneous fields Cavanagh  _et al._ (1996); Tannús and
Garwood (1997); Garwood and DelaBarre (2001). Consider a state transfer
problem from the ground state $|{\psi_{I}}\rangle$ of an initial Hamiltonian
${\cal H}_{I}$ to the ground state $|{\psi_{F}}\rangle$ of the final
Hamiltonian ${\cal H}_{F}$. The adiabatic method involves implementing a
control Hamiltonian of the form

$\displaystyle H(t)=(1-\omega(t)){\cal H}_{I}+\omega(t){\cal H}_{F}.$ (20)

The scalar parameter $\omega(t)$ is slowly (adiabatically) varied from
$\omega(0)=0$ to $\omega(T)=1$, such that the system remains in an eigenstate
of the instantaneous Hamiltonian $H(t)$ throughout Messiah (2014). As long as
there is no level-crossing, this method leads to a robust state transfer from
$|{\psi_{I}}\rangle$ to $|{\psi_{F}}\rangle$, since the field imperfections
can only affect the intermediate trajectory, but not the final state.

Figure 4: Different stages of adiabatic inversion.

Broadband inversion forms a simple, yet an intuitive example of the adiabatic
method (see Fig. 4). Imagine inverting a set of spin qubits of Larmor
frequencies $\delta_{i}$ from $|{00\cdots 0}\rangle$ to $|{11\cdots 1}\rangle$
using an RF field of amplitude $\omega$, frequency $\nu_{\mathrm{RF}}$, and
phase $\hat{x}$. The effective Hamiltonian in the rotating frame of the RF is
of the form

$\displaystyle
H_{\mathrm{eff}}(t)=\sum_{i}\Delta_{i}(t)\sigma_{z}/2+\omega\sigma_{x}/2,~{}~{}\mbox{with}~{}~{}\Delta_{i}(t)=2\pi\left[\delta_{i}-\nu_{\mathrm{RF}}(t)\right].$
(21)

This corresponds to a precession of the spin qubits about an effective field
$\vec{B}_{i,\mathrm{eff}}\propto\Delta_{i}(t)\cos\theta_{i}(t)\hat{z}+\omega\sin\theta_{i}(t)\hat{x}$,
where $\theta_{i}(t)=\tan^{-1}\left(\omega/\Delta_{i}(t)\right)$. Initially,
one starts with a highly off-resonant pulse, i.e.,
$|\nu_{\mathrm{RF}}(0)|\gg\delta_{i}$ and $|\Delta_{i}(0)|\gg\omega$, so that
$\theta_{i}(0)\sim 0$. Thus, the initial state $|{00\cdots 0}\rangle$ is an
eigenstate of $H_{\mathrm{eff}}(0)$, and the qubits are locked to the
effective field. Gradually the RF frequency is swept to on-resonance
($\theta=\pm\pi/2$) and slowly taken off-resonant far on the other side. This
way, all the qubits continue to get locked to the effective field and finally
reach the state $\theta=\pi$, which corresponds to the state $|{11\cdots
1}\rangle$. In the adiabatic limit, the inversion happens to all the spin
qubits with different Larmor frequencies, even in the presence of
inhomogeneous amplitude $\omega$.

Adiabatic methods are routinely being applied in several fields including
spectroscopy and imaging Tannús and Garwood (1997); Garwood and DelaBarre
(2001). Adiabatic quantum algorithms, wherein the solution of a task is
modeled as the ground state of a Hamiltonian, are also popular Albash and
Lidar (2018).

####  IV.1.2 Geometric phase method

If the state of a quantum system undergoes a cyclic evolution and returns to
the original state Berry (1988); Anandan (1992), then apart from a dynamical
phase, it acquires a geometric phase, that is purely dependent on the
geometric aspects of the path followed. The dynamical phase can be effectively
removed using a spin-echo type of time-reversal sequence leaving behind a pure
geometric phase Jones  _et al._ (2000). In the adiabatic limit, the geometric
phase is called the Pancharatnam-Berry phase Pancharatnam (1956); Berry
(1987). The geometric phase has long been observed and used in NMR Suter  _et
al._ (1987). Ekert and coworkers Ekert  _et al._ (2000) had proposed geometric
phase quantum gates and recognized them to be robust against certain types of
control errors. Universal quantum computing by non-abelian geometric phase
gates, or holonomic gates, was proposed by Zanardi and Rasetti Zanardi and
Rasetti (1999). Subsequently, there have been several experimental
demonstrations of geometric quantum gates, including in NMR Jones  _et al._
(2000), ion qubits Leibfried  _et al._ (2003), ultracold neutrons Filipp  _et
al._ (2009), superconducting qubits Berger  _et al._ (2013), as well as NV
centers Nagata  _et al._ (2018).

####  IV.1.3 Lyapunov method

Lyapunov method was originally used for stabilizing a dynamical system into an
asymptotic stable state Wikipedia contributors (2021); Isidori (1995) and is
particularly useful for quantum state transfer Grivopoulos and Bamieh (2003).
To explain the method, we shall consider a transfer from an arbitrary initial
state $|{\psi_{I}}\rangle$ to an eigenstate $|{\psi_{F}}\rangle$ of the system
Hamiltonian $H^{S}$ Hou  _et al._ (2012); Wang  _et al._ (2014). We model the
distance between the instantaneous state and the target state in terms of a
positive function, called Lyapunov function $V$, which is a continuously
differentiable and decreasing function, i.e., $\dot{V}\leq 0$. There can be
multiple ways to set up the Lyapunov function, one of which is

$\displaystyle
V(t)=\langle{P}|{\rho(t)}\rangle,~{}~{}\mbox{with}~{}~{}P=I-|{\psi_{F}}\rangle\langle{\psi_{F}}|,$
(22)

and $\rho(t)=|{\psi(t)}\rangle\langle{\psi(t)}|$ being the instantaneous
state. As the system reaches the target state $V(t)\rightarrow 0$. Now
invoking the Lyapunov constraint,

$\displaystyle\dot{V}(t)$
$\displaystyle=\langle{P}|{\dot{\rho}(t)}\rangle=-i\sum_{m}\omega_{m}(t)\langle{P}|{\left[H^{S}+{\cal
H}_{m},\rho(t)\right]}\rangle\leq 0~{}~{}~{}\mbox{(using Eqn. \ref{eq:vn})},$
$\displaystyle=\sum_{m}\omega_{m}(t)v_{m}(t)\leq
0~{}~{}~{}\mbox{with}~{}v_{m}(t)=-i\langle{\rho(t)}|{[P,{\cal
H}_{m}]}\rangle,$ (23)

where we have used $[P,H^{S}]=0$. There are different ways to satisfy the
above constraint. Optimal Lyapunov solutions are discussed in Ref. Hou  _et
al._ (2012). For instance, if the control amplitudes are constrained by
$|\omega_{m}(t)|\leq\omega_{m}^{\mathrm{max}}$, then one obtains the bang-bang
control form Hou  _et al._ (2012)

$\displaystyle\omega_{m}(t)=\begin{cases}-\omega_{m}^{\mathrm{max}}~{}&~{}v_{m}(t)>0,\\\
\omega_{m}^{\mathrm{max}}~{}&~{}v_{m}(t)<0,\\\ 0~{}&~{}v_{m}(t)=0.\end{cases}$
(24)

Thus the Lyapunov method provides a systematic way to generate quantum control
for state transfer from an arbitrary initial state to an eigenstate of the
system Hamiltonian $H^{S}$. It is also possible to generalize the Lyapunov
method for gate synthesis (eg. Ghaeminezhad and Cong (2018)). Wang et al Wang
_et al._ (2022) described a Lyapunov based bang-bang quantum control scheme
for cooling an optomechanical oscillator. Recently, Purkayastha Purkayastha
(2022) extended the Lyapunov method for linear open quantum systems.

###  IV.2 Direct Search Methods

Direct search methods rely on cleverly sampling the search space starting from
a random initial guess. The following methods employ a stochastic search
algorithm to reach an optimal solution.

####  IV.2.1 Strongly Modulating Pulses (SMP)

Fortunato et al Fortunato  _et al._ (2002) described generating SMPs using a
stochastic search with a split and search method as illustrated in Fig. 5.
Given a target operator $U_{F}$, it starts with just one random control
segment $\Omega_{1}^{(0)}$ with a sufficiently long duration $\tau_{1}^{(0)}$.
The Hamiltonian for the one-segment sequence is simply
$H^{(0)}_{1}=H^{S}+H^{\Omega_{1}^{(0)}}$, and the corresponding propagator is
$U_{1}^{(0)}=\exp(-iH^{(0)}_{1}\tau_{1}^{(0)})$. By maximizing the performance
function $\Phi$ (see Eq. 16) using a vector stochastic search, such as Nelder-
Mead simplex Nelder and Mead (1965), we obtain
$\\{\tau^{(1)}_{1},\Omega_{1}^{(1)}\\}$. Now we split it into two equal
segments
$\\{\tau^{(1)}_{1}/2,\Omega_{1}^{(1)};\tau^{(1)}_{1}/2,\Omega_{1}^{(1)}\\}$
and use them as the starting point for second iteration. The final solution
involves a control sequence $\Omega$ wherein each segment has a specific
duration, amplitude, frequency, as well as initial phase.

Figure 5: Split and search method used for generating SMP.

Although stochastic search is computationally expensive, it was successfully
applied in generating controls in several QIP tasks Boulant  _et al._ (2003);
Weinstein  _et al._ (2004); Mahesh and Suter (2006); Baugh  _et al._ (2006),
including QOC of the 12-qubit NMR system Negrevergne  _et al._ (2006).

####  IV.2.2 Simulated annealing (SA)

SA Černỳ (1985) is a metaheuristic algorithm that attempts to reach the global
optimum of the parameter space by gradually shifting from exploration mode to
exploitation mode. In the $i$th iteration of threshold-based SA, the solution
$\Omega^{(i)}$ with performance $\Phi(\Omega^{(i)})$ is compared against a
random neighborhood point $\Omega^{\prime}$ with performance
$\Phi(\Omega^{\prime})$. We define
$\delta\Phi^{(i)}=\Phi(\Omega^{(i)})-\Phi(\Omega^{\prime})$ and the threshold
function (see Fig. 6)

$\displaystyle\Delta^{(i)}=\min\left[1,T^{(i)}e^{-\delta\Phi^{(i)}/T^{(i)}}\right].$
(25)

Here, the temperature $T^{(i)}$ is an iteration-dependent parameter lowering
which causes the threshold function to approach zero. The selection rule is

$\displaystyle\Omega^{(i+1)}$
$\displaystyle=\begin{cases}\Omega^{\prime}&\mbox{if}~{}~{}\delta\Phi^{(i)}\leq\Delta^{(i)}\\\
\Omega^{(i)}&\mbox{otherwise.}\end{cases}$ (26)

Figure 6: Threshold function $\Delta^{(i)}$ vs temperature $T^{(i)}$ and
change in performance $\delta\Phi^{(i)}$.

Notice that the random solution is selected not only when it is a better
solution but also when it is slightly worse than the current solution. This is
the salient feature of SA that enables it to jump over local optima and reach
the best neighborhood. Eventually the algorithm should promote solutions
comparable to or better than the current solution. This is achieved by
gradually reducing the temperature parameter which makes the algorithm more
and more exploitative. Some recent applications of SA include solving quantum
circuit transformation problem Zhou  _et al._ (2020), preparing high fidelity
quantum controls Ram  _et al._ (2021), and optimizing quantum circuits for
simultaneous dense protocol Situ and He (2022).

####  IV.2.3 Chopped random basis optimization (CRAB)

In QOC, often high fidelity solutions may lie in a low dimensional subspace of
the larger dimensional search space Lloyd and Montangero (2014). This property
is exploited in algorithms like CRAB, where the control sequence is written as
a linear combination of sensibly chosen basis functions which are fewer in
numbers compared to the dimensionality of the original search space. CRAB was
first developed for optimizing time dependent density matrix simulations Doria
_et al._ (2011), and was later adapted for the optimization of quantum
processes Caneva  _et al._ (2011); Müller  _et al._ (2021). Here, the control
sequence is modeled in terms of a continuous orthogonal functional basis, such
as Fourier series, Lagrange polynomials, Hermite functions, etc. For example,
in terms of Fourier basis, the amplitude of the control sequence of duration
$T$ for channel $m$ can be modeled as

$\displaystyle\omega_{m}(t)$
$\displaystyle=\bar{\omega}_{m}+\sum_{k=1}^{K}\alpha_{km}\cos\phi_{km}(t)+\beta_{km}\sin\phi_{km}(t),$
(27)

Here $\bar{\omega}_{m}$ is the constant component, $\alpha_{km}$, $\beta_{km}$
are the Fourier coefficients, $K$ is the total number of harmonics, and
$\phi_{km}=2\pi kt(1+r_{km})/T$ with random numbers $r_{km}\in[-0.5,0.5]$
added to improve the convergence. Since the control sequence is not
discretized, one resorts to numerical methods such as Runge-Kutta integration
to calculate the propagator (Eq. 3). The performance function is then
maximized by finding the optimal parameters for the model function. For this,
CRAB relies on the direct search, such as Nelder-Mead simplex Riaz  _et al._
(2019). The CRAB algorithm has been successfully applied to control of Bose-
Einstein-Condensates Sørensen  _et al._ (2018), adiabatic population transfer
of dressed spin states Wu  _et al._ (2019a), quantum Szilard engine
optimization Sørdal and Bergli (2019), etc.

####  IV.2.4 Evolutionary Methods

Evolutionary methods, such as genetic algorithm (GA), are inspired from
biological evolution processes, such as reproduction, mutation, and survival
of fittest, which leads to the emergence of a strong breed in an evolving
population. In GA, the initial random population of genes goes through
crossover, mutation, and selection based on a fitness function. There have
been several implementations of GA based QOC, such as generating unitary and
nonunitary QOC Bhole  _et al._ (2016), for preparing singlet order in an
11-qubit register Khurana and Mahesh (2017b), and QOC of qutrits Zahedinejad
_et al._ (2014). Another variant of the evolutionary method, namely
differential evolution (DE), has also been used for gate control Zahedinejad
_et al._ (2014) and QOC of the open quantum system Ma  _et al._ (2015).

###  IV.3 Gradient Methods

We now look at optimization methods that systematically move towards a local
optimum by evaluating the local gradients.

####  IV.3.1 Gradient Ascent Pulse Engineering (GRAPE)

Khaneja et al. Khaneja  _et al._ (2005) proposed a gradient method for
iteratively generating a control sequence. Being a local search method, it
starts from a random sequence ${\Omega}^{(0)}$ of $N$ segments, each of
duration $\tau$, and the goal is to optimize the amplitudes $\omega_{mn}$. The
control Hamiltonian is ${\cal H}_{m}({\omega}_{mn})=\omega_{mn}{\cal A}_{m},$
where ${\cal A}_{m}$ is the control operator. In the $i$th iteration, each
segment is corrected according to

$\displaystyle\omega_{mn}^{(i)}$
$\displaystyle=\omega_{mn}^{(i-1)}+\epsilon\tau~{}g_{mn}^{(i)},$ (28)

where $\epsilon$ is the step size and $g_{mn}^{(i)}$ is the gradient (see Fig.
7).

Figure 7: Illustrating the gradients $g^{(i)}_{mn}$ for amplitudes
$\omega_{mn}$ in the $i$th iteration of GRAPE algorithm Khaneja  _et al._
(2005).

GRAPE is popular, thanks to the simple analytical expressions for the first-
order gradients Khaneja  _et al._ (2005).

$\displaystyle
g_{mn}^{(i)}=\begin{cases}\mbox{Im}\left\\{\langle{U_{F}}|{U_{n+1:N}{\cal
A}_{m}U_{1:n}}\rangle\langle{U_{1:N}}|{U_{F}}\rangle\right\\}~{}&~{}\mbox{gate
synthesis}\\\ -i~{}\langle{\widetilde{\rho}_{n}}|{\left[{{\cal
A}_{m}},{\rho_{n}}\right]}\rangle~{}&~{}\mbox{state transfer.}\end{cases}$
(29)

Here forward propagated state $\rho_{n}=U_{1:n}\rho_{I}U_{1:n}^{\dagger}$ and
backward propagated state
$~{}\widetilde{\rho}_{n}=U_{n+1:N}^{\dagger}\rho_{F}U_{n+1:N}$.

While GRAPE has been used for quantum control in multiple architectures
Rowland and Jones (2012), there are also numerous variants of GRAPE. Lucarelli
Lucarelli (2018) implemented the bandwidth limited GRAPE algorithm using
Slepian basis functions. Priya et al Batra  _et al._ (2020) reported Push-Pull
GRAPE wherein the optimization is driven by the attraction of target operators
as well as the repulsion from orthogonal operators. GRAPE for open quantum
systems is discussed in Boutin  _et al._ (2017). GRAPE has also been adopted
for optimal control of quantum measurement Egger and Wilhelm (2014).

The convergence rate of the GRAPE algorithm is limited by the first-order
approximation made in the gradient expression of Eq. 29. Moreover, the step-
size $\epsilon$ needs to be smartly chosen for a given control problem. These
limitations are overcome by the second-order method, which is computationally
expensive though. The quasi-Newton method namely,
Broyden–Fletcher–Goldfarb–Shanno (BFGS) is one of the popular second-order
algorithms. de Fouquieres et al De Fouquieres  _et al._ (2011) demonstrated
convergence acceleration via BFGS algorithm.

####  IV.3.2 Gradient optimization of analytic controls (GOAT)

While GRAPE is quite successful, the control sequences are discrete and
rugged. Like CRAB discussed earlier, one may rather prefer to express the
control sequence in a smooth analytical form. It also allows encoding the
solution in a lower dimensional parameter space. This is one of the benefits
of gradient optimization of analytic controls (GOAT) Machnes  _et al._ (2018).
The model analytical function may be an educated guess or randomly chosen. For
example, if one models the control sequence in terms of the superposition of
$K$ Gaussian pulses,

$\displaystyle\omega_{m}(t)=\sum_{k=1}^{K}\exp\left[-\frac{(t-\delta_{m,k})^{2}}{\sigma_{m,k}^{2}}\right].$
(30)

Here $m$th channel parameters $\alpha_{m}=\\{\delta_{m,k},\sigma_{m,k}\\}$ are
optimized to minimize the gate infidelity
$f(\Omega)=1-{\langle{U_{F}}|{U_{\Omega}}\rangle}/{\langle{U_{F}}|{U_{F}}\rangle}$.
The gradients are then given by Machnes  _et al._ (2018)

$\displaystyle\partial_{\alpha}F(\Omega)=-\mathrm{Re}\left[\frac{f^{*}(\Omega)}{|f(\Omega)|}\frac{\langle{U_{F}}|{\partial_{\alpha}U_{\Omega}}\rangle}{\langle{U_{F}}|{U_{F}}\rangle}\right].$
(31)

While there is no analytical solution to the above equation, one can use the
equation of motion $\partial_{t}U_{\Omega}(t)=-iH(t)U_{\Omega}(t)$ to obtain
the coupled system of equations

$\displaystyle\partial_{t}\left(\begin{array}[]{c}U_{\Omega}(t)\\\
\partial_{\alpha}U_{\Omega}(t)\end{array}\right)=-i\left(\begin{array}[]{cc}H&0\\\
\partial_{\alpha}H&H\end{array}\right)\left(\begin{array}[]{c}U_{\Omega}(t)\\\
\partial_{\alpha}U_{\Omega}(t)\end{array}\right),$ (38)

which can be solved by numerical forward integration, such as the adaptive
Runge-Kutta. Machnes et al Machnes  _et al._ (2018) demonstrated the GOAT
algorithm by generating the iSWAP gate for a pair of transmon qubits coupled
to a tunable bus resonator. Kirchoff et al demonstrated a time-optimal CNOT
gate on a pair of transmon qubits Kirchhoff  _et al._ (2018).

###  IV.4 Variational methods

Variational methods are widely used for constrained optimization and are less
sensitive to local optima compared to gradient methods. For example, the
Krotov algorithm that is based on the Lagrange multiplier exhibits a monotonic
convergent behavior Krotov (1995). It starts with the Lagrange multiplier

$\displaystyle B_{n}$
$\displaystyle=\begin{cases}U_{n+1:N}^{\dagger}U_{F}\langle{U_{F}}|{U_{1:N}}\rangle~{}&~{}\mbox{for
gate synthesis, and}\\\
U_{n+1:N}^{\dagger}\left\\{\rho_{F}U_{1:N}\rho_{I}+\kappa
U_{1:N}\right\\}~{}&~{}\mbox{for state-to-state transfer,}\end{cases}$ (39)

where $\kappa$ is a positive constant to ensure positivity of fidelity. In
addition to the control sequence $\Omega$, the Krotov method involves a co-
sequence $\widetilde{\Omega}$, both of which are initialized with the same
random guess, i.e., $\Omega^{(0)}=\widetilde{\Omega}^{(0)}$. These sequences
are propagated forward and backward respectively according to the rules
Maximov  _et al._ (2008); Reich  _et al._ (2012); Batra  _et al._ (2020)

$\displaystyle\omega_{mn}^{(i)}$
$\displaystyle=(1-\delta)\widetilde{\omega}_{mn}^{(i-1)}+\frac{\delta}{\lambda_{m}}\mathrm{Im}\langle{B_{n}^{(i-1)}}|{{\cal
A}_{m}U_{0:n-1}^{(i)}}\rangle$ $\displaystyle\widetilde{\omega}_{mn}^{(i)}$
$\displaystyle=(1-\eta)\omega_{mn}^{{(i)}}+\frac{\eta}{\lambda_{m}}\mathrm{Im}\langle{B_{n}^{(i)}}|{{\cal
A}_{m}U_{0:n}^{(i)}}\rangle,$ (40)

where $\lambda_{m}$ are penalty constants and other constants
$\eta,\delta\in[0,2]$. Application of Krotov optimization for the optimal
control of spin dynamics in NMR and dynamical nuclear polarization gained
significant attention Maximov  _et al._ (2008); Reich  _et al._ (2012).
Vinding et al. showed the use of Krotov in magnetic resonance imaging for
design of spatial selective RF pulses Vinding  _et al._ (2012). Hwang et al.
showed the control of a non-Markovian open quantum system for Z gate and
identity gate Hwang and Goan (2012). Krotov optimization has also been applied
for the control of BEC in magnetic microtraps Jäger  _et al._ (2014).

###  IV.5 Machine Learning Methods

With the ever increasing computational power, the machine learning (ML)
methods are finding applications everywhere, including QOC. Reinforcement
learning (RL), for example, is popularly used in a variety of problems of
finding the best strategy that maximizes a reward function Sutton and Barto
(2018). In QOC, RL can be modeled to find the strategy that maximizes the
fidelity reward. For example, Bukov et al demonstrated RL to control out-of-
equilibrium systems Bukov  _et al._ (2018). RL has also been applied for
problems such as state control An and Zhou (2019), gate control Zhang  _et
al._ (2019); Baum  _et al._ (2021), for generating controls robust against
certain types of errors Niu  _et al._ (2019), and for the control of
multilevel dissipative quantum systems An  _et al._ (2021). QOC with
supervised ML Wu  _et al._ (2019b); Zeng  _et al._ (2020) and with
convolutional neural networks trained through deep learning architecture for a
quantum particle in a disordered system Huang  _et al._ (2022) have also been
reported. Another type of ML, namely differential programming (DP), together
with a neural network was used for eigenstate preparation in a variety of
single and multi-qubit systems Schäfer  _et al._ (2020) as well as for the
control of quantum thermal machines Khait  _et al._ (2021). Coopmans et al.
showed the transport of magnons in a spin chain by combining DP with CRAB and
shortcut to adiabaticity protocols Coopmans  _et al._ (2022).

###  IV.6 Hybrid Methods

There have been several attempts to combine different algorithms to realize
hybrid algorithms which incorporate the best of the original methods. For
example, Machnes et al. Machnes  _et al._ (2011a) combined GRAPE and Krotov to
form the hybrid algorithm and implemented it in the DYNAMO package.

While GRAPE can get stuck in a local optimum, the simulated annealing (SA) is
good at jumping over it and finding a better neighborhood. Thus the hybrid
algorithm, SAGRAPE, combines the best of the two Ram  _et al._ (2021). As
explained in Sec. IV.2.2, an important step in SA in every iteration is to
scan the neighborhood points of the current solution, which is a bottleneck in
the standard SAGRAPE algorithm. This is where ML can bring about a significant
speedup. Priya et al Batra  _et al._ (2022) have reported the recommender
system (RS) expedited SAGRAPE (RSSAGRAPE), wherein both SA and GRAPE routines
of SAGRAPE are speeded-up by RS.

The variation principle-based Krotov algorithm can lead to monotonic increase
at low fidelity but gets slow at high fidelities. The gradient-based quasi-
newton BFGS can yield high fidelity solutions, but need more computational
resources. To this end, Tannor et.al Eitan  _et al._ (2011) proposed the
hybrid Krotov-BFGS (K-BFGS) algorithm.

For many QOC problems, high fidelity solutions lie in a low dimensional
subspace of the larger dimensional search space Lloyd and Montangero (2014).
This property is exploited in algorithms like CRAB where the control sequence
is written as a linear combination of sensibly chosen basis functions which
are fewer in number when compared to the dimensionality of the original search
space. Sorensen et.al Sørensen  _et al._ (2018) proposed a hybrid algorithm
called gradient optimisation using parameterisation (GROUP) that incorporates
both CRAB and GRAPE algorithms. Here, an analytical expression of the gradient
of the cost function w.r.t the previously mentioned linear coefficients is
derived and is used in place of Nelder-Mead to update the linear coefficients.
The resulting algorithm can produce higher fidelity controls for the same
number of functional evaluations as the CRAB algorithm.

##  V Summary and outlook

To summarize, we had set out three intentions in this review. First, to
introduce the basic aspects of QOC. Second, to point out various practical
aspects in optimizing QOC as well as implementing the control sequence. Third,
to provide a quick overview of the diverse methods used for QOC. We provided
brief descriptions of various QOC algorithms sorting them into five
categories, namely, asymptotic method, direct search, gradient method,
variational method, and machine learning. A quick comparison of these methods
is shown in Table. 2. Finally, we also described hybrid algorithms which
combine two or more methods to yield better performance.

Table 2: An overview of the various QOC methods discussed in the review. Method |  Algorithm |  Merits |  Limitations
---|---|---|---
Asymptotic evolution |  Adiabatic Pulses |  Analytical control; Robust against pulse errors |  Specialized applications; Lengthy sequences
Geometric Phase |  Analytical control; Robust against pulse errors |  Specialized applications; Lengthy sequences
Lyapunov |  Analytical control |  Specialized applications
Direct Search |  SMP |  Highly adaptable |  Sensitive to initial guess; Computationally inefficient
Simulated Annealing |  Can overcome local optima |  Relatively slow
CRAB |  Analytical control sequence |  Sensitive to chosen model
|  Evolutionary Algorithms |  Highly adaptable; attempts global search |  Needs heavy computational resource
Gradient Method |  GRAPE |  Numerically efficient |  Sensitive to initial guess
GOAT |  Analytical control sequence |  Sensitive to chosen model
Variational |  Krotov |  Monotonic convergence |  Needs higher computational resource
Machine Learning |  Reinforcement Learning |  Robust against local optima |  A general ML platform for QOC needs to be developed
Hybrid |  |  Can combine the best of two or more methods |  Demand complex coding and fine-tuning of parameters

Taking an outlook, we may expect further roles of hybrid algorithms as well as
machine learning in designing complex and robust control sequences for larger
systems with severe parameter constraints. Measurement based feedback control
(eg. Lu  _et al._ (2017)) which allows tailor-made controls incorporating
device characteristics is likely to become routine. Finally, a somewhat
futuristic, but promising development is the quantum assisted quantum control,
wherein quantum computers themselves are used to generate quantum controls
(eg. Policharla and Vinjanampathy (2021)).

#### Software packages

Simulation Package for Solid-state NMR spectroscopy (SIMPSON), an open source
package, simulates NMR experiments along with optimal control protocols such
as GRAPE Tošner  _et al._ (2009). SIMPSON package is available for different
OS at https://inano.au.dk/about/research-centers-and-
projects/nmr/software/simpson. Dynamic Optimization platform (DYNAMO), a
Matlab based software package contains GRAPE and Krotov algorithms along with
the option of comparing and benchmarking new algorithms Machnes  _et al._
(2011b). The codes for the same can be found at http://qlib.info. The Quantum
Toolbox in Python (QuTiP), an open source python package available at
https://qutip.org/download.html, offers a simulation of generic quantum system
with optimization techniques such as GRAPE and CRAB Johansson  _et al._ (2012,
2013). Goerz et al. have implemented the Krotov algorithm in the QuTiP
framework Goerz  _et al._ (2019). Recently, Teske et al. developed an open
source package QOPT for simulating quantum dynamics and robust quantum control
in conjunction with common experimental situations Teske  _et al._ (2022).
QEngine is a C++ library for the implementation of quantum control in ultra
cold atoms Sørensen  _et al._ (2019).

## Acknowledgments

The authors dedicate this review to the 80th birth day of Prof. Anil Kumar,
IISc, Bangalore, who is noted for his pioneering contributions to NMR
spectroscopy as well as NMR quantum computation. PB acknowledges support from
the Prime Minister’s Research Fellowship (PMRF) of the Government of India.
TSM acknowledges funding from DST/ICPS/QuST/2019/Q67.

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