[1]Faculty of Electrical Engineering, Technion - Israel Institute of
Technology, Haifa 3200003, Israel ††thanks<EMAIL_ADDRESS>
[2]Faculty of Mechanical Engineering, Technion - Israel Institute of
Technology, Haifa 3200003, Israel ††thanks<EMAIL_ADDRESS>
# PANSATZ: Pulse-based Ansatz for Variational Quantum Algorithms

Dekel Meirom  Steven H. Frankel

###### Abstract

We develop and implement a novel pulse-based ansatz, which we call PANSATZ,
for more efficient and accurate implementations of variational quantum
algorithms (VQAs) on today’s noisy intermediate-scale quantum (NISQ)
computers. Our approach is applied to quantum chemistry. Specifically, finding
the ground-state energy associated with the electron configuration problem,
using the variational quantum eigensolver (VQE) algorithm for several
different molecules. We manage to achieve chemical accuracy both in simulation
for several molecules and on one of IBM’s NISQ devices for the $H_{2}$
molecule in the STO-3G basis. Our results are compared to a gate-based ansatz
and show significant latency reduction - up to $7\times$ shorter ansatz
schedules. We also show that this ansatz has structured adaptivity to the
entanglement level required by the problem.

##  1 Introduction

Today’s quantum computers (QCs) are often described as noisy intermediate-
scale quantum (NISQ) computers due to the relatively low numbers of qubits
available (e.g. 10’s to 100’s) and the relatively high levels of noise
associated with them (e.g. decoherence and gate fidelity errors) [1, 2, 3].
These limitations result in circuits with short width and shallow depth.

To make use of these NISQ machines, a class of hybrid quantum-classical
algorithms are being developed that seek to leverage the relative strengths of
quantum and classical computers. The most common example of such algorithms
are variational quantum algorithms (VQAs) [4]. VQAs use a QC to prepare a
short-depth parameterized quantum circuit (PQC) representing a trial solution
or ansatz to the problem at hand. Measurements of a final quantum state are
used to calculate a cost function, which is then minimized on a classical
computer to estimate the problem solution. Prominent examples of VQAs include
the variational quantum eigensolver (VQE) for quantum chemistry and materials
applications [5, 6], quantum approximate optimization algorithm (QAOA) for
combinatorial optimization problems [7], and variational quantum linear solver
(VQLS) for linear algebra problems [8].

One of the main challenges associated with effective VQA implementations is
related to the design of a suitable PQC/ansatz that balances expressibility
and noise, while avoiding exponentially vanishing gradients of the cost
function, referred to as the barren plateau (BP) problem. The two main
categories of ansatz that have been considered include problem-inspired ansatz
(PIA) and hardware-efficient ansatz (HEA) [6]. PIA structure is primarily
determined by the details of the problem being solved. HEA structure is
determined by the properties of the target hardware. HEAs are designed to
reduce PQC depth while maintaining a general and expressive ansatz. There have
been a number of attempts [9] to improve the gate-based ansatz approach
including ADAPT-VQE [10] and Noise-Adaptive Search (QuantumNAS) [11].

Recently, the idea of combining quantum optimal control (QOC), a method by
which optimal pulses can be designed to improve qubit coherence and gate
fidelity, with VQA has been proposed [12]. A few recent studies have sought to
implement this and related ideas, proposing to bypass the PQC at the gate
level for a pulse-based state-preparation [13, 14], ansatz generation [15,
16], machine learning tasks [17] and better gate compilation using QOC
techniques tailored for VQAs [18, 19, 20, 21]. In this paper, we propose an
implementation of an ansatz at the pulse level, creating a new type of HEA
that displays reduced latency and better expressibility with minimal
additional parameters and optimization complexity.

###  1.1 Contributions

We developed a general purpose parameterized pulse-based ansatz with a
relatively small number of parameters. We demonstrated, using the VQE
algorithm, that such an ansatz outperforms gate-based hardware efficient
ansatz, which is widely used on today’s NISQ devices. Using our ansatz, we
were able to reach chemical accuracy on a real NISQ device. We believe that by
using such a pulse-based ansatz, along with state of the art error mitigation
techniques, this approach might bring us one step closer to solving real-world
problems using quantum computers.

##  2 Computational Approach

###  2.1 Gate-based ansatz

Most of today’s quantum computing logic is represented using quantum gates.
The implementation of such quantum gates on quantum hardware is done using
control pulses. In order to map between the desired logic of a quantum gate
and a sequence of control pulses, a series of experiments that select the
correct pulses has to be run on the hardware, which is usually referred to as
the process of calibration. Today’s QCs are noisy and unstable, which leads to
the requirement of repeating calibrations frequently to maintain high-fidelity
of the gate operations.

The complexity of the calibration process and the need to do it very
frequently limits the QC designers to only a small number of gates that will
be mapped into control pulses. This set of gates should be universal (to allow
universal computing) and is called the native gate set. Each quantum gate that
is not included in the native gate set must be first decomposed into a
sequence of native gates in a process termed transpilation. In most cases,
such decompositions are not unique, and finding the optimal one is
challenging. The process usually introduces redundancy and added latency in
the qubits manipulation compared to the case where the original gate was part
of the native gate set and had its own mapping to control pulses. An example
of this type of redundancy is shown in Fig.1.

The structure of the HEA tries to minimize the depth of the gates in their
decomposed format in order to lower the noise of execution. Each HEA layer
usually consists of an entangling layer, built from native two-qubit gates,
followed by a parameterized single-qubit gate layer featuring rotation angles
as parameters. In addition, there is another initial layer of parameterized
single-qubit gates. Henceforth, we will refer to an ansatz built from gates a
GANSATZ and use the Real Amplitudes Ansatz, which is commonly used in VQE
problems, as a baseline for comparison to our new approach.

Figure 1: Demonstrating redundancy in the control pulses in non-native gate
decomposition. (a) Qubit trajectory on the Bloch sphere which represents the
logic of $R_{x}(\frac{\pi}{4})$ directly. (b) Qubit trajectory on the Bloch
sphere of a decomposition of a $R_{x}(\frac{\pi}{4})$ gate into
$R_{z}(\theta)$ and $\sqrt{x}$ gates. (c) The logical decomposition of
$R_{x}(\frac{\pi}{4})$ gate into $R_{z}(\theta)$ and $\sqrt{x}$ gates. Figure
2: PANSATZ structure. The PANSATZ is built out from repeated $L$ layers, each
consisting of parameterized two qubit pulses ordered in two alternating layers
according to the device connectivity, followed by virtual $R_{z}$ gates and
single qubit pulses on all qubits. The initial layer consists of fixed single
qubit pulses which rotate the control qubits of the next layer to the XY
plane. The two qubit pulses have flat-top Gaussian shape, and are coloured in
light red, acting on the two qubits adjacent to the line it is drawn on. The
single qubit pulses have DRAG shape, and are coloured in red (which represents
the Gaussian part of the pulse) and blue (which represents the DRAG derivative
part of the pulse).

###  2.2 Our approach: pulse-based ansatz

The structure of our proposed pulse-based ansatz, which we will henceforth
refer to as PANSATZ, is similar to the GANSATZ structure, where each gate is
replaced with a parameterized pulse, including the two-qubit gates, which are
also parameterized as part of the PANSATZ. As opposed to the GANSATZ,
calibration is not performed and the pulses are not mapped into a logical
unitary. Therefore, the logical unitary of the pulse sequences in the ideal
case is unknown, and it is not trivial to trace the evolution of the qubits
state after each pulse. Similar to other HEAs, the PANSATZ strives to prepare
a quantum state while minimizing the incoherent noise associated with the
preparation process (as VQAs are resilient to coherent errors [22, 23], the
focus is only on incoherent errors). By trading off knowledge about the
unitary evolution in the noiseless case, the PANSATZ can have better
expressibility (with the same number of layers), as each part of the unitary
evolution can be parameterized, and lower incoherent noise compared to the
GANSATZ. Each pulse has many degrees of freedom which can be parameterized.
While keeping more degrees of freedom as parameters improves the
expressibility of the PANSATZ, it might also create over-parameterization and
introduce trainability issues. Therefore, fixing some of the degrees of
freedom as hyper-parameters and limiting the optimization process of the
algorithm to only a few parameters for each pulse is crucial for the
trainability of the ansatz. As the structure and properties of the control
pulses vary between different quantum hardware, the choice of the parameters
to optimize should be hardware dependent. Recently, a study focused on pulse
level algorithms of natural atoms technology was published [24].

Figure 3: Simulations results. The top insets of each figure are
representations of the molecular geometry, not drawn to scale. The error bars
on the simulation data are smaller than the size of the markers. The reference
on the FCI for each point is highlighted as blue tick marks. (a) Simulation
results of the VQE on $H_{2}$ molecule. (b) Simulation results of the VQE on
$HeH^{+}$ molecule. (c) Simulation results of the VQE on $LiH$ molecule. (d)
Deviation of the simulation result from the FCI result of the VQE on $H_{2}$
molecule. (e) Deviation of the simulation result from the FCI result of the
VQE on $HeH^{+}$ molecule. (f) Deviation of the simulation result from the FCI
result of the VQE on $LiH$ molecule. The transformation error is the error
introduced by reducing the number of qubits needed to encode the molecule’s
Hamiltonian and effectively reducing the active search space.

###  2.3 Superconducting qubit implementation

This study is focused on fixed-frequency superconducting qubits, primarily
because of their availability on devices accessible via IBM cloud access. The
driving Hamiltonian of each qubit for this hardware can be expressed in the
rotating frame using the rotating wave approximation (RWA) as [25]:

$\begin{split}H_{c}(t)=A(t)(e^{i((\omega_{q}-\omega_{d}(t))t+\phi(t))}\hat{a}+\\\
e^{-i((\omega_{q}-\omega_{d}(t))t+\phi(t)}\hat{a}^{\dagger})\end{split}$ (1)

where $A(t),\omega_{d}(t),\phi(t)$ are the time-dependent amplitude,
frequency, and phase of the driving microwave, $\omega_{q}$ is the frequency
of the qubit and $\hat{a}^{\dagger},\hat{a}$ are the bosonic creation and
annihilation operators. Previous studies have proposed to parameterize and
optimize most of the possible driving Hamiltonian’s time-dependent variables
[13, 15]. In our study, in order to avoid trainability issues and minimize the
number of parameters, we propose to keep a fixed shape (the envelope of the
waveform) to the pulses with only a few parameters to optimize, while the rest
are calibrated as hyper-parameters at the start of the algorithm based on the
given hardware.

Decoherence and dephasing are major sources of incoherent noise for this qubit
architecture. Hence, the duration of a quantum schedule dramatically affects
the amount of noise in the computation. Therefore, pulse duration was chosen
as the optimization parameter (as illustrated in Fig.2). This enables
exploration of the Hilbert space with the shortest possible schedule duration.
The pulse shape, driving frequency $\omega_{d}(t)$, amplitude and parameters
associated with that shape, were chosen at the beginning of the algorithm as
described below and kept fixed throughout the VQA computation. The phase
$\phi(t)$ of the driving pulse is manipulated using $R_{z}$ gates, allowing
the addition of only one parameter per qubit per ansatz layer, instead of an
added parameter for each pulse envelope.

In our implementation, we choose derivative removal by adiabatic gate (DRAG)
[26, 27] as the shape of the single-qubit pulses to minimize leakage into
higher energy levels of the device. The $\sigma$ and $\delta$ parameters of
the DRAG pulse were taken from the calibrated $X$ gate of the device and kept
fixed. We also use flat-top Gaussian functions as the shape of the cross-
resonance (CR) [28] pulses to enable smooth waveforms with maximum amplitude
for most of the pulse duration. The $\sigma$ and the rise-fall ratio of the
flat-top Gaussian are taken from the CR part of the CNOT gate of the device
and kept fixed. The frequencies of the pulses were chosen as the resonance
frequency of the qubit they are controlling and the amplitude was chosen in a
quick calibration process to find the maximum amplitude for short pulses which
creates negligible leakage. Negative duration was allowed in order to account
for unconstrained optimizers, while negative time was translated into negative
amplitude and the absolute value of the duration. Between the layer of the CR
pulses and the single qubit DRAG pulses, a layer of $R_{z}$ gates was added in
order to enable changing the relative phase of each qubit, as the phase in the
DRAG pulse is fixed. In superconducting qubits technology, such gates can be
implemented as virtual gates with zero duration [29], therefore they are
perfect gates that do not inject any noise to the system. We chose to keep the
initial layer of single qubit pulses fixed with pulses that rotate some of the
qubits to the XY plane of the Bloch sphere. The qubits that were chosen are
those which will act as control qubits in the first entangling layer (which
will be every other qubit if we assume linear nearest-neighbor connectivity).
This was done in order to create as much entanglement as possible from this
layer, as the creation of entanglement is usually the most computationally
time consuming. Although the structure of the PANSATZ is fixed, by allowing
some of the pulse duration parameters to be zero, some of the pulses and even
entire layers can be kept out of the schedule (as opposed to the HEA, where
the CNOT gates are not parameterized), adjusting the structure of the ansatz
during the optimization process, benefiting from an adaptive approach similar
to ADAPT-VQE [10].

Figure 4: Duration of the PANSATZ schedules compared to Real Amplitudes HEA
for $H_{2},HeH^{+}$ and $LiH$ molecules, based on numerical simulations. The
duration of the GANSATZ is constant because the ansatz has the same number of
gates for each atomic distance, and the duration of each gate is fixed.

###  2.4 Two qubit pulse implementation

The two qubit pulses are usually the most important, as they can create
entanglement between the qubits, but also the most time-consuming and error-
prone. Therefore, additional care should be taken in the optimization process
of these pulses. The effective two-level system Hamiltonian describing the CR
pulse can be approximated as the time-independent Hamiltonian [30]

$\begin{split}H_{CR}=\omega_{IX}IX+\omega_{IY}IY+\omega_{IZ}IZ+\\\
\omega_{ZI}ZI+\omega_{ZX}ZX+\omega_{ZY}ZY+\omega_{ZZ}ZZ\end{split}$ (2)

Where the dominant terms are the entangling term $ZX$ and the single-qubit
terms $ZI$ and $IX$ [31, 32, 33]. The $ZX$ term is crucial for creating
entanglement and is usually the only desired term of the CR pulse when
designing a CNOT gate, while the $ZI$ and $IX$ are usually referred as
unwanted terms and have the largest coefficients. In the PANSATZ case, there
is no target unitary for the CR pulse, only a search for the mapping between
the parameters of the pulse and the cost function. In our numerical
experiments, we observed that a relatively small change in a CR pulse duration
caused a large change in the cost function, mainly as a result of these heavy
weighted single-qubit terms. These frequent large changes in the cost function
harm the flow of the classical optimizer and lead to problems with the
convergence of the algorithm. Therefore, we changed the CR pulses into an echo
format [34, 33], which has a DRAG-shaped flip pulse (which needs to be
calibrated a priori) on the control qubit between the two halves of a CR flat-
top pulse. Such an echo cancels these single-qubit terms while keeping the
entangling term. This makes each entangling pulse longer than it needs to be
in the optimal case, but helps with the training process. This issue might be
resolved by creating a tailor-made optimizer, but this is beyond the scope of
this research.

##  3 Results

In order to test our method we implemented the standard VQE algorithm but with
our PANSATZ as the ansatz to find the ground state energy of different
molecules. We first implemented the algorithm as a simulation, using the
Qiskit-dynamics package [35], which utilizes the JAX array library to enable
accelerated GPU execution, to find the minimum energy of the $H_{2}$,
$HeH^{+}$ and $LiH$ molecules. Next, we implemented the algorithm on one of
IBM’s devices, ibm_lagos, to find the minimum energy of $H_{2}$. All of the
molecular Hamiltonians used in this work were computed in the STO-3G basis
using Qiskit-nature package. These Hamiltonians were converted into spin
Hamiltonians using parity transformation, and encoded into qubits after
utilizing symmetries in the Hamiltonian, to reduce the amount of qubits needed
in order to encode the solution to the problem. The $H_{2}$ and the $HeH^{+}$
Hamiltonians were represented using 2 qubits, while the $LiH$ Hamiltonian was
represented using 4 qubits, which was achieved by reducing the active search
space of the spin orbitals, as explained in [6]. Such reduction introduces an
error, especially in small atomic distances, but reduces the number of
required qubits to encode the Hamiltonian significantly, which allows reaching
better results. We compared our results to the full configuration interaction
(FCI) result with respect to the minimal STO-3G basis calculated by
diagonalizing the Hamiltonian. In both the simulation and on the actual IBM
device hardware we used $10000$ shots for each circuit run to measure the
expectation value with a small variance. In the run on ibm_lagos we used
tensored readout error mitigation [36, 37]. For these tasks, we used PANSATZ
with only 1 layer, consisting of fixed single-qubit pulses, followed by
parametrized two-qubit pulses and parametrized single-qubit pulses (as
described in the previous section). This structure generates 5 parameters for
the $H_{2}$ and $HeH^{+}$ molecules (while GANSATZ has 4 parameters) and 11
parameters for the $LiH$ molecule (while GANSATZ has 8 parameters). As initial
pulse parameters, we chose duration for the single qubit pulses so the final
state will be approximately the Hartree-Fock (HF) state. All the two-qubit
pulses were initialized with zero duration (as there is no entanglement at the
HF state). For example, the HF state for the $H_{2}$ molecule is the
$|01\rangle$ state, so all the parameters were initialized with zero duration
except the single qubit pulse of the first qubit. This pulse was initialized
with the same duration as the pulse in the fixed layer (which is calibrated to
create $\leavevmode\nobreak\ \frac{\pi}{2}$ rotation around the X-axis),
resulting in a $\pi$ rotation to the first qubit, yielding the HF state. We
achieved good agreement between the simulation results and the results from
ibm_lagos device (after readout error mitigation).

Figure 5: VQE results for $H_{2}$ molecule on $ibm\\_lagos$ device. (a) The
final ground energy found by the algorithm with and without readout error
mitigation. The error bars on the data are smaller than the size of the
markers. The reference on the FCI for each point is highlighted as blue tick
marks. (b) Deviation of the result from the FCI result with and without
readout error mitigation.

###  3.1 Simulation

We simulated fixed-frequency transmon qubits by using the following device
Hamiltonian -

$H=\sum_{k=1}^{N}(\omega_{k}\hat{a}_{k}^{\dagger}\hat{a}_{k}-\frac{\delta_{k}}{2}\hat{a}_{k}^{\dagger}\hat{a}_{k}^{\dagger}\hat{a}_{k}\hat{a}_{k})+\sum_{<kl>}g(\hat{a}_{k}^{\dagger}\hat{a}_{l}+\hat{a}_{l}^{\dagger}\hat{a}_{k})$
(3)

where $\omega$ is the qubit frequency, $\delta$ is the qubit anharmonicity,
$g$ is the neighboring qubit coupling strength and
$\hat{a}_{k}^{\dagger},\hat{a}_{k}$ are the bosonic creation and annihilation
operators. For the simulation, we used the values reported by IBM on
ibm_manila for these parameters. In order to make the Hilbert space finite, we
truncated the energy levels of the transmons at 3 levels to be able to take
into account leakage to the $|2\rangle$ level. We also simulated relaxation
and decoherence noise using the following dissipators:

$D_{0}=\sqrt{\Gamma_{0}}\cdot\sigma^{+}$ (4)
$D_{1}=\sqrt{\Gamma_{1}}\cdot(\hat{a}_{k}\hat{a}_{k}^{\dagger}-\hat{a}_{k}^{\dagger}\hat{a}_{k})$
(5)

where $\sigma^{+}$ is the Pauli ladder operator ($\sigma_{x}+i\sigma_{y}$). We
used $\Gamma_{0},\Gamma_{1}$ which are proportional to $T_{1}=T_{2}=100[us]$.
At the end of each calculation of qubit evolution, we sampled the final qubit
state with $10000$ shots in order to introduce shot noise caused by finite
sampling to the simulation. For comparison we also ran a GANSATZ simulation
with 1 layer using the IBM simulator. we used the same device parameters and
connectivity and inserted relaxation and decoherence noises of
$T_{1}=T_{2}=100[us]$, and simulated shot noise by sampling the results with
$10000$ shots. In the gate model simulator leakage can not be simulated, so we
assumed that there was no leakage. The duration parameter can be seen either
as a continuous parameter, or as a discrete parameter, with the smallest time
unit of the classical controller of the quantum processing unit as the
discrete unit (about $0.222[ns]$ for most of today’s IBM QCs). Therefore, we
tested two different optimizers in the simulation - simultaneous perturbation
stochastic approximation (SPSA), which is used for continuous parameters and
approximates the gradient of the parameter vector using only 2 measurements
regardless of size [38], and steepest-ascent hill climbing [39, 40], which is
used for discrete parameters. Both optimizers had similar accuracy, while the
steepest-ascent hill climbing converged with much fewer iterations. As shown
in Fig. 3, using PANSATZ we reached chemical accuracy ($0.0016$ Hartree)
compared to the FCI result, up to a standard deviation (calculated from the
variance in the expectation value calculation), across all atomic distances
studied for the $H_{2}$ and $HeH^{+}$ molecules. Because of the small number
of parameters, this was achieved with only a few tens of iterations in the
worst case. Throughout the optimization process of the parameters, the
duration of the schedule slowly increased as entanglement was added to the
prepared state by including also CR pulses (with non-zero duration) in the
schedule. For cases where more entanglement was needed in order to reach
chemical accuracy, the total duration of the schedule increased even further,
until the optimizer converged to the correct result. For example, the
entanglement of the ground state of the Hamiltonian of the $H_{2}$ molecule
after the parity transformation is higher for large atomic distances.
Therefore, as entanglement is usually the most time consuming to create in
superconducting qubits, the optimizer converged at a longer schedule duration
when solving these problems, as can be seen in Fig.4. Even the longest
duration schedule achieved by using PANSATZ is less than half the duration of
the equivalent GANSATZ. For the $LiH$ molecule, 1 layer is not expressive
enough for large atomic distances (although it is more expressive than 1 layer
of GANSATZ). Therefore, the optimizer converged with low entanglement state,
which encountered minimal noise due to the short duration and had better
results than having longer CR pulses. Because of the low noise, good results
were achieved in small atomic distances, but the algorithm solutions deviate
from the FCI as the atomic distance increases.

Figure 3 shows excellent agreement between the PANSATZ predictions of the
ground state energy in Hartree units versus interatomic distance in angstroms
to the FCI result.

Figure 6: Duration of the PANSATZ schedules results from $ibm\\_lagos$
compared to GANSATZ. The GANSATZ duration was taken as the duration of 1 layer
of Real Amplitudes HEA transpiled on the same device Figure 7: Convergence
plot of the steepest-ascent hill climbing optimization algorithm for solving
VQE for $H_{2}$ molecule with 0.7 angstrom atomic distance, run on
$ibm\\_lagos$.

###  3.2 Real Hardware

We used ibm_lagos, one of the IBM Quantum Falcon processors, to find the
ground energy of the $H_{2}$ molecule at various atomic distances. We used the
open-pulse feature [30] to create the algorithm ansatz at the pulse level, and
the PANSATZ structure described above. We used the steepest-ascent hill
climbing algorithm for the classical optimizer, as it converged with fewer
iterations in our simulations. We also used uncorrelated readout error
mitigation in our post processing of the hardware measurement results, which
is a scaleable method that can be used also when solving larger VQA problems.
The results are shown at Fig.5. The results were obtained in a single
convergence process, while convergence was declared by either reaching
chemical accuracy or reaching 30 iterations. The readout error mitigated
results show excellent agreement with the FCI results, including multiple
points which reached chemical accuracy. These results were achieved as a
direct result of the short duration of the PANSATZ schedule, as shown in
Fig.6, which was adaptive to the amount of entanglement needed for each atomic
distance. The GANSATZ duration was taken as the duration of 1 layer of Real
Amplitudes HEA compiled on the same device. By using a relatively small amount
of parameters, a discrete optimization algorithm, and initial parameters which
create a state close to the HF state, the PANSATZ reached the desired solution
within only a few iterations (example shown in Fig.7), giving hope for
convergence within a reasonable amount of iterations also in larger problems.

##  4 Summary and Discussion

We developed a parameterized pulse-based ansatz, which we call PANSATZ, to be
used with VQAs. We chose pulse duration as the lone parameter. We tested
PANSATZ in the context of VQE to find ground state energies of small molecules
on both simulation and actual IBM hardware. We achieved state of the art
results, reaching chemical accuracy with the raw expectation value results
(mitigating only readout errors), which, to the best of our knowledge, is an
achievement that has not been shown yet on superconducting quantum hardware.
Previous demonstrations of chemical accuracy on superconducting quantum
hardware [41, 42] used extensive post-processing error mitigation techniques
such as zero noise extrapolation (ZNE) [43, 44], probabilistic error
cancellation (PEC) [43] and purification [42], which might be used also with
PANSATZ to make the results even more resilient to noise and help solve larger
problems within the required accuracy. Adjustment of such mitigation
techniques to make them suitable for pulses (for example, using pulse
stretching instead of global folding in ZNE [45]) is left for further
research. Our experiments show significant latency reduction, resulting in
improved accuracy of PANSATZ over typical gate-based ansatzes. The PANSATZ
structure enables on-the-fly adaptation of the schedule latency depending on
the entanglement level required to solve the given problem, potentially
enabling simulations of larger molecules accurately. Our PANSATZ approach can
be used with other VQAs as an improvement to the HEA. The use of PANSATZ in
quantum algorithms where problem-inspired ansatze are used, such as QAOA,
remains to be considered. Using a hybrid gate-based and pulse-based ansatz can
be explored similarly to [46].

##  5 Acknowledgement

The authors gratefully acknowledge the financial support of the Israel Science
Foundation (ISF) on grant number 3457/21. We would like to thank Dr. Adi
Makmal for the fruitful technical discussions. We also gratefully acknowledge
the use of IBM Quantum services for this work and to advanced services
provided by the IBM Quantum Researchers Program. The views expressed are those
of the authors, and do not reflect the official policy or position of IBM or
the IBM Quantum team.

##  6 Data Availability

The underlying code used in this study is available on GitHub and can be
accessed via this link https://github.com/dekelmeirom/PANSATZ

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