# Pulse-based variational quantum optimization and meta-learning in
superconducting circuit

Yapeng Wang  Institute for Quantum Science and Technology, Department of
Physics, Shanghai University, Shanghai 200444, China  Yongcheng Ding
<EMAIL_ADDRESS>Institute for Quantum Science and Technology, Department
of Physics, Shanghai University, Shanghai 200444, China  Department of
Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644,
48080 Bilbao, Spain  Francisco Andrés Cárdenas-López  f.cardeans.lopez@fz-
juelich.de Forschungszentrum Jülich GmbH, Peter Grünberg Institute, Quantum
Control (PGI-8), Jülich, Germany  Xi Chen<EMAIL_ADDRESS>Instituto de
Ciencia de Materiales de Madrid (CSIC), Cantoblanco, E-28049 Madrid, Spain

###### Abstract

Solving optimization problems using variational algorithms stands out as a
crucial application for noisy intermediate-scale devices. Instead of
constructing gate-based quantum computers, our focus centers on designing
variational quantum algorithms within the analog paradigm. This involves
optimizing parameters that directly control pulses, driving quantum states
towards target states without the necessity of compiling a quantum circuit. In
this work, we introduce pulse-based variational quantum optimization (PBVQO)
as a hardware-level framework. We illustrate the framework by optimizing
external fluxes on superconducting quantum interference devices, effectively
driving the wave function of this specific quantum architecture to the ground
state of an encoded problem Hamiltonian. Given that the performance of
variational algorithms heavily relies on appropriate initial parameters, we
introduce a global optimizer as a meta-learning technique to tackle a simple
problem. The synergy between PBVQO and meta-learning provides an advantage
over conventional gate-based variational algorithms.

##  I Introduction

Quantum computing has emerged as a promising field with the potential to
revolutionize computational methodologies. Unique properties such as
superposition and quantum entanglement in quantum mechanics offer theoretical
speedups by embedding problems into quantum systems. With advancements in
experimental techniques, we now find ourselves in the noisy intermediate-scale
quantum (NISQ) era, where quantum computers demonstrate capabilities for
solving complex problems [1]. Among the most intriguing applications of
quantum computing in this era is quantum optimization [2]. Broadly, quantum
optimization entails utilizing quantum devices to determine the ground state
of a Hamiltonian that encodes a classical cost function. It finds its roots in
the adiabatic theorem, wherein a system is prepared in the ground state of a
simple Hamiltonian and evolved to the problem Hamiltonian, guaranteeing
convergence to the problem’s ground state if adiabatic criteria are met. This
concept gave rise to quantum annealing and the development of quantum
annealers within the analog quantum computing paradigm [3, 4, 5], obviating
the need for constructing quantum gates. Subsequently, it inspired the quantum
approximate optimization algorithm (QAOA) [6, 7] as one of the most renowned
variational quantum algorithms (VQAs) [8, 9]. Through incooperation with
classical optimizers, gate parameters within the quantum circuit are
iteratively optimized to minimize a cost function, typically the energy
expectation of a Hamiltonian. This showcases the utility of state-of-the-art
quantum devices across interdisciplinary fields [10, 11, 12, 13].

Various approaches have been proposed to enhance the performance of QAOA. It
operates as a bang-bang control, alternating between evolving the mixing and
problem Hamiltonians [14]. Optimal control theory enables analytical solutions
for optimal procedures within bounded operating times, featuring square-pulse-
based QAOA on two sides and smooth annealing in between [15]. This resembles
mimicking QAOA in the digital-analog quantum computing paradigm [16, 17],
where quantum gates and specific Hamiltonian evolutions coexist. Another
perspective focuses on the circuit level, akin to quantum machine learning
[18], proposing different structures as ansatz to enhance model performance.
For instance, the quantum alternating operator ansatz replaces trivial
transverse local operators with entangling operators tailored to specific
problems [19]. Additionally, adaptive QAOA dynamically selects the mixer from
a pre-selected operator pool [20]. Moreover, shortcuts-to-adiabaticity has
been introduced from quantum control to quantum computing [21], as parameter
searching in QAOA is equivalent to optimizing a discretized annealing
schedule. Approximate counterdiabatic terms can be easily implemented in
quantum circuits to suppress energy excitation, thereby enhancing the
performance of variational quantum optimization [22, 23, 24]. These recently
proposed QAOA variants, combined with classical optimizers, have been
benchmarked up to 28 qubits [25].

Here, we aim to introduce an alternative approach to variational quantum
optimization by concentrating on optimizing parameters that characterize
control pulses. The advantages of pulse-based methods over traditional gate-
based approaches are becoming increasingly evident. Pulse-based optimization
offers faster state preparation, simpler implementation, and greater freedom
in navigating the quantum state space. Recently, pulse-based models have been
proposed to enhance variational quantum algorithms, such as extending
variational quantum eigensolver (VQE) to Ctrl-VQE [26, 27], modifying ansatz
to PANSATZ [28], and exploring applications in quantum machine learning [29].
Previous related work in quantum optimization has explored analog versions of
QAOA and realized variational coherent annealing by introducing auxiliary
terms [30, 31]. Additionally, a comparison between gate-model and pulse-based
models has been conducted on bipotent quantum architectures where quantum
gates are not readily available for arbitrary qubits [32]. Building upon these
previous results, we propose pulse-based variational quantum optimization
(PBVQO), with the objective of solving quantum optimization problems at the
level of control pulses on realistic quantum devices. Furthermore, we
recognize that local optimization of variational quantum algorithms can be
computationally expensive, requiring additional circuit depth and numerous
measurements. By initializing parameters close to global minima, the number of
optimization iterations can be significantly reduced. This observation
motivates the exploration of meta-learning techniques, whose feasibility has
been demonstrated in gate-model QAOA with counterdiabaticity [33].

The remainder of the paper is organized as follows: In Sec. II, we introduce
the concept of PBVQO, which is a hardware-agnostic framework requiring single-
qubit operations and tunable two-body interactions. Then we test our method by
combining it with a digital-analog superconducting circuit architecture in
Sec. III.  In Sec. IV, we demonstrate our protocol through numerical
experiments aimed at solving the MAX-CUT problem with up to 8 qubits. To
tackle optimization challenges, we utilize a combination of BFGS, a quasi-
Newtonian optimizer for parameter updates, and genetic algorithms to pre-
select suitable parameter initializations. Additionally, in Sec. V, we present
a comparative analysis with conventional gate-based QAOA to highlight the
advantages of our approach. Finally, the conclusion and outlook are presented
in Sec. VI.

##  II Pulse-based optimization

We exemplify PBVQO by solving the MAX-CUT problem. At the hardware level,  we
assume that the hardware architecture can only directly realize two-body
neighboring interaction as an Ising spin chain or an Ising spin ring by
connecting the first and last qubit with periodic boundary conditions. To
avoid searching for composite pulses to implement effective many-body
interactions, we utilize the architecture to solve 2-regular graphs without
any disconnected cycles, meaning the degree of each node is 2, which can be
easily embedded into the architecture by topology. Thus, the problem is
equivalent to finding the ground state of an antiferromagnetic Ising ring.  We
assume that the effective hardware Hamiltonian includes only local spin
operator and neighbouring interaction, which can be expressed as:

$H_{\textbf{PBVQO}}=\sum_{j}\frac{\omega_{j}}{2}\sigma_{j}^{z}+P(t)\sigma_{j}^{y}\sigma_{j+1}^{y},$
(1)

where $P(t)$ is the pulse to be optimized, and
$\sigma_{j}^{y}\sigma_{j+1}^{y}$ can be replaced by other interaction terms
depending on the capabilities of the platform. This effective Hamiltonian is
feasible in various quantum devices without loss of generality. In principle,
the pulse ansatz $P(t)$ can be of any form, as long as it can be implemented
on the specefic quantum device. However, in practical implementation, not all
pulses are realizable, so a filter $F[P(t)]$ is required for pulse. For
simplicity, we illustrate the PBVQO framework by assuming that the optimal
pulse can be expressed as a series of trigonometric functions

$P(t)=\sum_{i=1}^{n}A_{i}\sin\bigg{[}(2i-1)\pi t+\phi_{i}\bigg{]},$ (2)

where $A_{i}$ are the pulse amplitudes, $t\in[0,T]$ is the operation time, and
$\phi_{i}$ are the phases.  The trigonometric functions resemble the annealing
schedule used in variational quantum annealing [34], though the mechanism is
not exactly the same. In this context, the series allows for the expression of
an arbitrary function by providing sufficient freedom with amplitudes and
phases.

On the contrary, the conventional QAOA for MAX-CUT problems prepares the
system in the product state $|+\rangle^{\otimes N}$, which is an eigenstate of
the mixing Hamiltonian (mixer) $H_{\text{mix}}=\sum_{j}\sigma_{j}^{x}$, and
then let the system evolve by alternate application of the problem Hamiltonian
$H_{p}=\sum_{\langle i,j\rangle\in\mathcal{G}}\sigma_{i}^{z}\sigma_{j}^{z}$
($\mathcal{G}$ is the graph to be solved) with the mixing ones for $p$ rounds.
In other words, we compute the evolution of the system through the product
$|\Psi_{f}\rangle=\prod_{j}^{p}\exp(-i\beta_{j}H_{\text{mix}})\exp(-i\gamma_{j}H_{p})|+\rangle^{\otimes
N}$. QAOA aims for finding the set of parameters $\\{\beta_{j},\gamma_{j}\\}$
that minimize the averaged energy of the problem Hamiltonian,
$\langle\Psi_{f}|H_{p}|\Psi_{f}\rangle$ so that $|\Psi_{f}\rangle$ be the
ground state of the problem Hamiltonian. Measuring the $Z$-Pauli strings of
the problem, we obtain the max-cut of the graph, if the optimization procedure
was successful. From the annealing perspective, finding the parameters
$\\{\beta_{j},\gamma_{j}\\}$ is equivalent to find the optimal schedule
function for a digitalized quantum annealing process

$H_{\textbf{QA}}=B(t)\sum_{j}\sigma_{j}^{x}+\Gamma(t)\sum_{\langle
i,j\rangle\in\mathcal{G}}\sigma_{i}^{z}\sigma_{j}^{z},$ (3)

where $\beta_{i}^{*}=B(i\delta t)\delta t$ and $\gamma_{i}^{*}=\Gamma(i\delta
t)\delta t$ are the optimal parameters describing the schedule.

We realize that in PBVQO, it is not necessary to prepare the system in the
state $|+\rangle^{\otimes N}$. Instead, it is more advantageous to start in
the state $|0\rangle^{\otimes N}$, which is an eigenstate of the mixing
Hamiltonian in our architecture, $\sum_{j}{(\omega_{j}}/{2})\sigma_{j}^{z}$.
Unlike the typical QAOA realization, we do not switch-off the mixing
Hamiltonian. Moreover, we will consider the two-body interactions to be
$\sum_{\langle i,j\rangle}\sigma^{y}_{i}\sigma^{y}_{j}$, which remain always
turned-on. Consquentely, we need to engineer the pulse sequences such that
both Hamiltonian encodes the ground state of the MAX-CUT Hamiltonian
$\tilde{H}_{p}=\sum_{\langle
i,j\rangle\in\mathcal{G}}\sigma_{i}^{x}\sigma_{j}^{x}$.

The workflow of PBVQO consists of the following steps: We initialize the pulse
ansatz by randomly selecting both amplitudes and phases. Then, the pulse is
filtered to obtain $F[P(t)]$. And calculate the dynamics of the mixing and
problem Hamiltonian, resulting in the state $|\Psi_{f}\rangle$. Afterwards, we
compute the average of the energy
$\langle\Psi_{f}|\tilde{H}_{p}|\Psi_{f}\rangle$ and we update the parameters
$\\{A_{i},\phi_{i}\\}$ using a classical optimizer, such as BFGS. When the
optimization converges according to our hyperparameters, we measure the system
on the $x$-axis, obtaining the classical configuration of the MAX-CUT problem.
We evaluate the PBVQO performance by computing the error rate $R$, defined as

$R=\biggm{|}\frac{\langle\Psi_{f}^{*}|\tilde{H}_{p}|\Psi_{f}^{*}\rangle-
E_{g}}{E_{g}}\biggm{|},$ (4)

where $E_{g}$ is the ground energy of $\tilde{H}_{p}$ obtained by numerically
diagonalizing the Hamiltonian.

##  III Hardware architecture

Figure 1: Schematic circuit representation of the model implementing the
PBVQO. Two charge qubits are coupled through a grounded SQUID. The only
tunable parameter is the external flux $\varphi_{\text{ext}}$ through the
SQUID, indicated by the red shaded region.

We hereby introduce the superconducting circuit architecture employed for
variational quantum optimization, which was originally proposed for digital-
analog quantum simulation [35]. The minimal structure consists of two charge
qubits and a grounded superconducting quantum interference device (SQUID) as a
coupler. Each charge qubit consists of a capacitance $C_{J\ell}$ parallel
connected to a Josephson junction with energy $E_{J\ell}$ and biased with an
external voltage source $V_{g\ell}$ as depicted in Fig. 1. Moreover, the SQUID
is fromed a closed loop embedded with two identical Josephson junctions of
capacitances $C_{s}/2$ and Josephson energies $E_{Js}/2$, respectively. Also,
the loop is threaded by an external magnetic flux $\varphi_{ext}(t)$. We
denote $\Phi_{1}$, $\Phi_{2}$, and $\Phi_{s}$ as the fluxes describing the
charge qubits and the SQUID respectively. The circuit Lagrangian reads

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\sum_{j=1,2,s}\bigg{[}\frac{C_{J_{j}}\dot{\Phi}^{2}_{j}}{2}+\frac{C_{g_{j}}(\dot{\Phi}_{j}-V_{g_{j}})^{2}}{2}+\frac{C_{c}(\dot{\Phi}_{j}-\dot{\Phi}_{s})^{2}}{2}$
(5) $\displaystyle+$ $\displaystyle
E_{J_{j}}\cos(\varphi_{j})\bigg{]}+\frac{C_{J_{s}}\dot{\Phi}^{2}_{s}}{2}+E_{J_{s}}^{\rm{eff}}[\varphi_{ext}(t)]\cos(\varphi_{s}),$

here the overdot notation represents derivation with respect to time.
Moreover,
$E_{J_{s}}^{\rm{eff}}[\varphi_{ext}(t)]=2E_{J_{s}}|\cos[\varphi_{ext}(t)]|$ is
the tunable Josephson energy of the SQUID with
$\varphi_{j}=\Phi_{j}/\varphi_{0}$ being the superconducting phase.

The idea of this architecture is to manipulate the coupling strength between
both charge qubit by modulating the external phase $\varphi_{ext}(t)$ without
exciting the SQUID, in this case we need to assume that the latter operates in
the so-called high-plasma frequency and low-impedance approximation [36, 37]
where it is possible to factorize the SQUID from the system dynamics similar
to perturbative diagonalization based on  Schrieffer-Wolff transformation
[38]. Then for eliminating the SQUID degree of freedom, we compute the Euler-
Lagrange equation of $\mathcal{L}$ obtaining

$\displaystyle
C_{\Sigma_{1}}\ddot{\Phi}_{1}-C_{c}\ddot{\Phi}_{s}+\frac{E_{J_{1}}}{\varphi_{0}}\sin(\varphi_{1})=0,$
(6a) $\displaystyle
C_{\Sigma_{2}}\ddot{\Phi}_{2}-C_{c}\ddot{\Phi}_{s}+\frac{E_{J_{2}}}{\varphi_{0}}\sin(\varphi_{2})=0,$
(6b) $\displaystyle
C_{\Sigma_{s}}\ddot{\Phi}_{s}-C_{c}(\ddot{\Phi}_{1}+\ddot{\Phi}_{2})+\frac{E_{J_{s}}^{\rm{eff}}[\varphi_{ext}(t)]}{\varphi_{0}}\sin(\varphi_{s})=0.$
(6c)

Here, $C_{\Sigma_{j}}$ is the total capacitance of each nodes. The high-plasma
frequency allows us to assume that the variation on the fluxes on the charge
qubits are larger than in the SQUID i.e.,
$\ddot{\Phi}_{1(2)}\gg\ddot{\Phi}_{s}$, moreover, the low-impedance regime
refers to almost no current flows on the SQUID such that we can operate the
SQUID  in the linearized regime [36]. These conditions permit us to express
$\varphi_{s}$ in terms of the other node variables.

$\displaystyle{\varphi_{s}}$ $\displaystyle=$
$\displaystyle\sum_{j}{-\frac{C_{c}E_{J_{j}}\sin{(\varphi_{j})}}{E_{J_{s}}^{\textrm{eff}}C_{\Sigma_{j}}}}.$
(7)

Now, we have to perform the same reasoning to the SQUID charge. To do so, we
compute the canonical conjugate momenta
$\vec{Q}=\vec{\nabla}_{\vec{\Phi}}\mathcal{L}$

$\displaystyle\vec{Q}=\begin{pmatrix}C_{\Sigma_{1}}\dot{\Phi}_{1}-C_{c}\dot{\Phi}_{s}-C_{g_{1}}V_{g_{1}}\\\
C_{\Sigma_{2}}\dot{\Phi}_{2}-C_{c}\dot{\Phi}_{s}-C_{g_{2}}V_{g_{2}}\\\
C_{\Sigma_{s}}\dot{\Phi}_{s}-C_{c}(\dot{\Phi}_{1}+\dot{\Phi}_{2})\end{pmatrix},$
(8)

and apply the high-plasma approximation for eliminating the terms proportional
to $\dot{\Phi}_{s}$, leading to

$\displaystyle Q_{s}$ $\displaystyle=$ $\displaystyle-
C_{c}\sum_{j}\frac{Q_{j}+C_{g_{j}}V_{g_{j}}}{C_{\Sigma_{j}}}.$ (9)

The conditions in Eq. (7) and in Eq. (9) eliminated the SQUID degree of
freedom such that the effective qubit-qubit Hamiltonian can be expressed as

$H=\sum_{j=1,2}\big{[}\mathcal{H}_{j}+\mathcal{H}_{{\rm{ctrl},j}}\big{]}+\mathcal{H}_{\text{c}},$
(10)

where $\mathcal{H}_{j}$ and are the effective charge qubit and control
Hamiltonian, whereas $\mathcal{H}_{\text{c}}$ corresponds to the interaction
Hamiltonian given by

$\displaystyle\mathcal{H}_{j}$ $\displaystyle=$
$\displaystyle\frac{Q_{j}^{2}}{2\tilde{C}_{J_{j}}}-E_{J_{j}}\cos{\varphi_{j}}+g[\varphi_{ext}(t)]\sin^{2}{(\varphi_{j})},~{}~{}~{}~{}$
(11a) $\displaystyle\mathcal{H}_{{\rm{ctrl},j}}$ $\displaystyle=$
$\displaystyle\tilde{n}_{g_{j}}(t)Q_{j},~{}~{}~{}~{}$ (11b)
$\displaystyle\mathcal{H}_{\text{c}}$ $\displaystyle=$ $\displaystyle
g[\varphi_{ext}(t)]\sin{(\varphi_{1})}\sin{(\varphi_{2})}.$ (11c)

where we define the effective capacitaces $\tilde{C}_{J_{j}}$ and
$\tilde{C}_{J_{s}}$ as

$\displaystyle\tilde{C}_{J_{j}}$ $\displaystyle=$
$\displaystyle\frac{C_{*}^{3}}{C_{k}(2C_{c}+C_{s})+C_{c}(C_{c}+C_{s})},$ (12)
$\displaystyle\quad\tilde{C}_{J_{s}}$ $\displaystyle=$
$\displaystyle\frac{C_{c}C_{*}^{3}}{(C_{c}+C_{1})(C_{c}+C_{2})},$ (13)

where
$C_{*}^{3}=C_{c}(C_{1}+C_{2})(C_{s}+C_{c})+C_{c}^{2}C_{s}+C_{1}C_{2}(2C_{c}+C_{s})$,
and $C_{j}=C_{g_{j}}+C_{J_{j}}$ ($j\in\\{1,2\\}$ and $k$ is the other
element). The dimensionless gate charges are given by

$\displaystyle\tilde{n}_{g_{j}}(t)$ $\displaystyle=$
$\displaystyle-\frac{C_{g_{j}}}{2e\tilde{C}_{J_{j}}}V_{g_{j}}-\frac{\tilde{C}_{J_{j}}C_{c}^{2}C_{g_{k}}}{2eC_{*}^{3}\tilde{C}_{J_{j}}}V_{g_{k}}.$
(14)

Finally, $g[\varphi_{ext}(t)]$ is the effective coupling strength defined as

$g[\varphi_{ext}(t)]=\frac{C^{2}_{c}E_{J_{1}}E_{J_{2}}}{E_{J_{s}}^{\textrm{eff}}(C_{1}+C_{c})(C_{2}+C_{c})}.$
(15)

We quantize the Hamiltonian by promoting quantum operators
$\hat{Q}_{j}=-2e\hat{n}_{j}$ and $\hat{\varphi}_{j}$ satisfying canonical
commutation relation $[\hat{n}_{j},\exp(\pm i\hat{\varphi}_{j})]=\exp(\pm
i\hat{\varphi}_{j})$, substituting Eq. (9) and (7) into the Hamiltonian (10),
we obtain for the charge qubit Hamiltonian

$\displaystyle\mathcal{H}_{j}=4E_{Cj}n_{j}^{2}-E_{J_{j}}\cos{\varphi_{j}}+g[\varphi_{ext}(t)]\sin^{2}{(\varphi_{j})},~{}~{}~{}$
(16)

where $E_{Cj}=e^{2}/2\tilde{C}_{J_{j}}$ is the charge energy. In the charge
basis representation, the Hamiltonian in Eq. (10) can be expressed in the two-
level approximation as follows

$\displaystyle
H=\sum_{j=1,2}\bigg{[}\frac{\omega_{j}}{2}\sigma_{j}^{z}+\Omega_{j}(t)(n_{0,1}|g\rangle\langle
e|+\text{c.c.})\bigg{]}+\frac{g[\varphi_{ext}(t)]}{4}\sigma_{1}^{y}\sigma_{2}^{y},$
(17)

where $\omega_{j}=E_{J_{j}}$ is the transition frequency of the $j$th charge
qubit. The effective circuit Hamiltonian has three different contributions;
the free energy term setting the transition frequency, the single-qubit
interaction with drive amplitude $\Omega_{j}(t)$ and the tunable interaction
term, note as the matrix element of the charge operator are real, we can
implement an X or Y rotation just by changing the relative phase of the
external charge offset $\tilde{n}_{g_{j}}(t)$. Additionally, in this
architecture, depending on the scheduling of the external voltages
$V_{g_{j}}(t)$ and the external flux $\varphi_{ext}(t)$ we are able to
dynamically switch off the two-qubit interaction while implementing the
single-qubit rotations and vice versa, allowing to implement the digital-
analog QAOA dynamics.

On the other hand, the superconducting architecture discussed here is also
applicable for implementing gate models by selecting drive frequencies that
activate/deactivate certain process in the Hamiltonian in Eq. (17). This can
be achieved by assuming that the external flux contains a constant DC and time
varying AC component of the form

$\varphi_{\text{ext}}(t)=\varphi_{DC}+\varphi_{AC}(t),$ (18)

here, we assume that $\varphi_{AC}(t)$ can be decomposed using the pulse
defined in Eq. (2)

$\varphi_{AC}(t)=A_{1}\cos{(\nu_{1}t+\tilde{\phi}_{1})}+A_{2}\cos{(\nu_{2}t+\tilde{\phi}_{2})},$
(19)

here, we impose the condition $|A_{1}|,|A_{2}|\ll|\varphi_{DC}|$ such that the
effective inductance can be expressed as

$\displaystyle\frac{1}{E_{J_{s}}^{\textrm{eff}}}$ $\displaystyle\approx$
$\displaystyle\frac{1}{\bar{E}_{J_{s}}}\left[1+\frac{\sin{({\varphi}_{DC})}}{\cos{({\varphi}_{DC}})}{\varphi}_{AC}(t)\right],$
$\displaystyle\bar{E}_{J_{s}}$ $\displaystyle=$ $\displaystyle
2E_{J_{s}}\left|\cos{({\varphi}_{DC})}\right|.$ (20)

Therefore, we reformulate the Hamiltonian (17) as

$H=\sum_{j=1,2}\bigg{[}\frac{\omega_{j}}{2}\sigma_{j}^{z}+\frac{\Omega_{j}(t)}{2}\sigma_{j}^{x}\bigg{]}+\left[g_{0}+g_{1}\varphi_{AC}(t)\right]\sigma_{1}^{y}\sigma_{2}^{y},$
(21)

where the always-on and tunable coupling strength reads

$\displaystyle g_{0}$ $\displaystyle=$
$\displaystyle\frac{C_{c}^{2}E_{J_{1}}E_{J_{2}}}{4(C_{1}+C_{c})(C_{2}+C_{c})\bar{E}_{J_{s}}},$
$\displaystyle g_{1}$ $\displaystyle=$
$\displaystyle\frac{C_{c}^{2}E_{J_{1}}E_{J_{2}}}{4(C_{1}+C_{c})(C_{2}+C_{c})\bar{E}_{J_{s}}}\frac{\sin{({\varphi}_{DC})}}{\cos{({\varphi}_{DC})}}\,.$
(22)

We now proceed by expressing the Hamiltonian in the interaction picture,
assumign that there is no drive $\Omega_{j}(t)=0$, and performing the rotating
wave approximation obtaining the Hamiltonian

$\displaystyle\hat{\mathcal{H}}_{I}=\frac{g_{1}}{4}\left(M_{-}\sigma_{1}^{x}\sigma_{2}^{x}+M_{+}\sigma_{1}^{y}\sigma_{2}^{y}-N_{+}\sigma_{1}^{x}\sigma_{2}^{y}+N_{-}\sigma_{1}^{y}\sigma_{2}^{x}\right),$
(23)

where $M_{\pm}=A_{1}\cos{\tilde{\phi}_{1}}\pm A_{2}\cos{\tilde{\phi}_{2}}$ and
$N_{\pm}=A_{1}\sin{\tilde{\phi}_{1}}\pm A_{2}\sin{\tilde{\phi}_{2}}$. This
approach allows us to engineer the effective two-body interaction required in
the algorithm by the adequate selection of the amplitudes and phases of the
external flux $\varphi_{AC}(t)$. For instances, we can engineer the standard
QAOA the mixing Hamiltonian $XX$ by setting
$\tilde{\phi}_{1}=\tilde{\phi}_{2}=0$ and $A_{1}=-A_{2}$ obtaining

$H_{XX}=\frac{g_{1}A_{1}}{2}\sigma_{1}^{x}\sigma_{2}^{x}=G\frac{A_{1}}{2\cos(\varphi_{DC})}\frac{\sin(\varphi_{DC})}{\cos(\varphi_{DC})}\sigma_{1}^{x}\sigma_{2}^{x}.$
(24)

Thus, this architecture allows us to implement directly the problem-
Hamiltonian dynamics $U(-i\beta\sigma_{1}^{x}\sigma_{2}^{x})$, making the
approach more efficient than compile the problem in terms of single-qubit
rotations and entangling CNOT gates. The price to pay is that the condition
$|A_{1}|\ll|\varphi_{DC}|$ bounds our maximal gating time, making the
algorithm susceptible to noise and decoherence. A naive approach for
accelerating the gating time relies upon setting DC component close to
$\varphi_{DC}=\pi/2$. Nevertheless, such approach is not valid because the
energy spectrum of the system shrinks, leading to divergencies [39]. Another
approach is to use the architecture for engineering the counter-diabatic (CD)
Hamiltonian of the QAOA [23, 22]: $XY+YX$ for other selection of the pulse
parameters. Thus, the scheduling of the whole QAOA may include the pulses for
the single-qubit rotations, the action of the entangling gates and the
inclusion of the CD evolution.

Notice that from the Hamiltonian perspective, we can implement PBVQO by
setting $P(t)=g[\varphi_{ext}(t)]/4$. However, the proposed trigonometrical
ansatz does not match such coupling because $g[\varphi_{ext}(t)]$ is positive.
Moreover, we may encounter errors due to the always-on coupling $g_{0}$ during
PBVQO. To address these issues, we can filter our pulses considering such
experimental constrains under the relation:

$\displaystyle F[P(t)]=\begin{cases}~{}G,&~{}~{}-G\leq P(t)<G\\\
|P(t)|,&~{}~{}\text{otherwise}\end{cases},$ (25)

where $G={C^{2}_{c}E_{J_{1}}E_{J_{2}}}/{8E_{J_{s}}(C_{1}+C_{c})(C_{2}+C_{c})}$
is the bound on the coupling strength.

##  IV Numerical experiments

###  IV.1 Baseline

PBVQO imposes no limit on the operation time; thus, control pulses can be
periodic in time, mimicking Floquet engineering protocols. In light of this,
our first step is to analyze the role of the pulse duration $T$ in our model’s
performances, making it a free parameter in our optimization. We employ the
BFGS algorithm as the classical optimizer, a quasi-Newton method for solving
unconstrained nonlinear optimization problems. It achieves gradient descent by
preconditioning the gradient without explicitly requiring the Hessian matrix.

Figure 2: (a) Error rate
$R=|({\langle\Psi_{f}^{*}|\tilde{H}_{p}|\Psi_{f}^{*}\rangle-E_{g}})/{E_{g}}|$
of PBVQO for 2-regular graph MAX-CUT problem with 8 qubits. The boxplot
represents the inter-quartile range (IQR) of the data, with the box spanning
from the first quartile to the third quartile, and a line indicating the
median. Whiskers extend from the box to the farthest data points within 1.5
times the IQR. Any data point beyond the whiskers is considered an outlier,
denoted by a cross in the figure. Each best model performance over $50$ random
initializations of different operation times $T$ is denoted by a star. (b)
Optimal variational pulses $F[P^{*}(t)]$ and corresponding external flux
$\varphi_{\text{ext}}(t)$ at $T=5$. It results in the best error rate of
$R=0.168$. Optimal parameters:
$\\{A_{1}^{*}=2.017,,~{}A_{2}^{*}=0.644,~{}A_{3}^{*}=1.384,~{}\phi_{1}^{*}=-0.141,,~{}\phi_{2}^{*}=-0.596,~{}\phi_{3}^{*}=-0.408\\}$.
Dimensionless parameters for simulation of quantum dynamics:
$\omega_{1}=\omega_{2}\ =\cdots=\omega_{8}=6$ and $G=1$.

In Fig. 2(a), we solve the MAX-CUT problem for a 2-regular graph using $50$
different randomized initial parameters in each set, with $n=3$ and 6 tunable
parameters in the control pulses (25). To illustrate the dispersion and
skewness of the data, we depict the distribution using box plots. We observe
that extending the pulse length does not improve model performance;
specifically, $T=1$ is sufficient for the problem, although the performance is
not entirely satisfactory. The BFGS algorithm often converges at a local
minimum, hindering optimal model performance and preventing the attainment of
the optimal solution of MAX-CUT after obtaining the classical string via
measurement on the $x$-axis. In Fig. 2(b), we plot the control pulse and
corresponding external flux for the best performance among all the
initializations, which achieves an error rate of $R=0.168$.

Figure 3: (a) Optimal variational pulses $F[P^{*}(t)]$ and corresponding
external flux $\varphi_{\text{ext}}(t)$ found by genetic algorithm for 2-qubit
problem. Optimal parameters:
$\\{A_{1}^{*}=0.307,~{}A_{2}^{*}=0.491,~{}A_{3}^{*}=4.202,~{}\phi_{1}^{*}=3.798,~{}\phi_{2}^{*}=3.253~{}\phi_{3}^{*}=3.441\\}$.
(b) Optimal variational pulses $F[\tilde{P}^{*}(t)]$ and corresponding
external flux $\varphi_{\text{ext}}(t)$ optimized by BFGS for 8-qubit problem,
which is initialized by $\\{A_{i}^{*},\phi_{i}^{*}\\}$ for 2-qubit problem
with maximal fitness. Optimal Parameters:
$\\{\tilde{A}_{1}^{*}=-1.668,~{}\tilde{A}_{2}^{*}=4.560,~{}\tilde{A}_{3}^{*}=6.861,~{}\tilde{\phi}_{1}^{*}=3.456,~{}\tilde{\phi}_{2}^{*}=3.919~{}\tilde{\phi}_{3}^{*}=5.113\\}$.
Dimensionless parameters for simulation of quantum dynamics are the same as
those before.

###  IV.2 Meta-learning

As we can see, the performance of variational algorithms heavily relies on
parameter initialization. Adequate initialization can bring parameters close
to the global minima, significantly reducing the iteration number. In the case
of QAOA, one can utilize $\beta_{i}=B(i\delta t)\delta t$ and
$\gamma_{i}=\Gamma(i\delta t)\delta t$ as digitized annealing schedules for
$B(t)$ and $\Gamma(t)$, for instance, $B(t)=1-t/T$ and $\Gamma(t)=t/T$,
instead of random initial parameters. However, the tunable pulse $P(t)$ does
not conform to the standard quantum annealing paradigm since it is periodic,
and the mixer is never turned off. In other words, there is no established
initialization strategy for PBVQO. We propose a hypothesis that the optimal
parameters $\\{A_{i}^{*},\phi_{i}^{*}\\}$ found for minimizing an easier
problem with an outstanding optimizer can serve as priors for a harder problem
with a mediocre optimizer. This realization of meta-learning for the harder
problem as parameter initialization, even if the optimal parameters are not
sufficiently close to the global minima, is our proposition.

###  IV.3 Genetic optimization

To verify our hypothesis, we identify the easiest problem as a two-qubit MAX-
CUT, which can be directly embedded in the minimal structure of our circuit
(10). As observed, BFGS is highly sensitive to parameter initialization,
making it suboptimal for our problem. Conversely, the genetic algorithm has
been utilized to optimize quantum circuits and algorithms, demonstrating its
ability for global search at the expense of considerable computational
resources. While employing the genetic algorithm for optimizing the harder
problem may be too complex, it is deemed acceptable for solving the two-qubit
case and deriving parameters with maximal fitness.

In Fig. 3(a), we present the optimal control pulse $F[P^{\ast}(t)]$ for the
two-qubit problem, achieving errorless ground state preparation. The
corresponding parameters $\\{A_{i}^{*},\phi_{i}^{*}\\}$ are then employed to
initialize the BFGS optimizer for the 8-qubit problem, resulting in an initial
error rate of $R=0.744$. This initialization leads to $F[\tilde{P}^{*}(t)]$
[see Fig.3(b)] that converges to $R=0.056$, which is remarkably close to the
exact ground state.

Figure 4: Histogram of BFGS (baseline), Meta-learning with BFGS, and Meta-
learning with genetic algorithm (GA) for 8-qubit QAOA.

Furthermore, we establish the statistical significance of meta-learning with
the genetic algorithm over BFGS, demonstrating that the example in Fig. 3 is
not cherry-picked. In Fig. 4, we depict the distribution of error rates $R$
for the baseline (BFGS for 8-qubit), meta-learning with BFGS, and meta-
learning with the genetic algorithm via histograms. In meta-learning, we solve
a two-qubit problem with the local optimizer BFGS and global optimizer GA,
respectively, finding the exact solution
$\langle\Psi_{f}|H_{p}|\Psi_{f}\rangle=-1$.  Although both optimizers achieve
good results on simpler problems, using two sets of optimal parameters as
initialization for the local optimizer BFGS to solve the more challenging
8-qubit problem leads to different model performances. Therefore, we observe
that meta-learning with the genetic algorithm outperforms the other two
approaches. This finding supports the notion that employing a global optimizer
to optimize an easy problem and using the result as parameter initialization
for a harder problem is effective. In constrast, utilizing a local optimizer
for an easier problem (see Meta-BFGS) does not overcome the local minimum in
the harder one, even if it also solves the easier problem well. By showcasing
the superiority of meta-learning with the genetic algorithm, we provide the
evidence supporting the efficacy of our approach in enhancing the performance
of PBVQO.

##  V Discussion

After presenting the results of the numerical experiments with PBVQO, we
proceed to compare them with conventional QAOA. As discussed in Sec. III, our
digital-analog superconducting architecture facilitates direct implementation
of XX rotation using gate Hamiltonian, avioding the need for CNOTs and local
rotations. To establish a fair comparison, we initially assess the error rate
of QAOA. Given that the performance of QAOA heavily depends on the depth $p$
(equivalent to optimizing $2p$ variables), we set $p=3$ for conventional QAOA,
aligning with the 6 parameters in the pulse ansatz of PBVQO. Testing with $50$
sets of random initial parameters, all of them converge to $R=0.25$ for the
$8$-qubit MAX-CUT problem.

Another critical aspect is runtime, as longer runtimes accumulate errors due
to quantum noise, particularly impactful in superconducting circuits. QAOA’s
runtime is directly related to single and two-qubit gate times, implemented by
corresponding gate Hamiltonians. The scale of Hamiltonians dictates gate times
by requiring more energetic input within a shorter operation time. Along this,
we calculate the energetic cost as [40]:

$C=\frac{1}{T}\int_{0}^{T}||H(t)||dt,$ (26)

where $||H(t)||$ represents the Frobenius norm of the total Hamiltonian of
PBVQO and the quantum gates in QAOA. A higher energetic cost indicates a
longer runtime, or alternatively, a more intense pulse if the operation time
is bounded. After averaging $50$ independent simulations, we find
$C_{\text{QAOA}}=488.76$ for QAOA, while PBVQO incurs almost half the cost at
$C_{\text{PBVQO}}=280.87$.

While demonstrating the capability of our superconducting architecture in
solving quantum optimization problems with PBVQO, surpassing conventional
QAOA, it’s essential to note that it is far from the optimal architecture for
MAX-CUT problems. For instance, the regular QAOA embedding requires all-to-all
connectivity thus, for being implemented in our platform it needs performing
swapping operations that add more complexity from the point of view of
compilation and pulse levels because now the pulses anzats requires learning
such information. Alternatives approaches will include flux-qubit or transmon
system coupled to tunable couplers [41, 42].

It is important to note that this pulse-based variational approach isn’t
limited to superconducting circuits or circuit quantum electrodynamics
architecture. The flexibility of our approach only requires a quantum platform
capable of implementing time-dependent couplings and single-qubit rotations.
Promosing candidates should be Rydberg atoms [43, 44, 45] where one- or two-
dimensional arrays of heavy alkali atoms such as Rubidium (Rb) and Cesium (Cs)
are confined in optical twezzers, and single-qubit gates can be implemented by
either microwave drives [46], optical stimulated Raman transitions or a
combination of microwave drives with magnetic field gradients [43] achieving
gate fidelity for single-qubit rotations of $\mathcal{F}=99.792\%$ [47].
Likewise, two-qubit interactions in Rb atoms has been implemented using the
the blockade effect with individual and separate addresing [43] and also with
local spin exchange with atoms in movable tweezers [48].

Similarly, trapped-ions [49, 50] are viable for implementing our QAOA
protocol. In this case, the microwave controlled single-qubit operations are
performed in the hyperfine energy levels, while the optical ones are made
using stimulated Raman transitions for inducing quadupolar transitions on the
$D\rightarrow S$ levels [49], these approaches have achieved single-qubit
fidelities around $\mathcal{F}=99.993\%$ [51] for microwave and
$\mathcal{F}=99.995\%$ [52] for optical induced gates, respectively.
Furthermore, there exist a plethora of ways for implementing two-qubit
interaction in the platform; from Coulomb interaction [49], using coherently
the motional degree of freedom of the quantized motion [53], and using such
degree of freedom as a mediator for implementing a multi-qubit interaction
[54]. Theses approaches have achieved gate fidelities of $\mathcal{F}=99.91\%$
[55] for hyperfine Mølmer-Sørensen gate.

##  VI Conclusion and outlook

In summary, we have introduced pulse-based variational quantum optimization
(PBVQO) as an extension to variational quantum algorithms in the NISQ era. Our
protocol is directly applicable to the superconducting circuit architecture
within the digital-analog quantum computing paradigm. Through experiments
solving the MAX-CUT problem, we have demonstrated PBVQO’s capability to
approximate the ground state of the problem Hamiltonian. Furthermore, we have
proposed meta-learning for parameter initialization in more challenging
problems, utilizing a genetic algorithm as the global optimizer. The positive
outcomes of meta-learning with the global optimizer have been illustrated
through benchmarking against baseline PBVQO and meta-learning with a local
optimizer. Additionally, we have conducted comparisons with conventional gate-
based QAOA and analyzed the mechanisms underlying its advantages.

Overall, the outlook for PBVQO is promising, with numerous open questions and
opportunities for future research and development. For instance, investigating
new optimization algorithms or adapting existing classical optimization
methods to better suit the requirements of pulse-level optimization could lead
to further improvements in performance and convergence speed. By extending
pulse-based approaches to other prominent variational algorithms, such as VQE,
one can potentially unlock new capabilities and address a wider range of
optimization problems. Finally, as PBVQO continues to evolve, rigorous
experimental validation and benchmarking against classical and quantum
algorithms are essential. Thus, future study can be dedicated to the
comprehensive experimental studies for validating the scalability, robustness,
and performance of pulse-based techniques on noisy quantum hardware platforms,
with error migitation [56].

###### Acknowledgements.

This work is supported by NSFC (12075145 and 12211540002), STCSM
(2019SHZDZX01-ZX04), and the Innovation Program for Quantum Science and
Technology (2021ZD0302302), HORIZON-CL4-2022-QUANTUM-01-SGA project 101113946
OpenSuperQPlus100 of the EU Flagship on Quantum Technologies, the Basque
Government through Grant No. IT1470-22. F.A.C.L. thanks to the German Ministry
for Education and Research, under QSolid, Grant no. 13N16149. This project has
also received funding from the European Union’s HORIZON Europe program via
project HORIZON-CL4-2021-DIGITALEMERGING-02-10 (No. 101080085 QCFD).

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