# Robustness of controlled Hamiltonian approaches to unitary quantum gates

Eoin Carolan  School of Physics, University College Dublin, Belfield, Dublin
4, Ireland  Centre for Quantum Engineering, Science, and Technology,
University College Dublin, Belfield, Dublin 4, Ireland  Barış Çakmak  College
of Engineering and Natural Sciences, Bahçeşehir University, Beşiktaş,
Istanbul, Türkiye  TUBITAK Research Institute for Fundamental Sciences, 41470
Gebze, Türkiye  Department of Physics, Farmingdale State College—SUNY,
Farmingdale, NY 11735, USA  Steve Campbell  School of Physics, University
College Dublin, Belfield, Dublin 4, Ireland  Centre for Quantum Engineering,
Science, and Technology, University College Dublin, Belfield, Dublin 4,
Ireland  Trinity Quantum Alliance, Unit 16, Trinity Technology and Enterprise
Centre, Pearse Street, Dublin 2, D02YN67  Dahlem Center for Complex Quantum
Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

###### Abstract

We examine the effectiveness and resilience of achieving quantum gates
employing three approaches stemming from quantum control methods:
counterdiabatic driving, Floquet engineering, and inverse engineering. We
critically analyse their performance in terms of the gate infidelity, the
associated resource overhead based on energetic cost, the susceptibility to
time-keeping errors, and the degradation under environmental noise. Despite
significant differences in the dynamical path taken, we find a broadly
consistent behavior across the three approaches in terms of the efficacy of
implementing the target gate and the resource overhead. Furthermore, we
establish that the functional form of the control fields plays a crucial role
in determining how faithfully a gate operation is achieved. Our results are
demonstrated for single qubit gates, with particular focus on the Hadamard
gate, and we discuss the extension to $N$-qubit operations.

##  I Introduction

Prompted by Feynman [1], the idea of using quantum properties of matter and
light to process information has given rise to an extensive research effort.
Beyond the implications for basic science, quantum information technologies
would entail a significant computational speed-up for particular applications,
allowing for solutions problems that are intractable with current technologies
based on classical systems [2, 3, 4]. These quantum advantages have been
theoretically predicted for a variety information processing tasks, such as
search and factoring algorithms, or quantum cryptography [5]. Experimentally
it is now possible to implement them in systems such as superconducting qubits
[6, 7], trapped ions [8], and photons [9].

Several approaches for universal quantum computation have been developed,
chief among them being measurement-based [10, 11, 12, 13], gate-based [1, 14],
and adiabatic models [15, 16]. The relative benefits and drawbacks of each
approach notwithstanding [2, 3, 4], gate-based quantum computation presents an
attractive method. Any computation can be implemented by a relativity small
set of gates on a qubit register [17, 14]. Indeed, small scale quantum devices
are providing remarkable platforms for simulation of quantum systems [18, 19,
20, 21], insights from which can be greatly enhanced by improving the
implementation of the basic building blocks, i.e. the quantum gates.

Achieving this aim necessitates coherent control of quantum systems [22, 23,
24, 25, 26, 27, 28]. Beyond the basic requirement of enacting the desired gate
operation, we must consider several additional factors to ensure the
scalability and reliability of these operations. Among these are the resources
necessary for their fast and accurate implementation [29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42], understanding the spoiling impact of the
environment [43, 44, 45], and the impact of operational errors [46, 47]. The
assessment of the energetic efficiency of these devices is crucial in their
design [32] and may enforce practical constraints for their implementation.
The interplay between the performance of a quantum computing machine and its
energetic efficiency determines a fundamental connection between quantum
information processing and thermodynamics [48, 49].

Following this edict, in this work we consider three approaches to implement
gate operations on quantum systems through controlled Hamiltonian dynamics. In
particular, we consider the auxiliary evolution approach introduced in Refs.
[50, 51], where a driven auxiliary system is is coupled to the computational
register upon which the operation is faithfully induced provided the evolution
is adiabatic. We augment this approach with techniques from shortcuts-to-
adiabaticity [22, 23], specifically counterdiabatic driving (CD) [52, 53, 54,
55] and Floquet engineering (FE) [56], that allow to arbitrarily speed up the
implementation. In addition to these techniques, we consider an inverse
engineering (IE) approach [57] where the computational register is directly
driven by external control fields. We examine these approaches, both in terms
of their resource overhead and their resilience to systematic errors stemming
from imperfect timekeeping and environmental effects. We find that all are
effective in achieving the target gate operation and we highlight the
importance that the choice of control pulse plays in all cases. However, more
significant differences emerge when considering other performance metrics and
therefore we find that the optimal choice of how to realise such controlled
quantum gates will ultimately be dictated by the constraints of a given
architecture.

##  II Preliminaries

###  II.1 Control protocols

Here we outline the three control techniques that are the focus of the present
work. As shown in Fig. 1 for the auxiliary evolution approach we consider two
approaches to speed up the dynamics (i) counterdiabatic driving (CD) and (ii)
Floquet Engineering (FE); we also consider a third controlled implementation
where the computational register is directly driven via (iii) inverse
engineering (IE).

Figure 1: (a) Shows the auxiliary control setting based on the protocol from
Ref. [51]. Here an auxiliary qubit is coupled to the computational register,
with control fields acting only on this auxiliary system. We assume the driven
qubit can also experience environmental effects while the computational qubit
is completely isolated. (b) Shows the setup for the Inverse Engineering
setting. The Hamiltonian is designed without any additional resources, and
thus the computational qubit is driven directly and can be subject to
environmental noise.

####  II.1.1 Auxiliary Evolution with Control

The first method we consider for implementing unitary gates is the adiabatic
approach [50] where an auxiliary qubit is coupled to a computational register
upon which we wish to perform the gate operation. By driving this auxiliary
qubit adiabatically, the desired gate operation can be effected on the
computational register. To that end, we consider the general total Hamiltonian
(for the register and auxiliary qubit)

$H(t)=\sum_{k}P_{k}\otimes H_{\phi_{k}}(t),$ (1)

where $P_{k}$ are projectors derived from the rotation axis of the desired
unitary acting on the computational register, while $H_{\phi_{k}}(t)$ is the
angle-dependent driving Hamiltonian acting on the auxiliary qubit, given by

$H_{\phi_{k}}(t)=-\left[\cos(\theta_{f}\lambda)\sigma_{z}+\sin(\theta_{f}\lambda)\left[\cos{(\phi_{k})}\sigma_{x}+\sin{(\phi_{k})}\sigma_{y}\right]\right].$
(2)

where $\lambda\\!\equiv\\!\lambda(t)$ is the time-dependent control parameter.
The angles $\phi_{k}$ are dictated by the specific gate being implemented and
we will examine an exemplary choice in the proceeding section.

This approach necessitates that the auxiliary qubit is driven adiabatically,
which in general requires long timescales leaving the system open to errors
from environmental effects. Counterdiabatic driving [52, 55] allows us to
arbitrarily speed up the evolution while still achieving perfect adiabatic
dynamics by introducing additional term(s) to the system Hamiltonian. The CD
term is equivalent to the addition of an adiabatic gauge potential to the
system Hamiltonian [26]. In general, the CD term for a Hamiltonian, $H_{0}$,
is given by

$H^{CD}=\dot{\lambda}\mathcal{A}_{\lambda}=i\sum_{m\neq
n}\frac{\left|m\right\rangle\langle m|\dot{H}_{0}\left|n\right\rangle\langle
n|}{E_{n}-E_{m}},$ (3)

where $\left|k\right\rangle$ and $E_{k}$ are the Hamiltonian’s instantaneous
eigenstates and eigenvalues. Evolving the system with the total Hamiltonian
$H\\!=\\!H_{0}+H^{CD}$ gives rise to a transitionless dynamics in finite time.
The exact evaluation of Eq. (3) requires the complete knowledge of the
spectrum of the system Hamiltonian at all times, often limiting its
applicability. However, Ref. [26, 56] propose to approximate the exact
adiabatic gauge potential given in Eq. (3) with a nested commutator expansion

$\mathcal{A}_{\lambda}^{(l)}=i\sum_{k=1}^{l}\alpha_{k}\underbrace{[H[H,...[H}_{2k-1},\partial_{\lambda}H]]],$
(4)

where $l$ denotes the order of the expansion and, for an arbitrary system in
the limit of $l\rightarrow\infty$, one obtains the exact expression given in
Eq. (3). The coefficients, $\alpha_{k}$, are determined by minimizing the
action

$\displaystyle S_{l}$ $\displaystyle=\text{Tr}[\hat{G}_{l}^{2}],$
$\displaystyle\hat{G}_{l}$
$\displaystyle=\partial_{\lambda}\hat{H}-i[\hat{H},\hat{\mathcal{A}}_{\lambda}^{l}].$
(5)

This approach is particularly effective when dealing with many-body systems as
it allows to truncate the complexity of the control fields [56]. For a single
two-level system, as will be the focus of the present work, we find that Eq.
(4) is already identical to the full counterdiabatic term, Eq. (3), for
$l\\!=\\!1$, i.e. only the first term in the sum is required to achieve
perfect control. However, our main interest in employing Eq. (4) is because it
provides a means to engineer a Floquet Hamiltonian which approximately mimics
the action of adiabatic gauge potential [56] and therefore opens up new
possibilities in terms of feasible experimental implementations [58].

Floquet theory allows to design an effective Hamiltonian that stroboscopically
mimics the dynamics of another, potentially more complex or experimentally
unfeasible Hamiltonian. In order to achieve this, we need only to oscillate
the original driven Hamiltonian and its derivative with respect to the driving
parameter. Such a Floquet Hamiltonian can stroboscopically recreate the
dynamics of the full CD Hamiltonian
$H\\!=\\!H_{0}+\dot{\lambda}\mathcal{A}_{\lambda}^{(l)}$ with a comparatively
reduced operator set. The explicit form of the Hamiltonian that implements
this is

$\begin{split}H^{FE}_{\phi}=&\left[1+\frac{\omega}{\omega_{0}}\cos(\omega
t)\right]H_{\phi}(\lambda)\\\
&+\dot{\lambda}\left[\sum_{k=1}^{\infty}\beta_{k}\sin((2k-1)\omega
t)\right]\partial_{\lambda}H_{\phi}(\lambda),\end{split}$ (6)

where $\beta_{k}$ are Fourier coefficients of the expansion of the Floquet
Hamiltonian that are chosen to match the terms of the adiabatic gauge
potential expansion, $\omega_{0}\\!=\\!2\pi/\tau$ is a reference frequency and
we take $\omega$ to be much greater than $\omega_{0}$, whose ratio defines the
number of driving cycles for the evolution.

####  II.1.2 Inverse Engineering

As an alternative approach to auxiliary control, we consider directly driving
the computational register. The evolution of a closed quantum system obeys the
time-dependent Schrödinger equation and an arbitrary initial state is
connected to a designated final state by a unitary operator,
$\left|\psi(t)\right\rangle\\!=\\!U(t)\left|\psi(0)\right\rangle$. The
Hamiltonian that generates such a unitary time evolution is determined by the
well-known relation

$H(t)=i\dot{U}(t)U^{{\dagger}}(t).$ (7)

It is possible to follow several approaches to inverse engineer the desired
unitary [59, 60, 57, 23], and hence the corresponding Hamiltonian. In this
work, we adopt the approach taken in Ref. [57] and express $U(t)$ in the
following form

$U(t)=\sum_{n}e^{i\pi\lambda_{m}(t)}\left|m(t)\right\rangle\langle m(t)|,$ (8)

where the set $\\{\left|m(t)\right\rangle\\}$ forms a complete orthonormal
basis, and $\lambda(t)$ has the initial condition $\lambda_{n}(0)\\!=\\!2l$
where $l\in\mathbb{Z}$ to ensure $U(0)\\!=\\!\mathds{1}$. By taking suitable
choices for the free parameters that define the orthonormal basis and local
phase information, we can construct a Hamiltonian that implements the desired
unitary behaviour in such a way that is not dependent on a particular initial
state. In what follows we construct the IE Hamiltonian such that $\lambda(t)$
is the driving parameter. The motivation for choosing IE is to showcase
another control technique. However, it is important to remark that the IE
approach prescribed above and “typical” counterdiabatic control methods are
intrinsically related [61]. Thus, the results reported for the IE case would
be qualitatively similar if instead CD driving were applied to the
computational qubit directly. What does differ is that with the IE approach we
do not start with a reference Hamiltonian a priori for which the transitions
need to be suppressed. Instead, one can separate out the adiabatic gauge
potential term from the resultant IE Hamiltonian after transforming basis.
Nevertheless, we remind that, as depicted in Fig. 1, the key difference in our
analysis is embodied by the two distinct settings where either an auxiliary
system is employed to achieve the gate versus when the computational system is
directly driven.

###  II.2 Figures of merit

####  II.2.1 Gate Infidelity

To characterise how faithfully a gate has been implemented we adopt the
average infidelity measure [44]

$J_{T}=1-\sum_{i=1}^{3}\frac{w_{i}}{tr[\rho_{i}^{2}(0)]}\mathcal{R}e\big{\\{}tr[U\rho_{i}(0)U^{{\dagger}}\rho_{i}(\tau)]\big{\\}},$
(9)

and consider the average of the Hilbert-Schmidt norm of the ideal evolution of
three specific initial states with the obtained state, weighted by $w_{i}$
with $\sum_{i=1}^{3}w_{i}=1$. Three initial states satisfying particular
conditions have been shown to be the minimum amount needed to address all the
possible errors and characterise a general unitary operation for an open
system evolution [44, 62]. For a single qubit, the following set satisfies the
necessary conditions [44]

$\displaystyle\rho_{1}(0)$ $\displaystyle=\begin{pmatrix}2/3&0\\\
0&1/3\end{pmatrix},\\!\\!\\!\\!$ $\displaystyle\rho_{2}(0)$
$\displaystyle=\begin{pmatrix}1/2&1/2\\\ 1/2&1/2\end{pmatrix},\\!\\!\\!\\!$
$\displaystyle\rho_{3}(0)$ $\displaystyle=\begin{pmatrix}1/2&0\\\
0&1/2\end{pmatrix}.$

The first state, $\rho_{1}$, checks errors in the fixed basis states, and
therefore does not signal any possible errors that are diagonal in this basis.
The second state, $\rho_{2}$, addresses this and indicates the off-diagonal
errors in the fixed basis. The third state, $\rho_{3}$, is chosen to ensure
that populations are conserved, important for an open system setting.
Depending on the choice of the weights in Eq. (9), it is possible to highlight
the effect of a source of an error on the infidelity over the others. For
simplicity and without loss of generality, throughout this work we choose
these weights to be equal, i.e. $w_{i}\\!=\\!1/3$.

####  II.2.2 Cost of Control

The addition of control terms to the Hamiltonian implies an overall increase
in resources needed to evolve the system. The analysis of this cost has been
the focus of several recent works [63, 64, 65, 66, 67, 68, 69, 70, 71], where
different quantifiers have been introduced depending on the physical
motivation. In this work, we adopt the cost measure introduced in [72, 63, 54]

$\mathcal{C}=\frac{1}{\tau}\int_{0}^{\tau}\left\lVert H\right\rVert dt,$ (10)

where $\left\lVert\cdot\right\rVert$ denotes the norm of the Hamiltonian of
interest, and for simplicity we consider the trace norm. It is important to
emphasize that, following the approach taken in [36], we take $H$ to be the
full Hamiltonian that generates the driven dynamics implementing the gate
operation, not just the external control term. In fact, notice that it is only
for the case of CD control where an explicit additional Hamiltonian term is
added to the bare Hamiltonian. For both the FE and IE approaches, control is
embedded into the same operators that appear in the bare Hamiltonian.
Therefore, it is necessary to consider the cost of the full Hamiltonian
generating the time evolution. This measure is well motivated by the
functional form of the physical driving fields [63, 71] and it has been shown
to have connections to a Landauer-type limit for the change in information
encoded computational states [31].

###  II.3 Systematic Errors

####  II.3.1 Timekeeping errors

The controlled dynamics require that the drives are implemented for a specific
length of time, which we denote by $\tau$. Since the control protocols are
designed to be effective regardless of the specific functional form of the
drive, this provides a useful additional degree of freedom for control
protocols [73, 36]. We consider the following ramp profiles that satisfy the
boundary conditions $\lambda(0)\\!=\\!0$ and $\lambda(\tau)\\!=\\!1$,

$\displaystyle\begin{aligned} &\lambda(t)=\frac{t}{\tau},&&\text{linear}\\\
&\lambda(t)=\frac{10t^{3}}{\tau^{3}}-\frac{15t^{4}}{\tau^{4}}+\frac{6t^{5}}{\tau^{5}},\qquad&&\text{polynomial}\\\
&\lambda(t)=\sin\left(\frac{\pi
t}{2\tau}\right).&&\text{sinusoidal}\end{aligned}$ (11)

We look to characterise the impact of timing errors in the drive, i.e. where
the duration of the driving field over- or under-shoots the intended target
time, $\tau$, by assessing the resulting impact on the gate infidelity, Eq.
(9). We note that these pulses are chosen to capture and compare certain pulse
characteristics. Indeed much work has been done in designing more complex ramp
profiles via optimal control and machine learning methods [74, 75, 76] seeking
to optimize to a variety of relevant cost functionals. Our analysis can
therefore provide useful information for the seed pulses to ensure robustness
to, e.g. time keeping errors, while exploiting more advance techniques to
explore a greater optimization landscape.

####  II.3.2 Environmental Errors

We will be interested in considering how faithfully the gate operation is
implemented when the controlled system is not completely isolated and
therefore prone to environmental effects. To that end, we model the time
evolution of the driven system with a Markovian master equation

$\dot{\rho}=-i[H,\rho]+\mathcal{D}(\rho),$ (12)

where $\mathcal{D}$ gives rise to a dephasing process

$\displaystyle\mathcal{D}(\rho)$
$\displaystyle=\gamma(\sigma_{z}^{\lambda}\rho\sigma_{z}^{\lambda}-\rho),$
(13)

where the superscript $\lambda$ symbolises our assumption that the environment
only affects the driven part of the system, i.e. for the CD and FE cases we
assume the environment acts only on the auxiliary qubit, while for IE it is
applied directly to the computational qubit(s).

##  III Single Qubit Gate

We begin assessing the controlled implementation of a single qubit gate which
realises the operation

$\alpha\left|n_{+}\right\rangle+\beta\left|n_{-}\right\rangle\to\alpha\left|n_{+}\right\rangle+\text{e}^{i\phi}\beta\left|n_{-}\right\rangle,$
(14)

where $\left|n_{\pm}\right\rangle$ forms a basis in which the desired unitary
gate simply performs a rotation of $\phi$. For the case of the auxiliary
evolution outlined in Sec. II.1.1, where the computational qubit is coupled to
an auxiliary system that is subject to the controlled drive, the combined
initial state is assumed to be
$\left|\Psi_{i}\right\rangle=\left(\alpha\left|n_{+}\right\rangle+\beta\left|n_{-}\right\rangle\right)\otimes\left|0\right\rangle$
with $\left|0\right\rangle$ and $\left|1\right\rangle$ being the eigenstates
of $\sigma_{z}$. In this setting, Eq. (1) becomes

$H(t)\\!=\\!\left|n_{+}\right\rangle\langle n_{+}|\otimes
H_{\phi_{+}}(t)+\left|n_{-}\right\rangle\langle n_{-}|\otimes
H_{\phi_{-}}(t),$ (15)

where the projectors are given by $\left|n_{\pm}\right\rangle\langle
n_{\pm}|=\left(\mathbf{1}\pm\vec{n}\cdot\vec{\sigma}\right)/2$, with $\vec{n}$
being the rotation axis of the unitary and $H_{\phi_{\pm}}(t)$ is as given in
Eq. (2). For a single qubit gate, $\phi_{+}$ is taken to be zero, and
$\phi_{-}$ is taken to be the angle of rotation corresponding to the desired
unitary. Adiabatically evolving under the Hamiltonian above yields the
following final state

$\left|\Psi_{f}\right\rangle=\alpha\left|n_{+}\right\rangle\otimes\left|\epsilon_{\phi_{+}}^{g}\right\rangle+\beta\left|n_{-}\right\rangle\otimes\left|\epsilon_{\phi_{-}}^{g}\right\rangle,$
(16)

where $\left|\epsilon_{\phi_{\pm}}^{g}\right\rangle$ is the ground state of
Eq. (2), given by

$\left|\epsilon_{\phi_{\pm}}^{g}\right\rangle=\cos\left(\frac{\theta_{f}\lambda}{2}\right)\left|0\right\rangle+\text{e}^{i\phi_{\pm}}\sin\left(\frac{\theta_{f}\lambda}{2}\right)\left|1\right\rangle.$
(17)

Choosing $\theta_{f}\lambda\\!=\\!\pi$ as the endpoint of the drive, we
deterministically find that the desired gate has been implemented on the
computational qubit and the auxiliary system is left in its excited state.
Similarly, we explicitly show in Appendix A that one can define a driving
scheme for the auxiliary system initiated in its excited state, therefore
allowing a sequence of gates to be implemented without necessitating the
auxiliary qubit to be re-initialized.

In what follows, we focus on the Hadamard gate as the target operation on the
computational qubit, which corresponds to choosing the projectors as
$|n_{\pm}\rangle\langle
n_{\pm}|=\frac{1}{2}(\mathbf{1}\pm\frac{1}{\sqrt{2}}(\sigma_{x}+\sigma_{z}))$,
accompanied by the rotations on the auxiliary qubit with phases
$\phi_{+}\\!=\\!0$ and $\phi_{-}\\!=\\!\pi$, respectively, although we remark
that our results are qualitatively consistent for other choices of single
qubit gates.

To drive faster than adiabatic timescales we compute the CD term, Eq. (3) for
Hamiltonian (2) giving

$H^{CD}_{\phi_{\pm}}(t)=\dot{\lambda}\frac{\pi}{2}[\sigma_{y}\cos(\phi_{\pm})-\sigma_{x}\sin(\phi_{\pm})].$
(18)

where we have taken $\theta_{f}\\!=\\!\pi$. Note that the CD term is used in
addition to the bare time dependent Hamiltonian in (15). It is straightforward
see that the associated adiabatic gauge potential, Eq. (4) is identical to Eq.
(18), with the variational coefficient $\alpha_{1}\\!=\\!-1/4$ determined from
minimizing the action, from which we readily determine the Hamiltonian giving
rise to the Floquet controlled evolution

$\hat{H}^{FE}_{\phi_{\pm}}=\left[1+\frac{\omega}{\omega_{0}}\cos(\omega
t)\right]\hat{H}_{\phi_{\pm}}(\lambda)+\dot{\lambda}\left[\omega_{0}\alpha_{1}\sin(2\omega
t)\right]\partial_{\lambda}\hat{H}_{\phi_{\pm}}(\lambda),$ (19)

where $\omega_{0}\\!=\\!2\pi/\tau$ is the reference frequency and
$\omega\\!=\\!N\omega_{0}$ with $N\\!\in\\!\mathbb{N}\\!\gg\\!1$. The Floquet
Hamiltonian replaces the time dependent bare Hamiltonians in (15).

The same gate operation can be captured by the inverse engineering approach
prescribed in Sec. II.1.2. We consider the unitary operator

$U_{1}(t)=\left|m_{+}(t)\right\rangle\langle
m_{+}(t)|+e^{i\pi\lambda(t)}\left|m_{-}(t)\right\rangle\langle m_{-}(t)|,$
(20)

where the basis states are defined as

$\displaystyle\left|m_{+}(t)\right\rangle$
$\displaystyle=\cos[\vartheta(t)/2]\left|0\right\rangle+e^{i\varphi(t)}\sin[\vartheta(t)/2]\left|1\right\rangle,$
(21) $\displaystyle\left|m_{-}(t)\right\rangle$
$\displaystyle=e^{i\varphi(t)}\cos[\vartheta(t)/2]\left|1\right\rangle-\sin[\vartheta(t)/2]\left|0\right\rangle.$

with parameters $\vartheta(t),\varphi(t),$ and $\lambda(t)$ that can be tuned
in order to define the desired gate operation. For a single qubit gate, the
driving Hamiltonian found from Eq. (7) takes the form [57]

$H(t)=\frac{1}{2}\vec{\omega}(t)\cdot\vec{\sigma},$ (22)

where the explicit form of the angular components are given in Appendix B. The
action of this Hamiltonian is to transform the input state
$\left|\mu(0)\right\rangle\\!=\\!a\left|0\right\rangle+b\left|1\right\rangle$
to the final state
$\left|\mu(t)\right\rangle\\!=\\!\alpha(t)\left|0\right\rangle+\beta(t)\left|1\right\rangle$
where the populations are

$\displaystyle\alpha(t)$
$\displaystyle=\frac{a(e^{i\pi\lambda(t)}+1)-(e^{i\pi\lambda(t)}-1)(a\cos\vartheta(t)+be^{-i\varphi(t)}\sin\vartheta(t))}{2},$
$\displaystyle\beta(t)$
$\displaystyle=\frac{b(e^{i\pi\lambda(t)}+1)+(e^{i\pi\lambda(t)}-1)(b\cos\vartheta(t)-ae^{-i\varphi(t)}\sin\vartheta(t))}{2}.$

One choice of parameters that gives the Hadamard gate are
$\varphi(t)\\!=\\!0$, $\vartheta(t)\\!=\\!\pi/4$, and ramping from
$\lambda(0)\\!=\\!0$ to $\lambda(\tau)\\!=\\!1$, which in turn gives the
populations of the final state as $\alpha(\tau)\\!=\\!(a+b)/\sqrt{2}$, and
$\beta(\tau)\\!=\\!(a-b)/\sqrt{2}$. The corresponding Hamiltonian that drives
our qubit is then given as

$H_{Had}^{IE}(t)=\frac{\pi\dot{\lambda}(t)}{2\sqrt{2}}(\sigma_{x}+\sigma_{z}).$
(23)

Figure 2: We show the trajectories of the qubits for both the CD and IE
protocols of the Hadamard gate. We take the initial computational state to be
$\left|+\right\rangle$. The yellow line corresponds to the path of the qubit
in the IE case. The green and orange lines correspond to the computational and
auxiliary qubits of the auxiliary evolution cases, respectively, which begins
and ends with a separable global state of the two qubits while at intermediate
times the reduced states of either qubit are mixed.

(a) (b) (c) (d)

Figure 3: (a) Final gate infidelity, Eq (9) as a function of total protocol
duration, for the Hadamard gate. The auxiliary evolution with counterdiabatic
(CD) control is shown in the lowermost green, dotted curve. Inverse
engineering (IE) performs similarly shown by the blue, dotted curve. The
topmost, black, dot-dashed curve corresponds to an uncontrolled auxiliary
evolution where the performance is several orders of magnitude worse. Floquet
engineered (FE) auxiliary control is shown in the red, solid curve and is
shown to be highly effective. (b) Dynamical gate infidelity for the Hadamard
gate with $\tau\\!=\\!1$ using the same styling as panel (a) to identify the
different control protocols. The inset captures the oscillations present in
the FE driving around the dynamics of the CD approach. (c) We plot the cost,
Eq. (10), of implementing the Hadamard gate vs. total protocol duration for
IE, CD control, FE with $\omega/\omega_{0}=200$ with the same colouring as
before. (d) Final gate infidelity for the Hadamard gate vs. timekeeping error,
$\epsilon$, for over- or under-shooting the intended ramp duration. The total
(ideal) ramp time is $\tau\\!=\\!1$. We show the performance for the linear
(orange), the polynomial (cyan), the sinusoidal (purple) ramps. In all panels
we fix $\omega_{0}\\!=\\!2\pi/\tau$, $\omega\\!=\\!200\omega_{0}$ for the FE
case.

Fig. 2 shows the trajectories for the various control approaches on the Bloch
sphere in the absence of any errors. While the IE qubit (rightmost, yellow)
follows a path on the Bloch sphere and therefore remains pure during the gate
operation, the auxiliary evolutions’ computational qubit cuts through the
Bloch sphere (straight, green line) connecting the initial state
$(\left|+\right\rangle)$ to the final one $(\left|0\right\rangle)$. The latter
observation shows that, although the initial and final states of both the
control and register qubits are pure in the auxiliary evolution approach,
during the dynamics they are mixed, which indicates that they become entangled
during the process. By having a detailed look at the Bloch vectors of the
driven qubits for the auxiliary evolution and IE, it is possible to see that
their $x$ and $z$-components are equal to each other at all times, and only
$y$-components differ (in fact, this component remains identically zero for
the auxiliary evolution’s qubit for this particular gate operation). Thus, the
path that this computational qubit takes is restricted to the $x-z$ plane and
the projection of the path of the IE qubit to the same plane is identical and
therefore, as we demonstrate explicitly below, the performance in terms of the
implementation (in)fidelity, Eq. (9) are identical for the different processes
despite their dynamics being distinct.

Fig. 3(a) shows the final target state infidelity for a Hadamard gate
operation implemented using the three control strategies and for comparison we
also show the uncontrolled auxiliary evolution (black, dashed) for a linear
ramp $\lambda(t)\\!=\\!t/\tau$. As expected, CD and IE both achieve perfect
implementations regardless of the timescale of the drive (bottom-most dotted
lines). The solid red curve corresponds to the FE Hamiltonian, Eq. (19). We
see that despite the approximate nature of the FE approach, provided that the
chosen parameters are within the relevant regime of applicability [56], this
approach is also highly effective in implementing the controlled evolution,
tracking the same dynamics as the CD approach and maintaining an improvement
of several orders of magnitude over the uncontrolled implementation.

In Fig. 3(b) we fix $\tau\\!=\\!1$ and examine the computational qubit’s
approach to the target state during the evolution. This serves to demonstrate
that despite the actual dynamics giving rise to distinct paths, the
effectiveness of all control protocols in terms of gate infidelity is the
same. The inset demonstrates that the FE drive is a remarkably accurate
approximation to the exact drive, showing small oscillations around the
desired trajectory. While Fig. 3(a) and (b) demonstrate that, at the level of
implementation, all control protocols are largely equivalent insofar as they
can faithfully achieve the desired unitary, we will see in the following some
qualitative differences emerge when we consider alternative performance
metrics.

We show the total cost of implementing the controlled gate operation,
quantified using Eq. (10), in Fig 3(c) and for simplicity we consider a linear
ramp for all protocols. To begin with, for very fast driving times,
$\tau\\!\to\\!0$, we are in the opposite limit of adiabatic evolution and the
energetic costs of all control techniques diverge. This observation is in
accordance with previous works [72, 77] which establish that the energetic
resources necessary to drive a system arbitrarily fast while keeping it in the
adiabatic manifold requires to have access to arbitrarily large energetic
resources. Naturally, for longer quench durations we asymptotically reach the
adiabatic limit of the time evolution and the cost decays proportionally to
$1/\tau$. Specifically, in the long time limit the CD cost asymptotically
approaches to $2\sqrt{2}$, which corresponds to an unavoidable energy cost
given by the energy change of bare Hamiltonian of the driven auxiliary qubit,
while for IE the cost vanishes in the asymptotic limit. On the other hand, for
the FE case, the leading term for the cost in the long time limit is
$2\omega/\omega_{0}$ and proportional to the frequency of the Floquet driving,
i.e. how many times the FE dynamics intersects with the true adiabatic
dynamics. This requirement for high frequency driving manifests in a higher
energetic cost for achieving the control.

We now turn our attention to timekeeping errors. For simplicity we focus on
the case of IE, but remark the conclusions are qualitatively similar for both
the auxiliary evolution cases as the dynamical overlap with the target states
for the protocols coincide. In Fig. 3(d) we (arbitrarily) fix $\tau\\!=\\!1$
and consider the performance of the different ramp profiles given by Eqs
(LABEL:ramps) where we allow for the ramp to over- or under-shoot the target
time by a factor proportional to $1\pm\epsilon$. A simple linear ramp is the
most susceptible to this type of error, with the infidelity rapidly growing as
$\epsilon$ increases. Thus, while the linear ramp has some notable advantages,
e.g. resulting in a time-independent control term for IE [cfr. Eq. (23)], this
comes at the expense of requiring potentially costly accurate timekeeping
[78]. In contrast, due to their smooth start and end points the polynomial and
sinusoidal protocols allow for more severe timekeeping errors while still
faithfully implementing the gate, with timing errors of up to $20\%$ still
achieving infidelities $\lesssim 10^{-4}$. This can be understood from the
behavior of these functions at their endpoints where the rate of change of the
associated driving field remains sufficiently small for $\epsilon\\!<\\!0.2$.
As a result, the amplitude of obtaining the desired final state, which is
given by $\sin^{2}(\theta_{f}\lambda/2)$ [see Eq. (16) and (17)] does not
significantly deviate from unity. These results are consistent with
complementary studies of different control problems [79] and demonstrates that
the flatness of the applied ramp around the target is an important feature to
have in terms of the robustness of the protocol.

The physical differences implied by the approaches become most apparent when
considering open system effects on state evolution. Fig. 4 presents our
results on the infidelity between the final state and the target state as a
function of the total gate implementation time, scaled with the decoherence
rate for a dephasing environment and the explicit trajectories of the qubits.
In Fig. 4(a), we plot the trajectories of each qubit for the CD and IE
Hadamard gate when the driven qubits are exposed to a dephasing channel. For
the CD case, not driving the computational register directly can allay much of
the spoiling effects of the environment. This can be understood since the
auxiliary evolution approaches necessitate that the driven system ends in
state $\left|1\right\rangle$, thus while the dephasing will leave the system
in a mixed state, it nevertheless can have a large overlap with the intended
target state of the auxiliary qubit which therefore still exhibits a good
performance. Since we assume the computational qubit in the CD case does not
directly feel the spoiling effects of the dephasing channel, it simply stops
along its ideal trajectory when the auxiliary qubit falls short of its target
state. In contrast, since we drive the computational qubit directly in the IE
case while also exposing it to the dephasing channel, we see that IE qubit
(yellow) starts in the $\left|+\right\rangle$ state and is drawn towards the
$z$-axis, away from its ideal unitary dynamics by the environment. Fig 4(b)
shows the final state infidelity for Hadamard gate as a function of the
dephasing strength. The CD case (blue) displays better final state infidelity
than the IE case (red) for all values of $\tau\gamma$. For larger gates, we
expect that this difference will further widen in favour of the CD case.
Despite the unfavourable cost scaling and relative complexity of the CD
Hamiltonian compared to IE methods, it represents a potential attractive
approach for robust gate implementation.

It is natural to consider extending the above framework to the implementation
of $N$-qubit gates. The preceding analysis can readily be performed for two-
qubit entangling gates, such as the controlled-phase gate [50, 57] (we provide
the explicit forms of the CD and IE Hamiltonians to implement such a gate in
Appendix C). A qualitatively similar behavior is observed: once again the
overall performance in terms of process infidelity is consistent across all
control approaches. Similarly, the effect of time-keeping errors is most
significant for ramps that do not have smooth end points. A notable difference
emerges when considering the energetic cost. While the auxiliary evolution
approaches involve driving only a single qubit, and therefore the cost is
essentially bounded, they can nevertheless facilitate a gate operation on an
arbitrary sized register. However, this comes at the price of a difficult to
realise Hamiltonian, Eq. (1). This is in contrast to the IE approach where the
register is controlled directly and, as might be expected, the complexity and
energy required to implement IE control on multiple qubits scales poorly with
the register size.

Figure 4: (a) Qubit trajectories for the Hadamard gate under a dephasing
channel, Eq. (12), where the channel acts on the driven qubit in each case
with $\tau\gamma\\!=\\!2$. Styling is same as in Fig. 2. As we dephase in the
$z$-basis, both the state of the qubit in the IE case (yellow) and that of the
auxiliary qubit in the CD case (orange) are pulled towards the $z$-axis. The
computational qubit of the CD case (green) is not directly affected by the
channel, and does not deviate from the ideal path, instead stopping along that
trajectory once the auxiliary qubit driving the evolution has decohered. (b)
We show the final gate infidelity, Eq. (9), for the dephasing channel for the
Hadamard gate, with upper, dotted red and lower, dashed blue curves correspond
to the IE and CD cases, respectively.

##  IV Conclusions

We have systematically analyzed the effectiveness CD, IE, and FE methods in
the Hamiltonian implementation of unitary quantum gates. As the figures of
merit for all considered methods, we have put the gate infidelity, the
energetic cost, susceptibility to imperfect timekeeping, and robustness
against the effects of environmental noise at the center of our discussion. We
have focused on the single qubit Hadamard gate and observed that all methods
can faithfully achieve the desired gate, however, show some notable
qualitative differences when examining performance metrics beyond target
fidelities. For example, the energetic overhead of FE is the highest among the
considered methods, due to the high-frequency driving necessary to achieve a
gate operation closer to the ideal. As for the imperfect timekeeping errors of
the desired driving time, we have observed a subtle dependence on how the
Hamiltonian is driven. Smoother ramping of the Hamiltonian results in a more
successful gate implementation, in case the desired driving time is over- or
under-shot. Finally, we have assumed that the driven qubit in CD and IE
methods is in contact with a dephasing environment, and seen that the latter
control technique is more adversely affected by such environmental spoiling
effects than the former due to the fact that in this case computational
degrees of freedom are affected by the noise. A qualitatively similar behavior
can also be observed for a finite temperature dissipative environment. We
considered several commonly employed ramp profiles in order to highlight the
natural robustness that each approach has under the same conditions and to
provide insight into the properties that robust pulses should contain, e.g.
smooth end points. This information can then be used to further enhance
performance through the tailoring of ramp profiles by, e.g. optimal control
techniques. However, the cost functionals can be optimized over multiple
metrics, such as energetic cost, pulse bandwidth, and robustness to noise to
name a few, and thus this rapidly becomes a complex problem.

We finally offer some comments on the applicability of these general
Hamiltonians in light of recent experimental work has been done to implement
transitionless (or superadiabatic) gates on promising candidate architectures,
such as NV centres [80], superconducting qubits [81, 82], and rare-earth ions
[83]. Indeed the possible universal gate sets generated by the inverse
engineering case discussed in this work presents an attractive prospect for
applicability, owing to the relatively simple forms and interactions present
and the potential to drive them with time-independent control fields. The
counterdiabatic driving case represents a departure from the typical approach
to implementing a gate as it makes use of an additional auxiliary resource to
mediate the driving. One may view gates in this setting as controlled gates,
the Hadamard gate is perfectly implemented on the register qubit if the
auxiliary qubit is driven to $\left|1\right\rangle$ and the identity is
performed on it if the auxiliary is found in $\left|0\right\rangle$. This
implementation therefore requires from a platform that can readily achieve
controlled-gates, e.g. trapped-ion systems [8]. That this method involves
inducing a phase difference between states for the computational qubit, and is
more robust to noise than the direct driving approach is reminiscent of
superadiabatic geometric quantum gates [84]. Indeed, utilising auxiliary
evolution to achieve the “superadiabatic” part of these processes could lend
further robustness to these proposals.

###### Acknowledgements.

EC acknowledges support from the Irish Research Council Project ID No.
GOIPG/2020/356 and the Thomas Preston Scholarship. BÇ is supported by The
Scientific and Technological Research Council of Turkey (TUBITAK) under Grant
No. 121F246. SC is acknowledges support from the Science Foundation Ireland
Starting Investigator Research Grant “SpeedDemon” No. 18/SIRG/5508 and the
Alexander von Humboldt Foundation. The authors thank Anthony Kiely and
Alessandro Ferraro for insightful discussions.

##  Appendix A Excited State Driving

For a single-qubit gate we assume that the auxiliary qubit is initially in the
excited state, which implies that we initialise the system in the state
$\left|\Psi_{i}\right\rangle=\left(\alpha\left|n_{+}\right\rangle+\beta\left|n_{-}\right\rangle\right)\otimes\left|1\right\rangle$.
Evolving adiabatically with the usual Hamiltonian gives the final state

$\left|\Psi_{f}\right\rangle=\alpha\left|n_{+}\right\rangle\otimes\left|\epsilon_{0}^{e}\right\rangle+\beta\left|n_{-}\right\rangle\otimes\left|\epsilon_{\phi_{-}}^{e}\right\rangle,$
(24)

where the excited state of the time dependent Hamiltonian is

$\left|\epsilon_{\phi}^{e}\right\rangle=-\text{e}^{i\phi}\sin\left(\frac{\theta_{f}\lambda}{2}\right)\left|0\right\rangle+\cos\left(\frac{\theta_{f}\lambda}{2}\right)\left|1\right\rangle.$
(25)

Though the eigenvector has a natural $U(1)$ symmetry, a local phase appears on
$\left|0\right\rangle$ due to our specification of the initial state of the
dynamics at $\lambda=0$. The translation $\phi\to-\phi$ allows us to perform
our gate with our auxiliary qubit initially in the excited state, returning it
to its ground state at the end of the process, potentially to be reused.

##  Appendix B Explicit form of the single qubit IE Hamiltonian

The driving Hamiltonian can be identified in the following way [57]

$H(t)=\frac{1}{2}\vec{\omega}(t)\cdot\vec{\sigma},$ (26)

where the vector components are given as

$\displaystyle\omega_{x}(t)=$
$\displaystyle(\cos\pi\lambda-1)\dot{\varphi}\cos\varphi\cos\vartheta\sin\vartheta$
$\displaystyle+[\dot{\varphi}\sin\vartheta\sin\pi\lambda+(\cos\pi\lambda-1)\dot{\vartheta}]\sin\varphi$
$\displaystyle+(\dot{\vartheta}\cos\vartheta\sin\pi\lambda+\pi\dot{\lambda}\sin\vartheta)\cos\varphi,$
$\displaystyle\omega_{y}(t)=$
$\displaystyle(\cos\pi\lambda-1)\dot{\varphi}\sin\varphi\sin\vartheta\cos\vartheta$
$\displaystyle+[\dot{\varphi}\sin\vartheta\sin\pi\lambda-(\cos\pi\lambda-1)\dot{\vartheta}]\cos\varphi$
$\displaystyle+(\dot{\vartheta}\cos\vartheta\sin\pi\lambda+\pi\dot{\lambda}\sin\vartheta)\sin\varphi,$
$\displaystyle\omega_{z}(t)$
$\displaystyle=-\dot{\vartheta}\sin\vartheta\sin\pi\lambda-(\cos\pi\lambda-1)\dot{\varphi}\sin^{2}\vartheta+\pi\dot{\lambda}\cos\vartheta.$
(27)

##  Appendix C Examples of a two-qubit gate

One typical example is the controlled-phase gate where we have two
computational qubits, namely control and target, and depending on the state of
the control qubit we apply a phase shift operation on the target qubit.
Explicitly, if we take the initial state of the computational qubits as

$\left|\psi_{i}\right\rangle=\alpha\left|0,n_{+}\right\rangle+\beta\left|0,n_{-}\right\rangle+\gamma\left|1,n_{+}\right\rangle+\delta\left|1,n_{-}\right\rangle,$
(28)

then performing the controlled-phase gate yields the following output state

$\left|\psi_{f}\right\rangle=\alpha\left|0,n_{+}\right\rangle+\beta\left|0,n_{-}\right\rangle+\gamma\left|1,n_{+}\right\rangle+\text{e}^{i\phi}\delta\left|1,n_{-}\right\rangle.$
(29)

we focus on the application of the controlled-$Z$ gate for which we have
$\phi\\!=\\!\pi$. Within the auxiliary evolution framework, the gate operation
can again be implemented by performing a drive on the additional ancilla
qubit. This requires a Hamiltonian that implements the local phase-shift onto
the $\left|1,n_{-}\right\rangle$ subspace while keeping the others fixed,
which has the following form [50]

$\displaystyle H_{C\pi}^{AE}(t)=$ $\displaystyle\left(|0\rangle\langle
0|\otimes\mathbf{1}+|1\rangle\langle 1|\otimes|0\rangle\langle
0|\right)\otimes H_{0}(t)$ (30) $\displaystyle+|1\rangle\langle
1|\otimes|1\rangle\langle 1|\otimes H_{\pi}(t),$

where the time-dependent driving Hamiltonians $H_{0}(t)$ and $H_{\pi}(t)$
applied on the ancilla qubit are given as in Eq. (2). As before, both CD and
Floquet control can be used to ensure adiabatic dynamics.

Similarly we can define the same gate with IE method

$U_{2}(t)=\sum_{k=1,2}|m_{k,+}(t)\rangle\langle
m_{k,+}(t)|+\text{e}^{i\pi\lambda_{k}(t)}|m_{k,-}(t)\rangle\langle
m_{k,-}(t)|.$ (31)

The evolution basis is similarly defined

$\left|m_{k,+}(t)\right\rangle=\cos[\vartheta(t)/2]\left|k-1,0\right\rangle+\text{e}^{i\varphi_{k}(t)}\sin[\vartheta(t)/2]\left|k-1,1\right\rangle,$
(32)
$\left|m_{k,-}(t)\right\rangle=\text{e}^{i\varphi_{k}(t)}\cos[\vartheta(t)/2]\left|k-1,1\right\rangle-\sin[\vartheta(t)/2]\left|k-1,0\right\rangle.$
(33)

We now have six parameters, with the restriction that $\lambda_{k}(0)=0$. All
appear in the final state of the system under the action of the unitary.
Through a suitable choice of the parameters we can design the Hamiltonian to
implement the desired unitary dynamics. For example, adopting the general
formalism above, we obtain the desired IE Hamiltonian that applies the
controlled-$Z$ operation as follows [57]

$H_{C\pi}^{IE}(t)=\frac{\pi\dot{\lambda}(t)}{4}(\mathbf{1}\otimes\sigma_{z}+\sigma_{z}\otimes\mathbf{1}-\sigma_{z}\otimes\sigma_{z}).$
(34)

Note that the implementation of the above Hamiltonian requires a dephasing
($Z-Z$) type interaction between the target and control qubits.

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