# Towards Advantages of Parameterized Quantum Pulses

Zhiding Liang<EMAIL_ADDRESS>University of Notre DameNotre DameINUSA ,
Jinglei Cheng<EMAIL_ADDRESS>Purdue UniversityWest LafayetteINUSA ,
Zhixin Song<EMAIL_ADDRESS>Georgia Institute of TechnologyAtlantaGAUSA ,
Hang Ren<EMAIL_ADDRESS>University of California, BerkeleyBerkeleyCAUSA
,  Rui Yang<EMAIL_ADDRESS>Peking UniversityBeijingChina ,  Kecheng Liu
<EMAIL_ADDRESS>Peking UniversityBeijingChina ,  Peter Kogge
<EMAIL_ADDRESS>University of Notre DameNotre DameINUSA ,  Tongyang Li
<EMAIL_ADDRESS>Peking UniversityBeijingChina ,  Yongshan Ding
<EMAIL_ADDRESS>Yale UniversityNew HavenUSA and  Yiyu Shi<EMAIL_ADDRESS>University of Notre DameNotre DameINUSA

###### Abstract.

The advantages of quantum pulses over quantum gates have attracted increasing
attention from researchers. Quantum pulses offer benefits such as flexibility,
high fidelity, scalability, and real-time tuning. However, while there are
established workflows and processes to evaluate the performance of quantum
gates, there has been limited research on profiling parameterized pulses and
providing guidance for pulse circuit design. To address this gap, our study
proposes a set of design spaces for parameterized pulses, evaluating these
pulses based on metrics such as expressivity, entanglement capability, and
effective parameter dimension. Using these design spaces, we demonstrate the
advantages of parameterized pulses over gate circuits in the aspect of
duration and performance at the same time thus enabling high-performance
quantum computing. Our proposed design space for parameterized pulse circuits
has shown promising results in quantum chemistry benchmarks.

††copyright: none

##  1\. Introduction

In recent years, significant advancements have been made in both hardware and
software development in the field of quantum computing. Quantum computing is a
highly promising emerging technology with the potential to solve complex
problems that classical computers cannot handle (Preskill, 2018a). Among
various technologies used to implement quantum computers, superconducting
quantum computers have shown promising results in various applications, such
as simulating quantum systems (Arute et al., 2020), optimization (Harrigan et
al., 2021), and machine learning (Huang et al., 2021). However, the technology
is still in its early stages and faces significant challenges such as
decoherence, noise, and scalability (Preskill, 2018a), which must be overcome
before it can become a practical tool for solving real-world problems.

In the Noisy Intermediate-Scale Quantum (NISQ) era, superconducting qubits
have emerged as the leading technology for developing quantum computers.
Superconducting quantum hardware offers several advantages over other types of
quantum computing technologies, including scalability, fast gate operations,
and flexibility (Cheng et al., 2023). Therefore, we choose superconducting
quantum computers as our platform for evaluating parameterized quantum pulses.
Although our techniques can be generally applied to other types of quantum
computers, such as trapped ion, we focus on superconducting systems due to the
constraints related to pulse-level access.

Figure 1.  Illustrations of parameterized pulse circuits. Parameters include
amplitude, angle, duration, and frequency. The amplitude of a quantum pulse
refers to the strength of the signal that is applied to the qubit. The
duration refers to the length of time that the pulse is applied to the qubit.
The angle determines the phase of the pulse. The frequency specifies the
frequency of the carrier signal.

Quantum programs are usually written in high-level programming languages, such
as Q#, Python, or Quil (Svore et al., 2018; Johansson et al., 2012; Smith et
al., 2016), which are designed to avoid low-level details about the physical
implementation of quantum computing. These languages typically provide
constructions of quantum circuits, gates, measurements, and other operations.
Once the quantum program is written in a high-level language, it needs to be
optimized and transformed into a format that can be efficiently executed on
real quantum hardware. This involves techniques such as gate fusion, gate
cancellation, and gate commutation, which are used to reduce the number of
gates and improve the circuit’s efficiency. The next step is to map the
optimized circuit onto the physical qubits of the quantum hardware. This
involves assigning each logical qubit in the circuit to a physical qubit on
the hardware, taking into account factors such as connectivity constraints and
the availability of resources such as gates and measurements. The overall goal
is to make quantum circuits compatible with hardware topology while minimizing
the total number of SWAP gates that needs to be inserted. Next, these mapped
circuits need to be decomposed into basis gates that are natively supported by
the quantum backend. At the current phase, the compilation remains at the gate
level. Pulse-level controls come into play when we need to interact with
physical qubits. For superconducting quantum computers, once the decomposition
stage is finished, the circuits that are composed of basis gates will be
“translated” into pulses. Finally, these pulses will be transmitted to
physical qubits as control signals.

Figure 2.  Illustrations of the function of parameterized single qubit pulse.
a) Samplings 5000 times for amplitude while fix angle as 0. b) Samplings 5000
times for angle while fix amplitude = 0.08. The operation of amplitude
indicates the rotation with respect to the X-axis and the change of angle is
corresponding to the rotation with respect to the Z-axis.

Recent studies (Melo et al., 2022; Egger et al., 2023; Peng et al., 2023; Koch
et al., 2022; Di Paolo et al., 2023; Gokhale et al., 2020; Shi et al., 2019;
Earnest et al., 2021; Li et al., 2022; Smith et al., 2022) have highlighted
the benefits of utilizing pulse-level controls over gate-level programming for
a specific set of quantum algorithms, particularly variational quantum
algorithms. The key insight is that the gate-level abstraction layer is
designed with precise calibration of basis gates for higher fidelities. The
calibration of these basis gates aims to minimize their deviation from the
ideal theoretical representation. However, the ansatz circuits in variational
quantum algorithms may not demand such accurate implementation. The ”training”
process of variational quantum algorithms can inherently correct such
imprecision. By incorporating parameterized pulses within ansatz circuits, the
search for the desired state in the Hilbert space can be achieved with much
shorter circuit latency in comparison to gate circuits. For instance, the
current implementation of a U3 gate on IBM’s quantum hardware would require
two well-calibrated pulses and one virtual-Z operation for execution, whereas
at the pulse level, a single pulse can achieve the same functionality.

At present, the majority of quantum programs are designed and executed at the
gate level, with a well-established workflow in place for profiling the
performance of quantum gates on quantum hardware (Aleksandrowicz et al., 2019;
Preskill, 2018b). Profiling the performance of quantum gates involves
measuring and analyzing the physical characteristics and behavior of the gates
used in a quantum circuit. This information can be leveraged to understand the
limitations and sources of errors in the gates, optimize their performance,
and improve the overall fidelity and accuracy of the quantum computation. By
understanding the performance of individual quantum gates, researchers can
optimize the design of quantum circuits to minimize error rates and improve
the overall performance of quantum algorithms (Wang et al., 2022a, b, c,
2021a; Hu et al., 2022; Jiang et al., 2021; Cheng et al., 2022; Wang et al.,
2022d; Qi et al., 2023; Pistoia et al., 2021; Li et al., 2021; Wang et al.,
2021b; Patel et al., 2022; Zhang et al., 2022). Profiling the performance of
quantum gates also enables benchmarking of different quantum hardware
platforms (Tomesh et al., 2022; Lubinski et al., 2021; Larocca et al., 2022),
and it is crucial for developing error correction and mitigation techniques
(Das et al., 2021; Hua et al., 2021; Reed et al., 2012). In summary, profiling
the performance of quantum gates is vital for improving the overall
performance and reliability of quantum computing systems. Consequently,
profiling parameterized pulses is crucial for the advancement of pulse-level
controls and pulse-level quantum programming. We employ various criteria as
metrics for evaluating parameterized pulses and examine different pulse
templates to investigate their differences. Our aim is to offer guidance for
the design and application of parameterized pulses in quantum circuits.

We attempt to build detailed guidance surrounding the topics related to
parameterized quantum pulses. To this end, our contribution includes:

  * •

A set of pulse-level design spaces provided that guides the design of
parameterized quantum pulses of good quality.

  * •

A group of criteria to profile the pulse-level design space and characterize
the property and power of parameterized quantum pulses.

  * •

Benchmark applications include quantum chemistry tasks and quantum finance
tasks to validate the effectiveness of proposed pulse-level design spaces.

The rest of the paper is organized as follows. We provide a concise overview
of parameterized quantum circuits and systems, followed by a discussion on
related work in Section 2. In Section 3, we outline various criteria for
pulse-level circuit design and demonstrate the results on benchmarks. In
Section 4, we propose the pulse-level design space which we examine with the
aforementioned criteria. Then, we employ the parameterized pulses in different
applications and obtain the performance results in Section 5. In Section 6, we
discuss the advantages and limitations of parameterized quantum pulses and
present an outlook for future work.

Figure 3.  Comparison of the expressivity of single qubit gate-level circuit
and single qubit pulse circuit. a) Four different single qubit gate-level
circuits: Rz gate, Rx gate, Rx + Rz circuit, and U3 gate (Rx + Rz + Rx), with
sequence duration 0dt, 320dt, 320dt, and 320dt, respectively. And a single
qubit pulse circuit that takes amplitude and angle as parameters, with a
sequence duration of 160dt. For each circuit, we use Qiskit-Dynamics for 5000
times sampling on Bloch Sphere. b) The histograms showcase the estimated
fidelities of the enumerated circuits, with a theoretical Haar-distributed
fidelity plotted as a reference.

##  2\. Parameterized Pulse Circuits

The investigation of pulse-level quantum computing has gained significant
attention from researchers. In this study, we focus primarily on parameterized
quantum pulses. Most prior applications of parameterized quantum pulses have
been focused on the calibration of a quantum computer’s basis gates. For
example, experiments on Rabi oscillations are carried out to calibrate the
angle and amplitude of single-qubit gates (Sheldon et al., 2016a). To
calibrate two-qubit gates, Hamiltonian tomography is employed, with the aim of
maximizing the proportion of ZX interaction, as it is a crucial element of the
CNOT gate (Sheldon et al., 2016b; McKay et al., 2018). Several runtime-
efficient and noise-resilient strategies have been proposed for parameterized
quantum pulses, allowing for precise calibration of multi-qubit gates (McKay
et al., 2017; Sheldon et al., 2016b; Danin and Tennie, 2023). Quantum optimal
control (QOC) is regarded as one of the most promising options for generating
parameterized pulses for a given unitary matrix. Nevertheless, QOC is
excessively costly (Khaneja et al., 2005; Caneva et al., 2011; Sivak et al.,
2021; Cheng et al., 2020; Giannelli et al., 2022).

As depicted in Fig. 1, typical pulse-level parameters for a superconducting
quantum computer consist of amplitude, angle, duration, and frequency.
Adjusting these parameters affects the driving Hamiltonian of the quantum
operation, thereby impacting the state of the quantum circuit (Magann et al.,
2021). Given an example to understand the parameters function in the single
qubit parameterized pulse as follows (Krantz et al., 2019):

(1) $\widetilde{H}_{d}=-\frac{\Omega}{2}V_{0}s(t)(I\sigma_{x}+Q\sigma_{y})$

where $s(t)$ is a dimensionless envelope function, so that the amplitude of
the drive is set by $V_{0s}(t)$. Thus, we know amplitude determines the
rotation speed, while the duration determines the rotation time. For the
function of the angle, refer to the definition of I and Q (Krantz et al.,
2019):

(2) $\displaystyle\vspace{-1mm}I=\cos(\varphi)\quad(the\ ‘in-phase’\
component)$ (3) $\displaystyle Q=\sin(\varphi)\quad(the\ ‘out-of-phase’\
component)\vspace{-6mm}$

$I$ is the in-phase pulse which corresponds to rotations around the $x$-axis,
while $Q$ is the out-of-phase pulse which corresponds to rotations about the
$y$-axis. $I$ and $Q$ are both dependent on parameter angle $\varphi$, it can
be inferred that the angle determines the position of the rotation axis in the
XY plane.

Prior research has investigated the optimization of pulse amplitude,
frequency, and duration to mitigate decoherence and attain superior
performance in calculating the ground state energy of chemical molecules
(Meitei et al., 2021; Liang et al., 2022a; Egger et al., 2023), in
classification tasks of machine learning (Liang et al., 2022c; Pan et al.,
2023; Meirom and Frankel, 2022), and in the max-cut problem of QAOA on
different quantum hardware including superconducting quantum computers (Liang
et al., 2022b; Ibrahim et al., 2022) and neutral atom quantum computers (de
Keijzer et al., 2023).

##  3\. Pulse-level Circuit Criteria

To measure the efficacy of pulse-level circuits and emphasize their
superiority over gate-level circuits, we present four metrics: expressivity,
entangling capability, effective parameter dimension, and sequence duration.
Previous research on parameterized quantum circuits (PQC) at the gate level
has utilized both expressivity and entangling capability (Sim et al., 2019;
Dür et al., 2001), where sampling random circuits are used as benchmarks. In
addition, effective parameter dimension is a criterion for analyzing redundant
parameters in gate-level quantum circuits. Nonetheless, our objective is to
investigate parameterized quantum circuits at the pulse level, namely
parameterized pulse circuits (PPC). To evaluate the effectiveness of
parameterized pulse circuits, we compare their deviation with that of random
circuits, as well as the deviation of PQC with random circuits. By shifting
from gate-level to pulse-level, we can achieve improvements in sequence
duration while maintaining similar expressivity.

###  3.1. Sequence Duration

The term sequence duration refers to the total time required for the execution
of a quantum program. Sequence duration plays a vital role in determining the
overall execution time of the program and can significantly impact its
performance. Given that the current decoherence time on quantum machines is
limited, sequence duration is a critical consideration when designing quantum
programs. Thus, optimizing the sequence duration involves adjusting the pulse
parameters, adding delays or idle pulses, or utilizing different pulse shapes
to minimize the overall duration and enhance program performance.

In our Qiskit-Pulses test cases, the sequence duration of a pulse schedule is
always defined using a basic unit dt, where dt is the duration of a single
sample of the arbitrary waveform generator, typically equal to 0.222 ns. This
duration is discretely mapped from the hardware level to the software level
using the unit dt. While the search space of duration and amplitude partially
overlap, they jointly determine the “strength” of pulses. As hardware is
usually more precise for amplitude control, the hardware control typically
selects a fixed duration and then performs amplitude control. We decide to
retain duration control because we aim to adaptively reduce the decoherence
time of the quantum algorithm.

Expressivity | Quito | Lima | Jakarta
---|---|---|---
expr_z | 3.8918 | 3.8918 | 3.8918
expr_x | 0.2344 | 0.1771 | 0.1752
expr_xz | 0.0222 | 0.0202 | 0.0177
expr_zyz | 0.0173 | 0.0169 | 0.0235
expr_amp | 0.1985 | 0.1913 | 0.1673
expr_ang | 0.1709 | 0.7790 | 0.1719
expr_pulse | 0.0138 | 0.0202 | 0.0157
Table 1. The expressivity of different single qubit circuits sampled on three
different backend models by Qiskit-Dynamics, expr_z, expr_x, expr_xz,
expr_zyz, expr_amp, expr_ang, expr_pulse correspond to Rz gate, Rx gate, Rx +
Rz circuit, Rx + Ry + Rz (U3 gate) circuit, single qubit pulse circuit with
amplitude sampled while fix angle as 0, single qubit pulse circuit with angle
sampled while fix amp as 0.08, general single qubit pulse circuit with angle
and amplitude all sampled. Figure 4. Two-qubit pulse circuit structure and
corresponding pulse schedule. a) Dressedpulse consists of four single-qubit
pulses (SQP) and one cross-resonance pulse (CR). b) Blockpulse consists of six
SQPs and two CRs.

###  3.2. Expressivity

Model |  Expr |  Ent |  Param |  EPD |  Dur
---|---|---|---|---|---
2QDressedpulse |  0.0259 |  0.1145 |  11 |  11 |  276-1328dt
2QDressedpulse_fixamp |  0.0183 |  0.1837 |  10 |  10 |  276-1328dt
2QDressedpulse_fixang |  0.0163 |  0.1115 |  10 |  10 |  276-1328dt
2QDressedpulse_fixduration |  0.0253 |  0.1146 |  10 |  10 |  276-1328dt
2QBlockpulse |  0.0225 |  0.1977 |  18 |  18 |  992-2496dt
2QBlockpulse_fixamp |  0.0161 |  0.3127 |  16 |  16 |  992-2496dt
2QBlockpulse_fixang |  0.0175 |  0.1933 |  16 |  16 |  992-2496dt
2QBlockpulse_fixduration |  0.0318 |  0.0619 |  16 |  16 |  992-2496dt
Universal2QGateCircuit |  0.0253 |  0.4011 |  15 |  9 |  4768dt
RXCX2QGateCircuit |  0.3971 |  0.4883 |  4 |  3 |  1696dt
Table 2. Performance metrics of various two-qubit pulse models and two-qubit
gate models are provided for comparison. The table lists the expressivity
(Expr), entanglement capability (Ent), number of parameters (Param), effective
parameter dimension (EPD), and duration of the quantum operations (Dur) for
each model. For pulse models, the amplitude, angle, and duration have been
sequentially fixed and the analysis performed.

The capacity of a circuit to generate pure states is defined as expressivity,
which is introduced in previous PQC works (Sim et al., 2019). By taking
samples from these states on the Bloch sphere, we can evaluate a quantum
circuit’s capacity to explore the sphere for a single qubit. This ability
extends to the quantum circuit’s ability to explore the Hilbert space when
multiple qubits are involved.

####  3.2.1. Single-Qubit Pulse

A single-qubit pulse is a microwave pulse that is precisely tuned and
controlled to manipulate the state of a single qubit in a quantum computing
system. The effectiveness of the operation depends on the critical parameters
of the pulse including amplitude, duration, and shape. Achieving precise and
efficient manipulation of the qubit is crucial for performing quantum
computations, which is why these parameters must be adjustable. We conduct our
experiments on IBM’s superconducting quantum computer. Specifically, we define
a single-qubit pulse whose duration and $\beta$ are obtained from the
calibrated X gate from the backend and serve as its fixed parameters. The
parameters are the amplitude and angle, which can fine-tune the operation on
the qubit. To explore the effects of these two parameters, we first fixed the
angle to 0 and randomly sampled the amplitude 5000 times to obtain Fig. 2 a).
We then fixed the amplitude at 0.08 and randomly sampled the angle 5000 times
to obtain Fig. 2 b). From the results, we observe changes in amplitude
correspond to rotations around the X-axis, while changes in angle correspond
to rotations around the Z-axis. This implies that we have the potential to use
a single qubit pulse with a short sequence duration to implement U3 gate
operations.

####  3.2.2. Expressivity of Single-Qubit Pulse and Gate

In this section, we provide a detailed comparison between various models of
single-qubit gates and our proposed model of single-qubit pulses, by defining
their expressivity as a benchmark. Firstly, we consider the RZ gate, which
rotates around the Z-axis when the initial state is $|0\rangle$, and all the
samples on the Bloch sphere are on the $|0\rangle$ state. Secondly, we
consider the Rx gate, which rotates around the X-axis when the initial state
is $|0\rangle$, and the samples are distributed along a latitude line on the
Bloch sphere. We then observe that the Rz+Rx circuit produces states that
better cover the Bloch sphere. Next, we express the U3 gate using the Rz+Ry+Rz
circuit, which can achieve rotation on the X, Y, and Z axes, resulting in
higher degrees of freedom and better expressivity. We then demonstrate the
single-qubit pulse with amplitude and angle as parameters, where amplitude
indicates the amount of rotation around the X-axis and angle has enabled the
rotation around the Z-axis. We observe that the sequence duration of the
single-qubit pulse is half of Rx, Rx+Rz, and Rz+Ry+Rz. Finally, as a
reference, we uniformly sample the single-qubit matrix to simulate the most
expressive circuit theoretically. Fig. 3 a) shows the simulated state on the
Bloch sphere after 5000 samples and the corresponding gate-level and pulse-
level circuits.

We follow the same definition of expressivity proposed in Ref. (Sim et al.,
2019) for both gate-level and pulse-level ansatz

(4)
$\operatorname{Expr}=D_{\mathrm{KL}}\left(P_{\mathrm{Ansatz}}(F;\boldsymbol{\theta})\|P_{\text{Haar}}(F)\right),$

where the Kullback-Leibler (KL) divergence (Kullback and Leibler, 1951) is a
measure of distance between two probability distributions.
$P_{\mathrm{Ansatz}}(F;\boldsymbol{\theta})$ is a distribution of state
fidelities between two randomly sampled parameterized states
$|\psi(\theta)\rangle$ and $|\psi(\phi)\rangle$ obtained from the ansatz

(5) $P_{\mathrm{Ansatz}}(F;\boldsymbol{\theta})\equiv
P(F=|\langle\psi(\theta)|\psi(\phi)\rangle|^{2}).$

The latter quantity $P_{\text{Haar}}(F)$ is also a state fidelity distribution
but for the ensemble of Haar random states. In this case, the analytical
probability density function (PDF) is known as (Życzkowski and Sommers, 2005)

(6) $P_{\text{Haar}}(F)=(N-1)(1-F)^{N-2},$

where $N$ is the dimension of the Hilbert space. If the calculated Expr is
closer to $0$ for one particular ansatz, we say it is more expressive than
other candidates since it is able to sample uniformly from the full Hilbert
space and hence approximate any possible state. This capability is essential
for variational quantum algorithms where we want to to train the ansatz to
generate a particular quantum state with limited prior information about that
target state.

In Fig. 3 b), the estimated confidence histograms for each circuit are
displayed, with the theoretical Haar distribution shown as a reference. The
bin size for generating the histograms was defined as 50, which affects the
accuracy of the Kullback-Leibler (KL) divergence calculation. However, by
keeping the settings consistent across all circuits, the observed results can
be fairly compared. The KL divergence is reported in Table 1 to quantify the
deviation, where lower KL divergence values correspond to circuit simulations
that are closer to the Haar distribution, representing circuits with stronger
expressive power or states closer to random states.

In Table 1, two out of three backend models, single-qubit pulses demonstrate
slightly better expressive power than single-qubit gate-level circuits. The
poor performance observed in the Lima case is attributed to unfavorable
backend settings. The parameter angle of a quantum pulse is equivalent to a
ShiftPhase operation, which causes the qubit state to rotate around the
Z-axis. We discovered that as the angle changes, the qubit state is unable to
fully rotate around the Z-axis, resulting in a decrease in expressivity on the
Lima backend. Generally speaking, single-qubit pulses exhibit stronger
expressive power than single-qubit gate-level circuits, although it is worth
noting that single-qubit pulses have only half the sequence length of Rx,
Rx+Rz, and U3 circuits.

Figure 5. Proposed pulse-level design space and the corresponding Pulse ID 1,
3, 5, and the proposed design space with fixed CR amplitude be marked as Pulse
ID 2, 4, 6. Random-generated pulses with the same number of parameters and the
same number of CR pulses as the proposed design space are introduced as Pulse
ID 7, 8, 9, 10, 11, 12, i.e., Pulse ID 7 has same number of parameters and CR
pulses with Pulse ID 1.

####  3.2.3. Multi-Qubit Pulse

A multi-qubit pulse is a quantum operation that acts on multiple qubits
simultaneously. The ability to manipulate entangled quantum states between
multiple qubits is crucial in quantum computing. The controlled-not (CNOT)
gate is a common example of a multi-qubit quantum operation at the gate level,
which can be implemented using multiple cross-resonance (CR) pulses and echoed
pulses. CR is a type of multi-qubit pulse where a pulse is applied to the
control qubit at the frequency of the target qubit. The cross-resonance
Hamiltonian can be expressed as:

$\displaystyle H$ $\displaystyle=Z\otimes A_{2}+I\otimes B_{2}$ (7)
$\displaystyle=a_{x}\hat{Z}\hat{X}+a_{y}\hat{Z}\hat{Y}+a_{z}\hat{Z}\hat{Z}+b_{x}\hat{I}\hat{X}+b_{y}\hat{I}\hat{Y}+b_{z}\hat{I}\hat{Z}.$

At the gate level, cross-resonance pulses need to be elaborately calibrated to
implement desired function. Specifically, we want to keep the $ZX$ term and
eliminate other interactions. Such calibration process requires techniques
including echoed CR and phase calibration as discussed in (Sheldon et al.,
2016b). At the pulse level, however, there is no need to remove terms other
than $ZX$. In this way, we trade precise controllability for more flexibility
and capacity to further explore the Hilbert space.

Figure 6. Q measurement result through 500 times sampling of 2QBlockpulse and
2QBlockpulse_fixamp.

###  3.3. Entanglement Capability

Entanglement is a unique quantum phenomenon that refers to the non-classical
interdependence between two or more qubits, even when they are located far
apart. In variational quantum algorithms and quantum machine learning,
generating shallow circuits with strong entanglement capabilities is
essential. Consequently, understanding how to measure a quantum circuit’s
entanglement ability is crucial in selecting an appropriate design space. This
helps to create more effective solution spaces for tasks such as ground state
preparation or machine learning benchmarks while also preserving the non-
trivial correlations in quantum data.

One commonly used standard for measuring the entanglement ability of a quantum
circuit is the Meyer-Wallach (MW) entanglement $Q$-measure. It can be employed
to quantify the global entanglement measure of pure-state qubits entanglement,
with the benefits of scalability and ease of computation. We define the
entanglement capability as the average $Q$-measure from an ensemble of
randomly sampled states

(8) $\text{ Ent }=\frac{1}{|S|}\sum_{\boldsymbol{\theta}_{i}\in
S}Q\left(\left|\psi_{\boldsymbol{\theta}_{i}}\right\rangle\right),$

where the Meyer-Wallach $Q$-measure is proposed to estimate the number and
types of entangled states an ansatz can generate. It can be calculated though
the average of the purity of each qubit (Brennen, 2003)

(9)
$Q(|\psi\rangle)=2\left(1-\frac{1}{n}\sum_{k=1}^{n}\operatorname{Tr}\left[\rho_{k}^{2}\right]\right),$

where $\rho_{k}$ is the reduced density matrix of the $k$-th qubit. Notice for
any product state $|\psi\rangle\otimes|\phi\rangle\otimes\cdots$, $Q=0$, while
$Q=1$ for the Greenberger-Horne-Zeilinger (GHZ) state
$|\psi\rangle_{\text{GHZ}}=(|0\rangle^{\otimes N}+|1\rangle^{\otimes
N})/\sqrt{2}$.

Figure 7. Illustration of the change in the expressivity of various pulse sets
as the number of qubits increases. The main plot shows the overall trend, with
a zoomed-in section highlighting the results of six proposed pulse-level
design spaces. The figure demonstrates that expressivity increases with an
increasing number of qubits until a small decrease in expressivity is observed
when the number of qubits reaches four, which may be due to insufficient
circuit depth. Additionally, the six subplots show the change in the
expressivity of each of the pulse-level design spaces as the number of layers
increases for four qubits.

In order to explore the entanglement capabilities of parameterized pulse
circuits, we constructed two types of two-qubit parameterized pulse circuits
as shown in the Fig. 4. We then sampled and calculated the entanglement
capabilities of the corresponding circuits. In particular, we use the
Hamiltonian tomography of the CR to investigate the impact of phase and
duration variations on the components of IX, IZ, and IY, and result in
altering the proportion of the ZX term, which directly affects the
entanglement power (Sheldon et al., 2016b; Alexander et al., 2020). Thus, it
is necessary to analyze each parameter’s effect in CR. As a result, we
measured the entanglement and expressivity capabilities while fixing the
amplitude, angle, and duration, respectively, to observe the effects of the
corresponding parameters.

In Table 2, we introduce two pulse models within the structure as shown in
Fig. 4, and fix each type of parameter sequentially. We also put two gate
models as baseline to compare, one is the Universal2QgateCircuit, a quantum
circuit composed of 18 gates from CNOT, Ry, and Rz being the minimal gate-
based circuit that simulates an arbitrary two-qubit unitary operator up to
global phase (Shende et al., 2004). Another gate model consists of four Rx
gates and one CNOT gate, where two of Rx gates are in front of the CNOT
implemented on the first and second qubit, respectively, and another two Rx
gates are behind the CNOT. From Table 2, fixing the amplitude
(2QDressedpulse_fixamp and 2QBlockpulse_fixamp) resulted in a significant
improvement in the entanglement capability and a slight improvement in
expressivity, while fixing the angle (2QDressedpulse_fixang and
2QBlockpulse_fixang) resulted in a small improvement in expressivity but did
not significantly affect the entanglement capability. Fixing the duration
(2QDressedpulse_fixduration and 2QBlockpulse_fixduration) resulted in a
negative impact in all metrics. Overall, Universal2QGateCircuit had the
strongest entanglement capability, while 2QBlockpulse_fixamp had the best
expressivity. Pulse models generally exhibit better expressivity and have
shorter sequence durations compared to gate models. Although the pulse-level
circuit exhibits a relatively low entanglement capability than the gate-level
circuit, it still possesses points with Q values approaching 1 as shown in the
Fig. 6. This characteristic is advantageous for variational quantum algorithms
(VQAs) tasks, as it allows us to search for and identify points with Q values
close to 1 during the training and optimization processes.

###  3.4. Effective Parameter Dimension

Figure 8. The main plot of this figure shows the entanglement capability of
various pulse sets as the number of qubits increases. The zoomed-in section of
the plot displays the results of six pulse-level design spaces proposed in
this study. The plot illustrates that as the number of qubits increases, the
entanglement capability continues to improve. The six subplots depict the
entanglement capability of each of the six pulse-level design spaces as the
number of layers increases, with the qubit count fixed at four. The subplots
demonstrate that with an increasing number of layers, the entanglement
capability of each pulse-level design space also continues to increase.

Some parameters of PQC can be eliminated without affecting its expressivity.
Effective parameter dimension (EPD) is a measure that quantifies the number of
independent parameters of quantum states generated by PQC (Haug et al., 2021).
It reflects how many independent directions the parameters can explore over
the Hilbert space.

EPD can be calculated through Quantum Fisher information (QFI) (Liu et al.,
2019), which characterizes the sensibility of the state generated by PQC to
its parameter changes (Bharti et al., 2022). Concretely, the QFI matrix is

(10)
$\mathcal{F}_{ij}=\operatorname{Re}\left(\left\langle\partial_{i}\psi(\boldsymbol{\theta})\\!\mid\\!\partial_{j}\psi(\boldsymbol{\theta})\right\rangle-\left\langle\partial_{i}\psi(\boldsymbol{\theta})\\!\mid\\!\psi(\boldsymbol{\theta})\right\rangle\left\langle\psi(\boldsymbol{\theta})\\!\mid\\!\partial_{j}\psi(\boldsymbol{\theta})\right\rangle\right)$

Here the gradient of parameters is calculated by finite difference:

(11)
$\partial_{i}\psi(\boldsymbol{\theta})\approx\left[\psi\left(\boldsymbol{\theta}+\epsilon\mathbf{e}_{i}\right)-\psi\left(\boldsymbol{\theta}-\epsilon\mathbf{e}_{i}\right)\right]/2\epsilon.$

EPD is the rank of QFI matrix:
$d_{\mathrm{eff}}=\operatorname{rank}\mathcal{F}(\boldsymbol{\theta})$, i.e.,
the number of linearly independent parameters over the design space.

The experiments presented in Table 2 suggests that all parameters of the
pulse-level model proposed in the Fig. 4 are valid. Compared to two-qubit gate
models, the parameter dimensions at the gate level are more susceptible to
redundancy, with Universal2QGateCircuit exhibiting six redundant parameters.
Based on our analysis in the previous section, we conclude that the richness
of EPDs is a significant contributing factor to the superior expressivity of
the PPC model over gate-based PQC within a shorter duration.

Pulse ID |  # of Params |  EPD |  # of CR Pulses |  Pulse Circuit Depth
---|---|---|---|---
1 |  (5N-3)L |  (5N-3)L |  (N-1)L |  NL
2 |  2(2N-1)L |  2(2N-1)L |  (N-1)L |  NL
3 |  (6N-3)L |  (6N-3)L |  (N-1)L |  4L
4 |  (5N-2)L |  (5N-2)L |  (N-1)L |  4L
5 |  (9N-7)L |  (9N-7)L |  (N-1)L |  (2N-1)L
6 |  (8N-6)L |  (8N-6)L |  (N-1)L |  (2N-1)L
Table 3. The estimated costs associated with the proposed design space at the
pulse level, which includes the number of parameters, effective parameter
dimension, two-qubit operations, and circuit depth, are expressed in terms of
two variables: N (number of qubits) and L (number of circuits).

##  4\. Proposed Pulse-level Design Space

###  4.1. Insights of Proposed Design Space

In this section, we introduce three different PPC model designs with a
polynomial scaling of parameters. The first PPC template is the hardware-
efficient pulse in Fig. 5 a, where we only apply two-qubit CR pulse on
physically connected qubits. Hence, such a pulse template is efficient in the
sense that requires least compiling resources to implement on the hardware.
The second pulse template is a decay layer pulse in Fig. 5 b. designed to
improve the trainability of PPC. The hope is to gradually reduce the operating
scope (involves fewer qubits) and hence alleviate the barren plateaus (BP)
phenomenon (McClean et al., 2018). Most studies on BP are conducted on the
gate-level, and similar research on the pulse level is minimally explored. The
last pulse template is the block-dressed CR pulse in Fig. 5 c. We first dress
the native two-qubit CR pulse with four single-qubit pulses on both sides to
enhance expressivity. Then, we arrange such a dressed CR pulse as a block and
loop through all the qubits. In general, one can always adopt neural
architecture search (NAS) methods to construct an automatic workflow to build
customized PPC models. But those approaches are usually too expensive to run
on real quantum hardware. The proposed pulse templates in this study are
aiming to make pulse-level research more accessible to the community.

###  4.2. Evaluation and Analysis of Design Space

In this section, we conduct an analysis of the proposed pulse-level design
space and summarize our observations and insights for each analyzed criterion.
These analyses are performed with respect to all evaluation criteria and
highlighted general trends. Before introducing our observations and insights,
we first conduct a detailed investigation using the criteria mentioned in the
previous section, in order to better understand the insights obtained from the
subsequent presentation of the comprehensive trends. We include evaluations of
expressivity, entanglement capability, as well as cost estimation for the
proposed pulse-level design space.

####  4.2.1. Expressivity Analysis.

Firstly, we measure the ability and potential of the proposed pulse-level
design space to explore the Hilbert space, while adding random-generated
pulses without any design space-based policy as a comparison to the
experiments. By evaluating the results of a series of expressivity tests (see
Fig. 7), the value of K-L divergence drops gradually as the test depth (the
number of qubits) increases, indicating that the expressivity of the pulse-
level design space increases progressively. As the number of qubits exceeds 4,
however, the value of K-L divergence increases slightly, indicating that one
layer of pulse-level design space is incapable of exploring this portion of
the Hilbert space. This is observed for all the proposed pulse-level design
spaces we described. In contrast, the trend of expressivity with the number of
qubits for pulses generated at random without policy is erratic and
unpredictable.

Figure 9. The application benchmark of the proposed pulse-level design space,
compared to random-generated PQC and two gate-level PQCs that are verified to
have good performance. a) displays the results for quantum chemistry
applications, with tasks involving the electronic structure of $H_{2}$ and
$LiH$ molecules, the bond lengths default to 0.735Å and 1.5Å, respectively. b)
displays the results for quantum finance applications, with tasks involving
portfolio optimization for 2 and 4 assets.

Then, we fix the number of qubits at four and gradually increase the width of
the test, that is, for the number of layers of PPC. In Fig. 7, for each of the
proposed pulse-level design spaces, increasing the number of circuit layers
significantly improves expressivity. In addition, we find that the design
space with a fixed CR amplitude is significantly more expressive than the
corresponding design space without a fixed CR amplitude, and that among all
design spaces, the design space with Pulse ID 2, which is the design space
Dressed CR with a fixed CR amplitude, obtains the best performance in
expressivity for a circuit with 1 layers and two, three, or four qubits,
respectively. When there are two circuit layers, the hardware-efficient pulse
with a Pulse ID 6 that fixes the amplitude of the CR gets the highest
expressivity performance. When there are three circuit layers, the Decay-layer
pulse with Pulse ID 4 that fixes the amplitude of CR has the best expressivity
performance.

####  4.2.2. Entanglement Capability Analysis.

Secondly, we tested the entanglement capability of the suggested pulse-level
design space to investigate and assess the strength of the proposed pulse-
level design space’s entanglement capability. Likewise, no design space-based
policy is added to the experiment for comparison purposes. According to Fig.
8, the entanglement capability of all pulse-level design spaces grows
progressively as the test depth (the number of qubits) rises. Meanwhile,
entanglement capability with respect to the number of qubits for pulses
generated at random without a policy is unpredictable. Then, we fix the number
of qubits at four and increase the test’s width progressively. From the six
subplots, we can conclude that the progressive growth of the layer of PPC has
a higher impact on the entanglement capability than the rise in qubits.
Experiments on entanglement capability revealed that the design space with a
fixed CR amplitude produces greater entanglement capability than the design
space without a fixed CR amplitude. In the case of a single-layer circuit, the
hardware-efficient pulse fixed CR amplitude with Pulse ID 2 can create the
strongest entanglement capability. In the design space without fixed CR
amplitude, the Decay-layer pulse with Pulse ID 3 provides the highest
entanglement capability. When the number of circuit layers increases to two
and three, the Decay-layer pulse with fixed CR amplitude and a Pulse ID 4
provides the best entanglement performance.

Applications | Benchmarks | # of Qubits | # of Pauli Strings | Measurement Bases | Entanglement Q
---|---|---|---|---|---
Quantum Chemistry | $H_{2}$ | 2 | 5 | X, Y, Z | 0.04914
$LiH$ | 4 | 100 | X, Y, Z | 0.00072
Quantum Finance | 2 Assets | 2 | 3 | Z | 0
4 Assets | 4 | 10 | Z | 0.00097
Table 4. A detailed characterization of several application benchmarks using
the number of qubits, number of Pauli strings, measurement basis, and
entangled qubits required to find the ground state of the problem.

####  4.2.3. Estimated Cost Analysis

For each given pulse-level design space, we estimate the cost of
implementation in Table 3 in terms of the number of qubits N and the number of
layers L of the circuit. We assess the cost of implementation for each pulse-
level design space, taking into account the number of parameters, effective
parameter dimension, number of CR pulses, and pulse width. We see a linear
relationship between the number of parameters, effective parameter dimension,
and number of CR pulses for all design spaces and the number of qubits. In
addition, the pulsed circuit depth of all design spaces is proportional to the
number of qubits.

##  5\. Application Benchmarking

###  5.1. Benchmark Applications

####  5.1.1. Quantum Chemistry.

Quantum chemistry studies the electronic structure and reaction
characteristics of molecules using the ideas and tools of quantum mechanics to
describe and calculate these properties. Quantum chemistry seeks to answer the
following challenge in particular: how to determine the ground state energy,
electronic structure, and features such as energy barriers for a given
molecule. Using the proposed pulse-level design space, we intend to achieve
the variational quantum eigensolver (VQE) to solve the challenge of
determining the ground state energy of molecules in quantum chemistry as part
of our application benchmarking. This process can be considered as the
solution of an eigenvalue problem for a Hamiltonian-like matrix, where the
Hamiltonian describes the energy of particle interactions in a molecule and
the eigenvalue is the ground state energy. The number of qubits, the number of
Pauli strings, the measurement base, and the theoretical requirements of the
ground state of the task for entanglement are stated for both molecular
problems in Table 4.

####  5.1.2. Quantum Finance.

Quantum finance refers to the multidisciplinary application of quantum
computing technology to the financial sector. Using quantum computers, it
intends to handle a variety of financial problems, including but not limited
to risk management, portfolio optimization, pricing of financial derivatives,
etc. For our application benchmarking, we have selected the problem of
investment optimization. Investment optimization is the construction of the
optimal portfolio to achieve a particular purpose, given an underlying
investment and a set of constraints. The risk factor is set to 0.5, and we are
provided the two assets and four assets, as well as the randomly produced
‘historical data’ for each, and apply information to form the quadratic
problem, then we map the variables in the quadratic problem to qubits so that
translate the problem to Ising Hamiltonian. Our objective is to achieve VQE
using pulse-level design spaces in order to obtain the ideal portfolio model.
The information about the quantum finance problems is also provided in Table
4.

###  5.2. Parameters Constraint Generator

Parameterized pulse circuits operate by specifying a sequence of pulses that
must adhere to hardware-imposed limitations. First, the pulse amplitude must
be expressed as a fraction of the maximum output voltage of the arbitrary
waveform generator (AWG) and must therefore be constrained to the interval
[-1,1] (Egger et al., 2023). Further experimentation by us has shown that the
amplitude does not need to occupy the entire [-1,1] interval to achieve the
desired oscillation period on the quantum state, and that the range may differ
slightly across different hardware backends. The variance between the overall
minimum and maximum values is typically around 0.35. For example, in Fig. 10,
the amplitude range is [0.1, 0.4] on $ibmq\\_guadalupe$, thus, the training
cost could be greatly reduced by selecting a suitable range of amplitude
instead of [-1,1]. Second, the pulse angle affects the orientation of the
rotation axis in the plane and thus oscillates within the interval [0,2$\pi$].
Third, the duration of a pulse, expressed in the unit of dt, must be a
multiple of 16 to be loaded in the AWG, and the duration must be kept long
enough to prevent the width of the top except for the rise-fall edge from
being negative. To impose these constraints, we have introduced a parameter
constraint generator to restrict the distribution range of all the parameters.
For amplitude, we benchmarked multiple backends and provided a lookup-table
(LUT) to find the amplitude range corresponding to a particular backend. For
angle, we set this parameter to be within the range [0,2$\pi$]. For the
duration, we restrict this parameter to the range [256,1024].

Figure 10.  Dynamics of quantum states of an SQP with amplitude in the range
of [0.1, 0.4] on $ibmq\\_guadalupe$.

###  5.3. Evaluation and Analysis

All the application benchmarks were conducted on Qiskit-Dynamics and Qiskit-
Aer. Fig. 9 depicts a comparison of task performance between the suggested
pulse-level design space, pulses generated at random, and the gate-based
approach. We chose TwoLocal, which consists of Ry gates and CZ gates, and
RealAmplitude, which consists of Ry gates and CX gates, for the gate-based
model. There is no significant difference between the proposed design space
and the randomly produced pulses for quantum finance tasks. We are able to
identify the design space with a fixed CR amplitude that performs better in
terms of expressivity and entanglement capability but yields poorer
application results. This phenomenon is due to the factor we discussed in the
previous section that our tested duration is mapped discretely to unit ‘dt’
from hardware to software, thus, the power of parameter ‘duration’ is limited.
Thus, the criteria could be effective for guiding circuit design the majority
of the time, but due to the fact that the proposed pulse-level design space
must be evaluated further, application benchmarking is essential. In addition,
the gate-based model generally obtains better performance than the pulse-based
model in these tasks.

As we evaluate the quantum chemistry tasks, we find in the $H_{2}$ task, the
random-generated pulse with few numbers of parameters (pulse ID 6 and 7,
within the same number of parameters of Hardware-efficient pulse) performs
badly. As for the $LiH$ task, the proposed pulse-level design space (Pulse ID
from 1 - 6) achieves way better results than random-generated pulses, and also
slightly better than the gate-based model. The reference ground state energy
is -8.9407H, and the estimated energy of Pulse ID 1, 3, and 5 are -8.7766H,
-8.8946H, and -8.8114H, respectively, the ‘TwoLocal’ gate-based ansatz obtains
-8.6442H, as for ‘RealAmplitude’ gate-based ansatz, the result is -8.5214H. In
terms of the sequence duration of the circuit, we have 979dt, 1835dt, and
1954dt for Pulse ID 1, 3, and 5, whereas, ‘TwoLocal’ and ‘RealAmplitude’ with
24800dt and 5600dt, respectively. In conclusion, the proposed pulse-level
design space performs up to 4.2% in terms of the accuracy of the selected
quantum chemistry problem with 67.2% latency advantage in duration, and up to
96% shortening in terms of duration with 1.5% accuracy advantage compared to
the selected gate-based design.

The different observations from the quantum finance task and the quantum
chemistry task are illustrated in Table 4. The portfolio optimization problem
with 2 assets and 4 assets are both too simple for VQEs, as they have a light
number of Pauli strings and nearly no entanglement requirement by the ground
state of the problem, and the measurement base is only on Z-axis. Thus, these
problems can be even solved by random-generated pulses with good performance
and lead to tiny differences that are hard to compare between models. In
contrast, quantum chemistry problems have much more Pauli strings, stronger
entanglement requirements, and more complex measurement bases than on all of
X, Y, and Z, and thus, we observe the expectations in the quantum chemistry
benchmark.

##  6\. discussion

The ability to manipulate quantum states of qubits at the pulse level holds
great promise, as it provides fine-grained control over the qubits.
Parameterized quantum pulse enables researchers to explore new design space
and generate more customized and complex quantum operations by leveraging the
specific physical properties of the quantum processors. For instance, we have
shown that a parameterized pulse template can generate highly entangled states
with a high degree of expressivity, indicating the potential of designing
parameterized pulse as the building block for the ansatz of variational
quantum algorithms. Furthermore, the pulse naturally allows for flexible
tuning with sufficient parameters, allowing for the exploration of Hilbert
space fully (Cariolaro and Pierobon, 2010). In terms of EPD, the high EPD
value demonstrated in our experiments also indicates the low redundancy of
parameterized quantum pulse, ensuring efficient utilization of experimental
resources.

However, pulse-level control also has its limitations. One challenge is the
complexity of designing pulse sequences for arbitrary quantum operations, as
pulse-level control heavily relies on the physical implementation of the
qubits. Consequently, designing parameterized quantum pulse requires a deep
understanding of the underlying interactions and may require significant
computational resources. Another challenge is the impact of noise and
decoherence on the system. Basis gates can be fine-tuned for parameterized
gates, but the calibration of arbitrary pulses is generally difficult. These
effects can limit the effectiveness of pulse-level control, especially when
dealing with large-scale quantum systems. The formalism of quantum pulses is
still an incomplete field, making it challenging to design quantum protocols
at the pulse level.

Figure 11.  Binary classification of random-generated datasets with 16
features. a) Encoding by 8 SQPs on two qubits. b) Encoding by 16 single-qubit
rotation gates on two qubits.

Moreover, we have encoded classical data using a parameterized pulse encoder,
which we initially believed had the potential to encode more data in a shorter
amount of time. However, as shown in Fig. 11, in a simple experiment, we found
that the gate-based encoder produced superior results, even with a shallow
circuit. Although encoding classical data with parameterized pulses remains
intriguing, this observation is unfavorable, and further research is
necessary. We are still possible gain some benefits from a parameterized pulse
encoder, for example, it is possible to encode time-sequence datasets by
encoding time information into the duration parameter in parameterized pulse.

To overcome the limitations of quantum pulse applications, researchers are
exploring new approaches that combine pulse-level and gate-level control. For
example, in this paper, we designed multiple pulse templates from gate-
inspired ansatz with good performance. Such an approach allows for the
flexibility and fine-grained control of pulse-level control while maintaining
the simplicity and generality of gate-level protocols. Another approach is to
use machine learning algorithms to optimize the pulse sequences for specific
quantum tasks. This approach can reduce the complexity of programming the
hardware and improve the overall performance of the quantum computation
(Castaldo et al., 2021).

Overall, the advantages and limitations of parameterized quantum pulses
highlight the need for continued research and development in this area to
fully realize the potential of quantum computing. In this paper, we have
evaluated parameterized quantum pulses from various perspectives, including
expressivity, entanglement capability, effective parameter dimension,
duration, and application-oriented benchmarking, which we hope will pave the
way for future studies of quantum pulses.

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##  Appendix A Acknowledgement

We would like to express our gratitude for the utilization of IBM Quantum
services and PaddleQuantum in conducting this research. Additionally, our
appreciation extends to Dr. Yao Lu for the insightful discussions of this
work.