# Quantum Computer Benchmarking via Quantum Algorithms

Konstantinos Georgopoulos<EMAIL_ADDRESS>School of
Computing, Newcastle University, Newcastle-upon-Tyne, NE4 5TG, United Kingdom
Clive Emary  Joint Quantum Centre Durham-Newcastle, School of Mathematics,
Statistics and Physics, Newcastle University, Newcastle-upon-Tyne, NE1 7RU,
United Kingdom  Paolo Zuliani  School of Computing, Newcastle University,
Newcastle-upon-Tyne, NE4 5TG, United Kingdom

###### Abstract

We present a framework that utilizes quantum algorithms, an architecture aware
quantum noise model and an ideal simulator to benchmark quantum computers. The
benchmark metrics highlight the difference between the quantum computer
evolution and the simulated noisy and ideal quantum evolutions. We utilize our
framework for benchmarking three IBMQ systems. The use of multiple algorithms,
including continuous-time ones, as benchmarks stresses the computers in
different ways highlighting their behaviour for a diverse set of circuits. The
complexity of each quantum circuit affects the efficiency of each quantum
computer, with increasing circuit size resulting in more noisy behaviour.
Furthermore, the use of both a continuous-time quantum algorithm and the
decomposition of its Hamiltonian also allows extracting valuable comparisons
regarding the efficiency of the two methods on quantum systems. The results
show that our benchmarks provide sufficient and well-rounded information
regarding the performance of each quantum computer.

##  I Introduction

Benchmarking quantum computers aims to determine the performance of a quantum
computing system under an appropriate set of metrics. Within the Noisy
Intermediate-Scale Quantum (NISQ) era [1], benchmarking the capabilities and
performance of quantum computers when executing quantum programs is of
paramount importance, especially for assessing their scalability.

An intuitive approach to benchmarking quantum computers is establishing a set
of quantum programs and measuring the performance of a quantum computer when
executing each one. Such a work gains more merit as bigger quantum computers
are built. There are various advantages to this approach, for example
benchmarking the limits and behaviour of a quantum machine within a scaling,
computationally intensive environment and particularly testing the system when
performing a “real-world” task. Various companies appear to favor such an
approach. IonQ for example tested their quantum computer using the Bernstein-
Vazirani [2] and the Hidden Shift [3, 4] algorithms. The metric for
performance was the likelihood of measuring the correct output [5]. On the
other hand, Google focused on the problem of quantum sampling and achieved
results they claim demonstrated quantum supremacy [6]. While the chosen
application was not particularly useful in a “real-world” scenario, it
excelled at demonstrating the computing power of the system. So a question
arises, highlighting the difficulty of creating quantum program benchmarks:
which benchmarks are more insightful?

The different competing quantum technologies pose a major challenge. The
technologies have different topologies and thus have unique strengths and
weaknesses. For example, the connectivity of an ion-trap computer provided a
large advantage on some benchmarks over a superconducting quantum computer
[7].

Clearly, systems can behave differently on different benchmarks, which
introduces the issue of invested interests, i.e., using benchmarks that are
expected to perform well on a current system [8]. While impressive, current
quantum computers are very small compared to the computers we hope to build in
the coming years. Hence, current benchmarks are also relatively simple
compared to truly useful programs. While running smaller versions of real-
world applications introduces error, and is accepted in classical
benchmarking, this is exacerbated for quantum computers. Entirely new issues
may be introduced when scaling up and it is difficult to say whether
performances measured today are good indicators of future performance. For
example, IonQ’s computer [5] has all $11$ qubits fully-connected. This
configuration is possible at this scale, but this might not be true for a
system with hundred or a thousand qubits. Such a system will likely require
multiple fully-connected groups of qubits and communication will need to be
orchestrated between them [9]. This introduces additional complexity which is
not found in small-scale benchmarks.

Our previous question can be refined as: which quantum algorithms would be
useful for program benchmarking quantum computers in the future? Algorithms
such as quantum Markov chains [10], Shor’s [11], Grover’s [12] and quantum
chemistry [7, 13, 14] are some obvious examples. Even if, for the most part,
these algorithms will remain out of reach for near-term quantum computers,
there is much to be gained from analyzing their scalability and reaction to
noise. Currently, classical-quantum hybrid algorithms [15, 16, 17, 18, 19] are
popular due to their ability to make use of the limited resources of NISQ
computers. Another example of an algorithm that holds great potential for
benchmarking are quantum walks, due to their susceptibility to noise and clear
quadratic advantage over classical random walks [20].

Within this paper we use quantum algorithms to benchmark three of the newer
IBM superconducting quantum computers: the $5$-qubit Bogota and Santiago and
the $7$-qubit Casablanca machines. The choice of quantum algorithms is
crucial, as we want them to be (i) scalable, in order to be able to face the
challenge of the growing number of qubits in quantum computers, (ii)
predictable in a way that allows us to recognize its noisy behaviour, and
(iii) demonstrate clear quantum advantage. With these criteria in mind, we
choose five algorithms: discrete-time quantum walks [21], continuous-time
quantum walks [22], a circuit simulating the continuous-time quantum walk by
decomposing its Hamiltonian to a sequence of Pauli gates, quantum phase
estimation [23] and Grover’s algorithm [12]. Finally, we note that using a
continuous-time quantum algorithm and its Hamiltonian decomposition for
benchmarking a (digital) quantum computer has not been carried out before, to
the best of our knowledge. For the experiments we make use of the IBM Qiskit
development kit [24, 25] to simulate and execute the quantum circuits.

The paper is organized as follows. Section II introduces the preliminary
methods necessary for the benchmarking process. Section III defines the three
benchmark metrics that result from the benchmarking process as well as
presents a concrete framework for program benchmarking quantum computers.
Moving on, the experimental process and the benchmark results for the chosen
quantum computers are layed out in Section IV before finally concluding the
paper in Section V.

##  II Preliminary Methods

This section introduces the noise model used to simulate the behaviour of
quantum computers. Additionally, it offers a brief discussion on the five
quantum algorithms we use for benchmarking. Finally, it presents the quantum
computers we are interested in benchmarking and the mathematical foundation of
the final benchmark metrics.

###  II.1 Unified Noise Model

To approximate the noisy behaviour of quantum computers we use the unified
noise model (UNM) we have recently developed [26]. This model combines three
sources of error: (i) hardware infidelities in the form of gate, state
preparation and measurement errors, (ii) decoherence in the form of thermal
relaxation and (iii) dephasing of the qubits. The experiments in [26] show
that the UNM performs very well at approximating the behaviour of the IBMQ
$15$-qubit Melbourne computer and better than other state of the art noise
models.

The main characteristic of the UNM is its architecture awareness: the
architectural graph that encompasses all the information regarding the
connectivity of the qubits within the quantum processing unit (QPU) gets
encoded within the model itself. Additionally, the model uses a number of
noise parameters (see Table 1) calibrated from the machine itself, i.e.,
parameters that express the error rates of the gates, state preparations and
measurements as well as the time it takes for the qubits within the QPU to
decohere and dephase. Each noise parameter is unique and corresponds to each
qubit individually or pair of qubits.

Parameter | Error Type | No. Parameters
---|---|---
$p_{r}$ | Gate error rates | $r$
$p_{m}$ | State preparation error rates | $m$
$p_{s}$ | Measurement error rates | $s$
$T_{1}$ | Thermal relaxation times | $n$
$T_{2}$ | Dephasing times | $n$
Table 1: The noise parameters and number of noise parameters for each type of
error within the UNM; $n$ is the number of qubits in the system, $m$ is the
number of qubits that are measured, $s$ is the number of state preparations
that occur and $r$ is the number of distinct types of gates implemented in the
architecture, each considered once per qubit or pair of qubits.

###  II.2 The Quantum Algorithms

Here we give a quick overview of the quantum algorithms we use for
benchmarking, i.e., (i) discrete-time quantum walks (DTQW), (ii) continuous-
time quantum walks (CTQW), (iii) Pauli decomposition of the CTQW Hamiltonian
(PD) (iv) quantum phase estimation (QPE) and (v) quantum search (QS). These
algorithms are chosen according to three major characteristics that are
interesting for benchmarking:

  * •

Scalability. The algorithm should be able to scale up (or down) and run on
increasingly larger quantum systems.

  * •

Predictability. The algorithm should produce a result that is easily
predictable. An important addition to predictability is noise susceptibility:
the algorithm should provide a result whose distortion under the effects of
noise is easily distinguishable from the ideal evolution.

  * •

Quantum advantage. The algorithm should provide a computational speed-up over
its classical counterpart or, in other words, represent a possibly relevant
real-world application.

##### Discrete-time Quantum Walks.

Quantum walks (DTQW) are the quantum mechanical analogue of a classical random
walk on a graph or a lattice [21, 27, 28]. They exhibit intrinsic properties
that render their evolution easily predictable and highly susceptible to
noise, making them an ideal candidate for benchmarking. First of all,
discrete-time quantum walks exhibit modular behaviour [27, 29]. This
characteristic describes the modular relationship between the parity of the
number of coin-flips of the walk, the initial state and the current position
of the walker, a property that gets violated in a noisy environment [27].
Secondly, quantum walks propagate quadratically further than classical random
walks [21, 20] thus showing clear quantum advantage over their classical
counterparts.

Finally, quantum walks are a highly scalable process. The size of the state-
space of a quantum walk (i.e., the number of states that the walk traverses,
represented by the number of qubits in the relevant register) can easily
increase to match the size of the quantum computer we are interested in
benchmarking.

##### Continuous-time Quantum Walks.

Continuous-time quantum walks (CTQW) were first introduced by Fahri and Gutman
in [22]. This algorithm, much like the DTQWs, have an easily predictable
quantum evolution that is highly susceptible to quantum noise, but exhibit
very different characteristics to the discrete case. First of all, the CTQW
evolution is determined by a Hamiltonian, $H$, instead of a coin-flip and is
driven by a unitary of the form $e^{-iHt}$. Unlike the DTQWs, continuous-time
quantum walks do not exhibit modular behaviour. Although like DTQWs they
feature a quadratic increase in the walker’s propagation [30, 31].

Finally, CTQWs are an easily scalable process as adding qubits to circuit can
scale up the size of the state-space. To the best of our knowledge, this is
the first work that uses a continuous-time quantum algorithm to benchmark the
performance of a digital quantum computer. There are a few ways that a
continuous-time quantum walk can be implemented on a gate-based (i.e.,
discrete) quantum computer, for example simulating the CTQW evolution using a
DTQW [32] or approximating the CTQW through a decomposition of the unitary
$e^{-iHt}$ to a sequence of universal gates. Within this paper, the latter
approach is utilized.

##### Pauli Decomposition of CTQW Hamiltonian.

Decomposition of quantum Hamiltonians to a set of universal gates is a well-
studied area of research [33, 34, 35, 36, 37]. For this paper, we are
interested in decomposing the Hamiltonian of the CTQW using the well-known set
of Pauli matrices as the universal gate set. A further detailed analysis of
this process is presented in the following Section II.3.

This procedure, often called Hamiltonian simulation, adheres to the criteria
for a good benchmarking process via the algorithm it decomposes. In other
words, since the CTQW is suitable for benchmarking, so is the circuit that
implements the decomposition of the CTQW Hamiltonian. Furthermore, it provides
added value to this research since one can evaluate the performance of the
quantum computer when executing the Hamiltonian simulation of a quantum
process (i.e., CTQW), as well as compare the decomposition with the original
algorithm.

##### Quantum Phase Estimation.

The quantum phase estimation (QPE) algorithm is used to estimate the phase (or
eigenvalue) of an eigenvector of a unitary operator. More precisely, given an
arbitrary quantum operator $U$ and a quantum state $\ket{\psi}$ such that
$U\ket{\psi}=e^{2i\pi\theta}\ket{\psi}$, the algorithm estimates the value of
$\theta$, given an approximation error [38, 23, 39].

Within this paper we exploit three characteristics of QPE that make it
interesting for benchmarking. First of all, it is a scalable algorithm as
further accuracy of the result (i.e., the estimated phase) can be obtained by
increasing the number of qubits in the system. Furthermore, the result of the
QPE is predictable and highly susceptible to quantum noise. Finally, QPE
offers clear quantum advantage, achieving an exponential speed-up over known
classical methods, rendering the algorithm one of the most important
subroutines in quantum computing and serving as the building block of major
quantum algorithms, like Shor’s [11] or the HHL algorithms [40].

##### Quantum Search.

Grover’s algorithm [12] describes a process for searching for a specific item
within a database. Within this paper we use quantum search (QS) to look for a
specific number $s$ within a set of numbers
$\mathcal{S}=\\{0,\dots,2^{n-1}\\}$, where $n$ is the number of qubits within
the quantum system that participate in the computation.

Quantum search represents an ideal algorithm for benchmarking quantum
computers. The algorithm can scale up to search for an item within a larger
database simply by adding qubits to the relevant quantum register. The result
is easily predictable, as it is simply the item (or number, in our case)
sought, as well as highly susceptible to noise (for example, the wrong result
might appear due to noise). Additionally, QS can speed up an unstructured
search problem quadratically, thus making it a very alluring application for
quantum computers. Finally, Grover’s algorithm can serve as a general trick or
subroutine to obtain quadratic runtime improvements for a variety of other
algorithms through what is called amplitude amplification [41].

###  II.3 Pauli Decomposition of CTQW Hamiltonian

In general, an arbitrary Hamiltonian $H$ of size $N\times N$, where $N=2^{n}$
and $n$ the number of qubits in the system, can be decomposed into a sequence
of Pauli operators of the set
$S=\\{\sigma_{I},\sigma_{X},\sigma_{Y},\sigma_{Z}\\}$ with well-known matrix
representation as

$H=\sum_{i_{1},\dots,i_{N}=I,x,y,z}\alpha_{i_{1},\dots,i_{N}}\left(\sigma_{i_{1}}\otimes\dots\otimes\sigma_{i_{N}}\right),$
(1)

where

$\alpha_{i_{1},\dots,i_{N}}=\frac{1}{N}\operatorname{tr}\left[\left(\sigma_{i_{1}}\otimes\dots\otimes\sigma_{i_{N}}\right)\cdot
H\right].$

For this paper we implement a CTQW on an $N$-cycle with Hamiltonian defined as
$H_{\text{qw}}=\gamma A=\frac{1}{d}A=\frac{1}{2}A$, where $\gamma=1/d$ is the
hopping rate between the two adjacent nodes in the cycle (i.e., node degree
$d=2$) and $A$ is the adjacency matrix. Note that, for the physical purposes
of this evolution, the rate $\gamma_{n|i}$ has dimensions $1/t$, where $t$ is
the time duration of the CTQW evolution, the Hamiltonian has the appropriate
energy dimensions and, for simplicity, $\hbar=1$.

The next step is to construct the unitary evolution operator from the
Hamiltonian. This can be done by exponentiating the Hamiltonian as $e^{-iHt}$
where $H$ is a sum of terms of the form of equation (1). It is important here
to consider two things. First of all, during matrix exponentiation, a
decomposition of the form $e^{-i(H_{1}+H_{2})t}=e^{-iH_{1}t}e^{-iH_{2}t}$,
where $H_{1}$ and $H_{2}$ are Hermitian operators, is possible if $H_{1}$ and
$H_{2}$ commute, i.e., $H_{1}H_{2}-H_{2}H_{1}=0$. This rule is naturally
expanded for more than two matrices on the exponent.

Secondly, in the case that not all matrices in the exponent commute, the
unitary operator resulting from the Hamiltonian exponentiation needs to be
decomposed using the Lie-product formula [33] as

$e^{-i(H_{1}+H_{2}+\dots)t}\approx\left(e^{-iH_{1}t/r}e^{-iH_{2}t/r}\dots\right)^{r}$
(2)

where, for our case, $H=\sum_{i}H_{i}$ is the Pauli decomposition of the
Hamiltonian to a sequence of Hermitian terms. This formula will create an
approximation of the Hamiltonian with bounded error depending on $r$ [33, 35].
To ensure that the Pauli Hamiltonian decomposition exhibits error at most
$\epsilon$, the bound $r$ can be taken as [33]

$r=\mathcal{O}\left((||H||t)^{2}/\epsilon\right),$

where $||H||$ is the norm of the Hamiltonian $H$ and $t$ is the continuous-
time duration of the quantum evolution.

Thus, using the above methodology, we can now decompose the unitary that
describes the continuous-time evolution of the CTQW (i.e., $e^{-iHt}$) in a
sequence of Pauli operations that approximate said evolution. As the Pauli
gates form a universal gate set, the resulting quantum circuit can be
implemented on the gate-based quantum processors.

###  II.4 The Quantum Computers and Circuits

Within this paper we are interested in benchmarking three computers, the
$5$-qubit Bogota, Santiago and the $7$-qubit Casablanca machine. They all
exhibit a quantum volume $V_{Q}=32$ [25, 42].

##### Quantum computer architectures.

Throughout our research we find that it is essential for the benchmarking
procedure to be architecturally aware. This is also reflected by our choice of
the UNM, an architecture aware noise model [26]. Figure 1 shows the qubit
connectivity of the quantum computers we benchmark in this paper.

---
(a)
(b)
Figure 1: Architectural graphs of the (a) IBMQ $5$-qubit Bogota and Santiago
machines and (b) the IBMQ $7$-qubit Casablanca machine. A node in the graph
represents a physical qubit whereas an edge represents a connection between a
pair of qubits.

##### Quantum circuits and characteristics.

For the implementation of quantum walks, we use a gate efficient approach that
is based on inverter gates, as shown in [43]. The QPE circuit is based on [44]
and heavily relies on quantum Fourier transform (and its inverse) [45] to
estimate the relevant eigenvalue. For QS we make use of two approaches to the
implementation, one with ancilla qubits (QSa) and one without (QSn) [46].

For the continuous-time quantum walk, the Hamiltonian is automatically
implemented by the Qiskit API when submitted for execution on the quantum
computer. The Pauli decomposition (PD) of the Hamiltonian can be easily
implemented on the quantum computers as it already maps the continuous-time
Hamiltonian to a discrete basis gate set decomposition.

We identify four quantum circuit characteristics that are of interest for
benchmarking:

  1. i

the number of gates in the circuit,

  2. ii

the number of active qubits, or qubits that are utilized by the quantum
circuit (also called workspace),

  3. iii

the depth of the circuit, i.e., the longest path between the start of the
circuit and a measurement gate and

  4. iv

the runtime of the circuit on the quantum computer. Table 2 shows those
characteristics of the quantum circuits that implement each quantum algorithm.

| Machine | No. Gates | Size of Workspace | Depth | QC Runtime $\pm$ s.d. (ms)
---|---|---|---|---
$5$-qubit Bogota | $47$ | $3$ | $35$ | $2.41\pm 0.03$
$5$-qubit Santiago | $47$ | $3$ | $35$ | $2.34\pm 0.02$
$7$-qubit Casablanca | $47$ | $3$ | $35$ | $2.54\pm 0.05$
(a) Discrete-time quantum walk circuit characteristics.
| Machine | No. Gates | Size of Workspace | Depth | QC Runtime $\pm$ s.d. (ms)
---|---|---|---|---
$5$-qubit Bogota | $19$ | $2$ | $13$ | $2.34\pm 0.04$
$5$-qubit Santiago | $19$ | $2$ | $13$ | $3.01\pm 0.07$
$7$-qubit Casablanca | $19$ | $2$ | $13$ | $2.62\pm 0.01$
(b) Continuous-time quantum walk circuit characteristics.
| Machine | No. Gates | Size of Workspace | Depth | QC Runtime $\pm$ s.d. (ms)
---|---|---|---|---
$5$-qubit Bogota | $243$ | $2$ | $183$ | $2.42\pm 0.04$
$5$-qubit Santiago | $243$ | $2$ | $183$ | $2.28\pm 0.06$
$7$-qubit Casablanca | $243$ | $2$ | $183$ | $3.76\pm 0.04$
(c) Pauli decomposition of CTQW circuit characteristics.
| Machine | No. Gates | Size of Workspace | Depth | QC Runtime $\pm$ s.d. (ms)
---|---|---|---|---
$5$-qubit Bogota | $93$ | $4$ | $66$ | $2.54\pm 0.01$
$5$-qubit Santiago | $97$ | $4$ | $72$ | $2.26\pm 0.03$
$7$-qubit Casablanca | $100$ | $4$ | $75$ | $2.68\pm 0.06$
(d) Quantum phase estimation circuit characteristics.
| Machine | No. Gates | Size of Workspace | Depth | QC Runtime $\pm$ s.d. (ms)
---|---|---|---|---
$5$-qubit Bogota | $497$ | $4$ | $358$ | $2.70\pm 0.08$
$5$-qubit Santiago | $479$ | $4$ | $343$ | $2.56\pm 0.04$
$7$-qubit Casablanca (ancilla) | $788$ | $6$ | $503$ | $2.81\pm 0.04$
$7$-qubit Casablanca (no ancilla) | $465$ | $4$ | $336$ | $2.74\pm 0.03$
(e) Quantum search circuit characteristics.
Table 2: Quantum circuit characteristics for the five quantum circuits. No.
gates and size of workspace are the number of gates and active qubits in the
circuit respectively; depth of the circuit is the longest path between the
start of the circuit and a measurement gate; QC runtime is the approximate
average execution time of the circuit on the quantum computer along with
standard deviation (s.d.).

###  II.5 Benchmark Indicators: Hellinger Distance

The end results of the benchmarking process will be in the form of a
comparison of the quantum computer output distribution with the distributions
resulting from the unified noise model simulations of each machine and the
ideal evolution.

To compare the probability distributions and generate the benchmarks, we use
the Hellinger distance (HD) [47].

######  Definition 1 (Hellinger distance).

For probability distributions $P=\\{p_{i}\\}_{i\in[s]}$,
$Q=\\{q_{i}\\}_{i\in[s]}$ supported on $[s]$, the Hellinger distance between
them is defined as

$h(P,Q)=\frac{1}{\sqrt{2}}\sqrt{\sum_{i=1}^{k}\left(\sqrt{p_{i}}-\sqrt{q_{i}}\right)^{2}}.$
(3)

The Hellinger distance is a metric satisfying the triangle inequality. It
takes values between $0$ and $1$ (i.e. $h(P,Q)\in[0,1]$) with $0$ meaning that
the two distributions are equal. Additionally, it is easy to compute, easy to
read and it does not depend on the probability distributions having the same
support. The last property is particularly useful since in many ideal output
distribution of quantum circuits the probability mass is concentrated on a few
states.

##  III Framework for Program Benchmarking

The desired benchmarks correspond to comparisons (via the Hellinger distance)
between the evolution of the quantum system on the computer, the noisy
simulation and the ideal case. To achieve a more memorable notation, in the
following analysis the symbol $q$ denotes the quantum computer, $i$ the ideal
evolution and $n$ the noisy simulation. For example, the subscript $q|n$
denotes a value that corresponds to the difference between the quantum
computer evolution ($q$) and the noisy evolution ($n$). We define three
distances of interest, or otherwise, three benchmark metrics, as follows.

######  Definition 2 (alpha benchmark).

The Hellinger distance between the probability distribution of the quantum
computer evolution, $Q$, and the ideal distribution, $D$, namely
$h_{\text{id}}(Q,D)$, is the alpha benchmark, with notation $\alpha_{q|i}$:

$\alpha_{q|i}\equiv h(Q,D).$ (4)

######  Definition 3 (beta benchmark).

The Hellinger distance between the probability distribution of the quantum
computer evolution, $Q$, and the distribution resulting from the noisy
simulations, $N$, namely $h_{\text{nm}}(Q,N)$, is the beta benchmark, with
notation $\beta_{q|n}$:

$\beta_{q|n}\equiv h(Q,N).$ (5)

######  Definition 4 (gamma benchmark).

The Hellinger distance between the distribution resulting from the noisy
simulations, $N$, and the ideal distribution, $D$, namely
$h_{\text{sm}}(N,D)$, is the gamma benchmark, with notation $\gamma_{n|i}$:

$\gamma_{n|i}\equiv h(N,D).$ (6)

As mentioned in Section II.5, the Hellinger distance is a metric that
satisfies the triangle inequality. Hence, since the benchmarks established in
the above definitions describe the pairwise Hellinger distances between three
probability distributions (the quantum computer, $Q$, the simulated evolution,
$N$, and the ideal evolution, $D$), it is possible to derive a relationship
between $\alpha_{q|i}$, $\beta_{q|n}$ and $\gamma_{n|i}$ via the triangle
inequality, through the following Lemma (whose proof follows simply from the
triangle inequality for a metric).

######  Lemma 1 (benchmarks triangle inequality).

Given probability distributions $Q$, $N$ and $D$, as established in the
benchmark definitions, the pairwise Hellinger distances between those
distributions, i.e., $h(Q,D)$, $h(Q,N)$ and $h(N,D)$ follow the triangle
inequality:

$h(Q,D)\leq h(Q,N)+h(N,D).$

Thus, the relevant benchmarks will also follow the triangle inequality as:

$\alpha_{q|i}\leq\beta_{q|n}+\gamma_{n|i}$ (7)

The above Lemma effectively means that, according to the benchmark
definitions, the deviation of the quantum computer evolution from the ideal
($\alpha_{q|i}$) will never be greater than the sum of the expected (i.e.,
simulated) evolution derived by the levels of noise within the machine and the
distance between the simulated and ideal distributions
($\beta_{q|n}+\gamma_{n|i}$). In other words, defining the benchmark metrics
using the Hellinger distance allows us to quantify the confidence on the
estimated level of noise during the execution of the quantum circuit.

Following the definitions and relationship between the benchmarks, we present
a framework for program benchmarking quantum computers as the sequence of the
six steps below:

Step 1

Select benchmark method(s). The selection of quantum algorithms that will be
used for benchmarking should adhere to the three criteria described in Section
II.2: (i) scalability, (ii) predictability and noise susceptibility and (iii)
quantum advantage.

Step 2

Quantum noise model and simulator. Select or implement a noise model that
approximates the noisy evolution within the quantum machine and a simulator
that can execute the noise model.

Step 3

Run experiments. Design and execute a suitable number of experiments of the
benchmark method(s) on the quantum computer. This step also includes
calibrating the noise parameters that encapsulate the level of noise within
the quantum computer at the time of the experiments.

Step 4

Simulate the noisy evolution. Use the noise model in order to simulate the
noisy evolution of the quantum computer using the calibrated noise parameters
from the time of the experiment, as described in Step 3.

Step 5

Simulate the ideal evolution. This can be done either through simple noise-
free simulations or by calculating the probabilities through the quantum
statevector.

Step 6

Calculate the benchmarks. The final benchmarks $\alpha_{q|i}$, $\beta_{q|n}$
and $\gamma_{n|i}$ are the Hellinger distances between the quantum computer,
the UNM and the ideal evolution in the setting described in Definitions 2, 3
and 4.

Following the benchmarking framework we can extract meaningful results from a
series of comparisons between the benchmark metrics, $\alpha_{q|i}$,
$\beta_{q|n}$ and $\gamma_{n|i}$. The $\beta_{q|n}$ benchmark essentially
describes how closely the noise model simulates the behaviour of the quantum
computer. The $\alpha_{q|i}$ benchmark shows how far the behaviour of the
quantum computer falls from the noise-free case thus giving an estimate of the
overall computer performance under the effects of noise. From the comparison
between the $\beta_{q|n}$ and $\alpha_{q|i}$ benchmarks we can extract
valuable information: if $\beta_{q|n}<\alpha_{q|i}$ the noise levels in the
quantum computer are closer to the estimated ones from the noise simulations;
on the opposite case the computer operates closer to the ideal evolution. In
the latter case the quantum computer behaves more efficiently with lower level
of noise than expected, thus giving us more confidence regarding the
computational result.

The $\gamma_{n|i}$ benchmark, even though it can be used as an estimate of the
noise levels in the quantum computer, does not give any relevant information
on its own. The value comes when considering the $\gamma_{n|i}$ benchmark
together with the $\alpha_{q|i}$ benchmark. First of all, the closer the
values of $\alpha_{q|i}$ and $\gamma_{n|i}$ are, the smaller the value of
$\beta_{q|n}$ (if $\beta_{q|n}=0$ then $\alpha_{q|i}=\gamma_{n|i}$ and vice
versa). Additionally, if $\alpha_{q|i}>\gamma_{n|i}$, then the noise model,
and hence the noise parameters, underestimate the levels of noise during the
quantum computer evolution, with the opposite being true if
$\alpha_{q|i}<\gamma_{n|i}$. Moreover, the absolute difference
$\lvert\alpha_{q|i}-\gamma_{n|i}\rvert$ can quantify this noise over- or
underestimation. Precisely, the smaller the absolute difference the smaller
the error in estimation. This information is useful to the benchmarking
process as it further highlights whether the machine is more or less noisy
than estimated while also showing the efficiency of the machine calibration
techniques.

Finally, using the triangle inequality from Lemma 1 we can make further
interesting remarks. From equation (7) we find that
$\beta_{q|n}\geq\alpha_{q|i}-\gamma_{n|i}$. Furthermore, from the above
analysis we know that a comparison between $\alpha_{q|i}$ and $\gamma_{n|i}$
can tell us whether the calibrated noise parameters over- or underestimate the
level of noise during the evolution on the quantum computer, and the absolute
difference $\lvert\alpha_{q|i}-\gamma_{n|i}\rvert$ can give an indication of
the scale of the error in estimation. Importantly, the above inequality does
not hold for absolute values when subtraction takes place (i.e., for an
inequality of the form
$\lvert\beta_{q|n}\rvert\geq\lvert\alpha_{q|i}-\gamma_{n|i}\rvert$ when
$\gamma_{n|i}>\alpha_{q|i}$). Considering this, we can interpret the triangle
inequality as a measure of the confidence on the calibrated parameters
encapsulating a picture of the noise that is accurate enough to provide a good
estimation of the quantum evolution, expressed as follows:

  * •

If $\beta_{q|n}\geq\lvert\alpha_{q|i}-\gamma_{q|i}\rvert$, then the over- or
underestimation of the noise is small enough to provide estimates of the
quantum evolution with high confidence.

  * •

If $\beta_{q|n}<\lvert\alpha_{q|i}-\gamma_{q|i}\rvert$, then the calibrated
parameters generate low confidence on the levels of noise.

##  IV Experiments and Results

###  IV.1 Experimental Setup

For the benchmarking experiments we run the quantum circuits that implement
the chosen quantum algorithms. In the case of the DTQW we run one step of the
algorithm (i.e., one coin-flip) on a workspace (i.e., number of active qubits
used by the circuit) of three qubits with initial state $\ket{0}$, as previous
work shows that this configuration is satisfactory for errors to take place
and the behaviour of the quantum walk to evolve in a predictable manner [27].
For the CTQW and its PD we use a small two-qubit state-space, as the Pauli
decomposition gets excessively large for a three-qubit state space or bigger.
The algorithms are implemented for a continuous time of $t=3$. Our QPE routine
is tailored to estimate a phase of $\theta=2\pi/3$ using a workspace of four
qubits with theoretical probability of success estimated at $0.688$. Finally,
the QS implementation performs an unstructured search for the element
$\ket{s}=\ket{10}$ (decimal) within a four qubits state-space i.e., a dataset
containing numbers $\ket{0}$ to $\ket{15}$. The QS algorithm is run for three
iterations and shows a theoretical probability of success estimated at $0.96$.
Table 3 shows more comprehensively the initial configuration for each
algorithm.

Algorithm | No. Qubits | Size of Workspace | Iterations/Duration | Probability of Success
---|---|---|---|---
DTQW | $2$ | $3$ |  $1$ (one coin-flip) |  $p_{\text{succ}}=0.5$ in states $\ket{1}$ and $\ket{3}$
CTQW | $2$ | $2$ | $t=3$ |  $p_{\ket{2}}=0.99$, $p_{\ket{1}}=p_{\ket{3}}=0.005$
PD | $2$ | $2$ | $t=3$ |  $p_{\ket{2}}=0.99$, $p_{\ket{1}}=p_{\ket{3}}=0.005$
QPE | $3$ | $4$ | $1$ |  $p_{\text{succ}}=0.688$ in state $\ket{3}$
QSa | $4$ | $6$ | $3$ |  $p_{\text{succ}}=0.96$ in state $\ket{10}$
QSn | $4$ | $4$ | $3$ |  $p_{\text{succ}}=0.96$ in state $\ket{10}$
Table 3: The initial configuration for the quantum walk (DTQW), continuous-
time quantum walk (CTQW) and its Pauli decomposition (PD), quantum phase
estimation (QPE) and quantum search algorithms with ancilla (QSa) and without
ancilla (QSn) for the benchmarking experiments. No. qubits is the number of
qubits in the state space, size of workspace is the number of active qubits
utilized by the circuit, iterations is the number of repetitions of the
quantum circuit and the probability of success is the theoretical probability
of the algorithm to give us the correct (or expected) result.

We run each algorithm independently $100{,}000$ times (see Section II.2) on
the three chosen computers (see Section II.4) and as a simulation using the
UNM (see Section II.1) and the ideal simulator [24] to reproduce the noisy and
the ideal quantum evolutions respectively. Finally, we compute the
$\beta_{q|n}$, $\alpha_{q|i}$ and $\gamma_{n|i}$ benchmarks, as defined in
Section III. The results are shown on Table 4 for each machine and are also
visualized in Figure 2.

Figure 2: Visualization of the three benchmarks, $\alpha_{q|i}$, $\beta_{q|n}$ and $\gamma_{n|i}$ for each of the three quantum computers (mapped to the left-hand side $y$-axis) when executing the five quantum algorithms. | Machine | $\alpha_{q|i}$ | $\beta_{q|n}$ | $\gamma_{n|i}$ | $\lvert\alpha_{q|i}-\gamma_{n|i}\rvert$ | QC Runtime (sec) | Sim. Runtime (sec)
---|---|---|---|---|---|---
$5$-qubit Bogota | $0.287$ | $0.025$ | $0.291$ | $0.004$ | $240.7$ | $3{,}331.5$
$5$-qubit Santiago | $0.321$ | $0.054$ | $0.362$ | $0.041$ | $234.2$ | $3{,}119.3$
$7$-qubit Casablanca | $0.269$ | $0.076$ | $0.331$ | $0.062$ | $254.5$ | $4{,}626.6$
(a) Quantum walk algorithm on two qubits.
| Machine | $\alpha_{q|i}$ | $\beta_{q|n}$ | $\gamma_{n|i}$ | $\lvert\alpha_{q|i}-\gamma_{n|i}\rvert$ | QC Runtime (sec) | Sim. Runtime (sec)
---|---|---|---|---|---|---
$5$-qubit Bogota | $0.139$ | $0.096$ | $0.234$ | $0.095$ | $234.4$ | $3{,}026.9$
$5$-qubit Santiago | $0.154$ | $0.035$ | $0.184$ | $0.030$ | $301.7$ | $2{,}905.7$
$7$-qubit Casablanca | $0.247$ | $0.044$ | $0.239$ | $0.008$ | $262.1$ | $3{,}035.2$
(b) Continuous-time quantum walk algorithm on two qubits.
| Machine | $\alpha_{q|i}$ | $\beta_{q|n}$ | $\gamma_{n|i}$ | $\lvert\alpha_{q|i}-\gamma_{n|i}\rvert$ | QC Runtime (sec) | Sim. Runtime (sec)
---|---|---|---|---|---|---
$5$-qubit Bogota | $0.436$ | $0.041$ | $0.467$ | $0.031$ | $242.7$ | $11{,}764.9$
$5$-qubit Santiago | $0.514$ | $0.084$ | $0.465$ | $0.049$ | $228.5$ | $11{,}618.6$
$7$-qubit Casablanca | $0.577$ | $0.039$ | $0.612$ | $0.035$ | $376.4$ | $12{,}861.3$
(c) Pauli decomposition of the CTQW algorithm on two qubits.
| Machine | $\alpha_{q|i}$ | $\beta_{q|n}$ | $\gamma_{n|i}$ | $\lvert\alpha_{q|i}-\gamma_{n|i}\rvert$ | QC Runtime (sec) | Sim. Runtime (sec)
---|---|---|---|---|---|---
$5$-qubit Bogota | $0.469$ | $0.177$ | $0.351$ | $0.118$ | $254.2$ | $6{,}287.1$
$5$-qubit Santiago | $0.488$ | $0.144$ | $0.369$ | $0.119$ | $226.8$ | $6{,}304.2$
$7$-qubit Casablanca | $0.524$ | $0.267$ | $0.351$ | $0.173$ | $268.3$ | $7{,}741.1$
(d) Quantum phase estimation algorithm on three qubits.
| Machine | $\alpha_{q|i}$ | $\beta_{q|n}$ | $\gamma_{n|i}$ | $\lvert\alpha_{q|i}-\gamma_{n|i}\rvert$ | QC Runtime (sec) | Sim. Runtime (sec)
---|---|---|---|---|---|---
$5$-qubit Bogota | $0.754$ | $0.014$ | $0.752$ | $0.002$ | $270.4$ | $22{,}834.3$
$5$-qubit Santiago | $0.751$ | $0.051$ | $0.738$ | $0.013$ | $256.7$ | $21{,}923.2$
$7$-qubit Casablanca (ancilla) | $0.760$ | $0.040$ | $0.752$ | $0.008$ | $280.1$ | $36{,}425.7$
$7$-qubit Casablanca (no ancilla) | $0.763$ | $0.061$ | $0.738$ | $0.025$ | $274.1$ | $22{,}889.9$
(e) Quantum search algorithm on four qubits.
Table 4: Results from benchmarking the three machines for each of the five
algorithms. The benchmark indicators $\alpha_{q|i}$, $\beta_{q|n}$ and
$\gamma_{n|i}$ are the HD between the quantum computer and the UNM, the
quantum computer and the ideal distribution and the UNM and the ideal
distribution respectively. QC runtime is the cumulative execution time in
seconds for $100{,}000$ iterations of each algorithm on the respective
machine; Sim. runtime is the time it takes to simulate for $100{,}000$
iterations the behaviour of each machine when executing the algorithms.

###  IV.2 Results

The results of the benchmark experiments are presented in Table 4 and below
for each of the algorithms used.

It is important at this point to emphasize an advantage of the structure
presented for program benchmarking quantum computers. The use of the Hellinger
distance to define the three benchmarks ($\alpha_{q|i}$, $\beta_{q|n}$ and
$\gamma_{n|i}$) allows us to perform a comparison of the performance of
different quantum algorithms on the same basis, i.e., a dimensionless
quantity. In other words, through the proposed framework, apart from measuring
the efficiency of a quantum processor, we can gather additional information
and compare the efficiency of circuit implementations for the selected
algorithms. This is done here for the two approaches on implementing the CTQW
on a gate-based computer, one that is done through the IBMQ API and one that
use the Pauli decomposition of the CTQW Hamiltonian (see Section II.3).

####  IV.2.1 Discrete-time Quantum Walk Benchmarking

The DTQW implementation produces a circuit with a relatively small depth and
number of gates. Our $\beta_{q|n}$ benchmark measures how closely a quantum
computer operates to the expected noise levels (indicated by the noise
parameters and simulated by the UNM). Thus, in the small circuit case of the
DTQW, the Bogota machine operates much closer to the expected evolution than
the other computers with the smallest $\beta_{q|n}$.

The $\alpha_{q|i}$ and $\gamma_{n|i}$ benchmarks lead to a coherent picture of
the overall performance of the quantum computers. In the DTQW case, the
$\alpha_{q|i}$ benchmark is relatively small compared to the larger
algorithms. This means that the quantum computers are not as erroneous as in
the larger circuit cases, an expected result. A comparison between the
$\alpha_{q|i}$ and $\gamma_{n|i}$ benchmarks shows that
$\alpha_{q|i}<\gamma_{n|i}$ for all the machines, indicating that the the
calibrated parameters overestimate the levels of noise during the evolution of
the circuit. Finally, $\beta_{q|n}\geq\lvert\alpha_{q|i}-\gamma_{q|i}\rvert$
for all quantum computers, which implies high confidence of our noise level
estimates.

####  IV.2.2 Continuous-time Quantum Walk Benchmarking

The CTQW circuit is the smallest circuit we use for benchmarking, which leads
us to expect that the machine evolution will be reasonably close to the ideal
case for this circuit. Evidently, the $\alpha_{q|i}$ benchmark is by far the
smallest for the CTQW algorithm on all three machines. As
$\alpha_{q|i}<\gamma_{n|i}$ for the Bogota and Santiago machines, we can say
that the noise parameters overestimate the noise for those two computers,
whereas they slightly underestimate the noise for the Casablanca computer.

For the CTQW, the Santiago machine operates closer to the expected levels of
noise according to its error rates, with the smallest $\beta_{q|n}$. Similarly
to the DTQW, we are confident in the calibrated noise as
$\beta_{q|n}\geq\lvert\alpha_{q|i}-\gamma_{q|i}\rvert$.

####  IV.2.3 Pauli Decomposition Benchmarking

Unlike the CTQW circuit, its Pauli decomposition is a large and quite deep
circuit, which means that we should expect a very noisy evolution. The small
$\beta_{q|n}$ values show that the quantum computers operate relatively close
to expectations.

As shown by the large values of $\alpha_{q|i}$ for all quantum computers, the
PD circuit is quite error-prone compared to the two previous cases of the
quantum walk. Thus, an important conclusion we can safely draw is that, for a
two-qubit continuous-time quantum walk, the Pauli decomposition of its
Hamiltonian leads to a more complex and noisier circuit than the decomposition
to base gates done automatically through the IBMQ API.

The $\gamma_{n|i}$ benchmarks show that the parameters overestimate the noise
of the Bogota and Casablanca machines and underestimate the noise within the
Santiago computer. Furthermore,
$\beta_{q|n}\geq\lvert\alpha_{q|i}-\gamma_{q|i}\rvert$ inspires high-
confidence on our noise estimates. This result is important as, through our
benchmarks, we produce evidence on the performance of two different techniques
for implementing the same algorithm.

####  IV.2.4 Quantum Phase Estimation Benchmarking

Moving on, the QPE circuit is slightly more complex than the DTQW circuit. An
overall bigger $\beta_{q|n}$ benchmark on every machine indicates that the QPE
circuit execution deviates slightly more from the expected evolution compared
to the other algorithms. This could be the result of random fluctuations of
the calibrated parameters.

Following the $\alpha_{q|i}$ benchmark, we can extract similar results for the
quantum computers as the PD circuit benchmarks, finding that the larger values
indicate that the quantum computer performance is hindered by the size of the
circuit. Interestingly, we see that $\alpha_{q|i}>\gamma_{n|i}$ for all the
quantum computers, showcasing that the calibrated parameters underestimate the
noise and we can expect noisier results from all the quantum computers for
computations of similar size. We have high-confidence of our calibrated
parameters as $\beta_{q|n}\geq\lvert\alpha_{q|i}-\gamma_{q|i}\rvert$ for all
experiments.

####  IV.2.5 Quantum Search Benchmarking

Finally, the quantum search circuit represents the largest implemented
algorithm and thus, we expect it to also be the noisiest. Following the above
methodology, we find that the $\beta_{q|n}$ benchmark shows the smallest
values for each machine, indicating that the quantum computers operated close
to the predicted evolutions. The $\alpha_{q|i}$ and $\gamma_{n|i}$ benchmarks
are quite close, increasing our confidence on the estimated noise levels.
Nevertheless, it is safe to conclude that the machines in the case of the QS
exhibit intense levels of noise, as the $\alpha_{q|i}$ benchmarks are very
large.

A quick comparison between the $\alpha_{q|i}$ and $\gamma_{n|i}$ shows that
the calibrated parameters offer a very good picture of the noise within the
quantum computer in this experiment.

####  IV.2.6 Analysis

A comparison of the $\alpha_{q|i}$ and $\gamma_{n|i}$ benchmarks shows that
when executing the DTQW and CTQW circuits, the machines are less error-prone
than the UNM and calibrated parameters imply, with the opposite being true for
the PD, QS and QPE. This informs us that the machines are better at executing
small circuits, an expected result. Additionally, a larger numerical
difference between $\alpha_{q|i}$ and $\gamma_{n|i}$ indicates a more
efficient computer behaviour. For example, Casablanca is the most efficient in
terms of noise for the DTQW case as $\alpha_{q|i}<\gamma_{n|i}$ with the
biggest difference. Similarly, in the QS case the Casablanca machine is the
least efficient with $\alpha_{q|i}>\gamma_{n|i}$. Here, as we use two
different implementations of the QS algorithm, we can also compare them with
each other. The benchmarks show that the QS circuit with ancilla is closer to
the expected evolution by the UNM, shown by $\beta_{q|n}$, and to the ideal
evolution, shown by $\alpha_{q|i}$, although not by much, shown by
$\alpha_{q|i}$ versus $\gamma_{n|i}$.

The results from the above analysis are crucial as they can lead towards the
selection of a machine appropriate to specific circuit needs. Additionally, we
provide a comparison between the continuous-time quantum walk circuit and its
Pauli decomposition. This fact is clearly reflected on the benchmarks, as the
values of $\alpha_{q|i}$ are much lower for the CTQW circuit, while both
circuits operate within expectations (small $\beta_{q|n}$ values) and are not
massively over- or underestimated by their noise parameters (small
$|\alpha_{q|i}-\gamma_{n|i}|$). Thus, we can easily conclude that, for the
two-qubit case, a Pauli decomposition of the CTQW Hamiltonian leads to a less
efficient circuit.

Overall, we can concentrate the general benchmarking results as follows. In
the smallest circuits (i.e., CTQW and DTQW case) the $\beta_{q|n}$ benchmarks
show how close the computers operate to the expected levels of noise. The
Casablanca machine shows the best $\alpha_{q|i}$ benchmark, i.e., it is the
closest to the ideal evolution, closely followed by Bogota and with Santiago
being the furthest away. The UNM and the calibrated parameters always
overestimate the noise in this case as $\alpha_{q|i}<\gamma_{n|i}$, thus
showing that all the quantum computers exhibit relatively low levels of noise
when executing small circuits. In the slightly deeper circuit of the QPE, the
Bogota machine outperforms the others, followed by Santiago and Casablanca. In
this case though the $\beta_{q|n}$ benchmark indicates slightly bigger
deviation from the expected level of noise which is biggest on the Casablanca
machine. The UNM and calibrated parameters always underestimate the noise in
the quantum computer as $\alpha_{q|i}>\gamma_{n|i}$, meaning that the quantum
computers are more error-prone. Lastly, in the largest QS and PD circuits, the
benchmarks indicate that all the machines operate close to the noise model
with low $\beta_{q|n}$ benchmarks and exhibit very low performance as the
$\alpha_{q|i}$ benchmarks are large, getting close to $1$ for the QS. The
noise model slightly underestimates the noise, but in this case of very deep
circuits, the benchmarks show that the machines will not produce any
meaningful results, an expected outcome.

A visual representation of the comparison between the distributions resulting
from the quantum computer executions and the ideal simulations for each
quantum algorithm are shown in Appendix A. A further comparison between the
quantum computer and the individual UNM distributions for each machine is
given in Appendix B.

##  V Discussion and Conclusions

In this paper we have presented an approach to benchmarking quantum computers
using scaling, high-level quantum algorithms considered as attractive “real-
world” problems. We have defined three benchmark metrics, each highlighting
different aspects of the machine’s efficiency either as a standalone or
through comparisons between them. Each benchmark metric describes the
difference between two quantum evolutions and together they follow the
triangle inequality.

In order to better present the main characteristics of our benchmarks, we
streamline the discussion as a comparison with the vastly used metric of
quantum volume [42], which quantifies the expected size of a circuit that can
be reliably run on a quantum computer. In contrast, our program benchmarks
describe the performance of the quantum computer when running a specific
circuit itself. This approach has advantages and disadvantages over
architecture-neutral benchmarks. First of all, our benchmark metrics show the
exact performance of a QPU (and in extend, the quantum computer itself) when
running the quantum circuit. Additionally, they highlight the difference with
an expected (noisy) simulated evolution and an ideal (noise-free) simulated
evolution, a result that better identifies the weaknesses of the machine in a
more structural manner. More specifically, the metrics allow us to realise the
manner and intensity that the computer deviates from the simulated evolutions.
Finally, all three machines we benchmark use the same QPU technology and
exhibit the same quantum volume of $32$. On the other hand, our benchmarking
process exhibits different metrics and results for each quantum computer.
Thus, our benchmarks capture the performance of each QPU more thoroughly and
provide us with a more detailed representation of their performance.

On the con side, our benchmarking process is slower compared to the
calculation of the quantum volume. This is an expected outcome as we need to
run a number of experiments on the quantum computer as well as the noisy
simulations. Moreover, the flip-side of the architecture-specific nature of
our metrics dictates that each computer will exhibit different benchmarks when
executing different algorithms.

Thus, we can conclude that our architecture-specific program benchmarks
showcase the performance of a quantum computer in a “real-world” environment,
as well as highlight the performance when running a specific algorithm and
carry out meaningful comparisons between related circuits (e.g., the CTQW vs
the Pauli decomposition of its Hamiltonian). Additionally, we can gather
information on whether the calibrated parameters over- or underestimate the
levels of noise during the quantum evolution. On the other hand, architecture-
neutral benchmarks like the quantum volume are more generic and excel at
showcasing the limitations of QPUs when running arbitrary quantum circuits.

It is also noteworthy that the benchmarking experiments carried out during
this work are not suitable for a comparative analysis regarding the efficiency
of each qubit topology (see Figure 1). One reason for this is the relatively
small size and limited flexibility of the computers themselves. Nevertheless,
when benchmarking larger quantum computers in the future, we believe that
using our framework, one could focus, for example, on searching which
architectures are more efficient for a specific task.

In conclusion, our work has shown that using quantum algorithms to benchmark
quantum computers in a well-structured environment can stress different
aspects and very informatively highlight the performance of a quantum
computer. Additionally, it provides a way to compare the efficiency of
different circuits used to implement the same task. To the best of our
knowledge, this is the first work that uses a continuous-time quantum
algorithm to benchmark the performance of a digital quantum machine. The
results show that, for small state spaces of the continuous-time quantum walk,
the Pauli decomposition does not produce an efficient circuit. This result is
expected as the complexity of a Hamiltonian operating on two qubits is very
small.

##  VI Acknowledgements

This work was supported by the Engineering and Physical Sciences Research
Council, Centre for Doctoral Training in Cloud Computing for Big Data, United
Kingdom [grant number EP/L015358/1].

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##  Appendix A Visual Comparison of the Quantum Computer Distributions

In order to enhance the results from the benchmarking process we offer in
Figure 3 a visual comparison of the probability distributions that produce the
Hellinger distances of Table 4. Each individual graph in the Figure portrays
the probability distributions for the execution of one of the quantum
algorithms used for benchmarking (see Section II.2). It also includes the
probability distribution of the ideal evolution of the respective quantum
algorithm.

|
---|---
(a) Quantum walk distributions. |  (b) Continuous-time quantum walk distributions.

|
---|---
(c) Pauli decomposition of CTQW distributions. |  (d) Quantum phase estimation distributions.

---
(e) Quantum search distributions.

Figure 3: Comparison of the probability distributions for each of the
algorithms when executed on the quantum machines and when simulated in a
noise-free environment.

##  Appendix B Comparison of Probability Distributions

In the context of visual comparisons, we present a series of figures that show
the difference between the probability distributions of each quantum computer
benchmarked and the unified noise model (UNM). Each figure includes a set of
plots that represent each algorithm used for benchmarking, i.e., for the
discrete-time quantum walk (Figure 4), the continuous-time quantum walk
(Figure 5), the Pauli decomposition of the continuous-time quantum walk
Hamiltonian (Figure 6), the quantum phase estimation algorithm (Figure 7) and
the quantum search algorithm (Figure 8). Finally, each figure contains a plot
of the probability distributions resulting from the UNM simulations and the
ideal evolution for each quantum algorithm, thus including every meaningful
comparison that derives from our benchmarks.

|
---|---
(a) |  (b)
|
(c) |  (d)
Figure 4: Comparison between the probability distributions of the discrete-time quantum walk execution on each quantum machine: (a) the IBMQ $5$-qubit Bogota, (b) IBMQ $5$-qubit Santiago and (c) IBMQ $7$-qubit Casablanca machines; (d) comparison between the UNM simulations for each machine and the ideal distribution. |
---|---
(a) |  (b)
|
(c) |  (d)
Figure 5: Comparison between the probability distributions of the continuous-time quantum walk execution on each quantum machine: (a) the IBMQ $5$-qubit Bogota, (b) IBMQ $5$-qubit Santiago and (c) IBMQ $7$-qubit Casablanca machines; (d) comparison between the UNM simulations for each machine and the ideal distribution. |
---|---
(a) |  (b)
|
(c) |  (d)
Figure 6: Comparison between the probability distributions of the Pauli decomposition of the continuous-time quantum walk execution on each quantum machine: (a) the IBMQ $5$-qubit Bogota, (b) IBMQ $5$-qubit Santiago and (c) IBMQ $7$-qubit Casablanca machines; (d) comparison between the UNM simulations for each machine and the ideal distribution. |
---|---
(a) |  (b)
|
(c) |  (d)
Figure 7: Comparison between the probability distributions of the quantum
phase estimation execution on each quantum machine: (a) the IBMQ $5$-qubit
Bogota, (b) IBMQ $5$-qubit Santiago and (c) IBMQ $7$-qubit Casablanca
machines; (d) comparison between the UNM simulations for each machine and the
ideal distribution.

|
---|---
(a) |  (b)
|
(c) |  (d)

---
(e)

Figure 8: Comparison between the probability distributions of the quantum
search execution on each quantum machine: (a) the IBMQ $5$-qubit Bogota, (b)
IBMQ $5$-qubit Santiago and (c) IBMQ $7$-qubit Casablanca machines; (d)
comparison between the UNM simulations for each machine and the ideal
distribution.


††footnotetext:  27-8-2024

#  $H^{p}$-Norm estimates of the partial derivatives and Schwarz lemma for
$\alpha$-harmonic functions

Adel Khalfallah  Department of Mathematics, King Fahd University of Petroleum
and Minerals, Dhahran 31261, Saudi Arabia<EMAIL_ADDRESS>and  Miodrag
Mateljević  M. Mateljević, Faculty of mathematics, University of Belgrade,
Studentski Trg 16, Belgrade, Republic of Serbia<EMAIL_ADDRESS>
###### Abstract.

Suppose $\alpha>-1$ and $1\leq p\leq\infty$. Let $f=P_{\alpha}[F]$ be an
$\alpha$-harmonic mapping on $\mathbb{D}$ with the boundary $F$ being absolute
continuous and $\dot{F}\in L^{p}(0,2\pi)$, where
$\dot{F}(e^{i\theta}):=\frac{dF(e^{i\theta})}{d\theta}$. In this paper, we
investigate the membership of $f_{z}$ and $f_{\overline{z}}$ in the space
$\mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$, the generalized Hardy space. We
prove, if $\alpha>0$, then both $f_{z}$ and $f_{\overline{z}}$ are in
$\mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$. If $\alpha<0$, then $f_{z}$ and
$f_{\overline{z}}\in\mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$ if and only if
$f$ is analytic. Finally, we investigate a Schwartz Lemma for
$\alpha$-harmonic functions.

###### Key words and phrases:

Poisson integral, $\alpha$-harmonic mapping, Hardy space, Bergman space

###### 2010 Mathematics Subject Classification:

Primary: 30C62, 31A05; Secondary: 30H10, 30H20.

##  1\. Preliminaries

We denote by $\mathbb{D}$ the unit disk and $\mathbb{U}:=\partial\mathbb{D}$
the unit circle. For $z=x+iy\in\mathbb{C}$, the two complex differential
operators are defined by

$u_{z}=\partial_{z}(u)=\frac{1}{2}\left(u_{x}-iu_{y}\right)~{}\mbox{and}~{}u_{\bar{z}}=\overline{\partial}_{z}(u)=\frac{1}{2}\left(u_{x}+iu_{y}\right),$

where $u$ is a complex valued function on $\mathbb{D}$.

###  1.1.  $\alpha$-harmonic functions

For $\alpha>-1$, we denote by $\Delta_{\alpha}$ the weighted Laplace operator
corresponding to the so-called standard weight
$w_{\alpha}(z)=(1-|z|^{2})^{\alpha}$, that is, for $z\in\mathbb{D}$,

(1.1)
$\displaystyle\Delta_{\alpha,z}=\partial_{z}w_{\alpha}(z)^{-1}\overline{\partial}_{z}.$

The weighted Laplacians of the form (1.1) were first systematically studied by
Garabedian in [4].

Let $g\in\mathcal{C}(\mathbb{D})$ and $f\in\mathcal{C}^{2}(\mathbb{D})$. Of
particular interest to us is the following inhomogeneous $\alpha$-harmonic
equation in $\mathbb{D}$:

(1.2) $\displaystyle\Delta_{\alpha}(f)=g.$

We also consider the associated _Dirichlet boundary value problem_ of
functions $f$, satisfying the equation (1.2),

(1.3) $\displaystyle\left\\{\begin{aligned}
\Delta_{\alpha}(f)&=g\;\;\;\;\text{in}\;\mathbb{D},\\\
f&=f^{\ast}\;\;\text{on}\;\mathbb{T}.\end{aligned}\right.$

Here the boundary data $f^{\ast}$ is a distribution on $\mathbb{T}$, i.e.
$f^{\ast}\in\mathcal{D}^{\prime}(\mathbb{T})$, and the boundary condition in
(1.3) is understood as $f_{r}\to f^{\ast}\in\mathcal{D}^{\prime}(\mathbb{T})$
as $r\to 1^{-}$, where

$f_{r}(e^{i\theta}):=f(re^{i\theta}),\quad r\in[0,1).$

If $g\equiv 0$ in (1.2), the solutions to (1.2) are said to be
$\alpha$-harmonic functions. Obviously, $0$-harmonic functions are simply
harmonic functions.

In [14], Olofsson and Wittsten showed that if an $\alpha$-harmonic function
$f$ satisfies

$\lim_{r\to
1^{-}}f_{r}=f^{\ast}\in\mathcal{D}^{\prime}(\mathbb{T})\;\;(\alpha>-1),$

then it has the form of a _Poisson type integral_

(1.4) $\displaystyle
f(z)=\mathcal{P}_{\alpha}[f^{\ast}](z)=\frac{1}{2\pi}\int^{2\pi}_{0}P_{\alpha}(ze^{-i\theta})f^{\ast}(e^{i\theta})d\theta$

in $\mathbb{D}$, where

$P_{\alpha}(z)=\frac{(1-|z|^{2})^{\alpha+1}}{(1-z)(1-\overline{z})^{\alpha+1}}$

is the $\alpha$-harmonic Poisson kernel in ${\mathbb{D}}$.

Moreover,

$P_{\alpha}(z)=\sum_{k=-\infty}^{\infty}e_{\alpha,k}(z),\quad z\in\mathbb{D},$

where

$e_{\alpha,k}(z)=z^{k},\quad k\geq 0,$

and

$e_{\alpha,-k}(z)=\frac{1}{B(k,\alpha+1)}\left(\int_{0}^{1}t^{k-1}(1-t|z|^{2})^{\alpha}dt\right)\overline{z}^{k},\quad
k\geq 1,$

see [14].

In particular, we obtain

$e_{\alpha,k}(z)=\mathcal{P}_{\alpha}[e^{ik\theta}](z),\quad k\in\mathbb{Z}.$

In addition, we have

(1.5)
$\overline{\partial}e_{\alpha,-k}(z)=\frac{1}{B(k,\alpha+1)}(1-|z|^{2})^{\alpha}\,{\overline{z}}^{k-1},\quad
k\geq 1,$

where $B$ is the beta function, see [14, Lemma 1.1].

Recall that

$M_{\alpha}(r)=\frac{1}{2\pi}\int_{\mathbb{U}}|P_{\alpha}(ze^{-it})|dt.$

The mapping $M_{\alpha}$ is increasing on $[0,1)$ and

(1.6) $\lim_{r\to
1}M_{\alpha}(r)=c_{\alpha}=\frac{\Gamma(\alpha+1)}{\Gamma^{2}(\alpha/2+1)}.$

See [10, 13] for related discussions.

###  1.2. Generalized Hardy spaces

For $p\in(0,\infty]$, the generalized Hardy space
$\mathcal{H}^{p}_{\mathcal{G}}(\mathbb{D})$ consists of all measurable
functions from $\mathbb{D}$ to $\mathbb{C}$ such that $M_{p}(r,f)$ exists for
all $r\in(0,1)$, and $\|f\|_{p}<\infty$, where

$M_{p}(r,f)=\left(\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{p}\,d\theta\right)^{\frac{1}{p}}$

and

$\|f\|_{p}=\begin{cases}\displaystyle\sup\\{M_{p}(r,f):\;0<r<1\\}&\mbox{if
}p\in(0,\infty),\\\ \displaystyle\sup\\{|f(z)|:\;z\in\mathbb{D}\\}&\mbox{if
}p=\infty.\end{cases}$

The classical Hardy space $\mathcal{H}^{p}(\mathbb{D})$ (resp.
$h^{p}(\mathbb{D})$) is the set of all elements of
$\mathcal{H}^{p}_{\mathcal{G}}(\mathbb{D})$ which are analytic (resp.
harmonic) on $\mathbb{D}$. (cf. [2, 3]). Obviously,
$\mathcal{H}^{p}_{\mathcal{G}}(\mathbb{D})\subset L^{p}(\mathbb{D})$ for each
$p\in(0,\infty]$. We denote by $h^{p}_{\alpha}(\mathbb{D})$ the corresponding
Hardy space for $\alpha$-harmonic functions.

Denote by $L^{p}(\mathbb{T})~{}(p\in[1,\infty])$ the space of all measurable
functions $F$ of $\mathbb{T}$ into $\mathbb{C}$ with

$\|F\|_{p}=\begin{cases}\displaystyle\left(\frac{1}{2\pi}\int_{0}^{2\pi}|F(e^{i\theta})|^{p}d\theta\right)^{\frac{1}{p}}&\mbox{if
}p\in[1,\infty),\\\
\displaystyle\sup\\{|F(e^{i\theta})|:\;\theta\in[0,2\pi)\\}&\mbox{if
}p=\infty.\\\ \end{cases}$

It is well-known that if $F$ is absolutely continuous, then it is of bounded
variation. This implies that for almost all $e^{i\theta}\in\mathbb{T}$, the
derivative $\dot{F}(e^{i\theta})$ exists, where

$\dot{F}(e^{i\theta}):=\frac{dF(e^{i\theta})}{d\theta}.$

In this paper, we consider the following problem :

For $\alpha>-1$, under what conditions on the boundary function $F$ ensure
that the partial derivatives of its $\alpha$-harmonic extension
$f=\mathcal{P}_{\alpha}[F]$, i.e., $f_{z}$ and $f_{\overline{z}}$, are in the
space $\mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$ (or $L^{p}(\mathbb{D})$),
where $p\in[1,\infty]$?

It is worth noting that a similar problem for harmonic functions was treated
in [15] and improved in [1, 9].

Theorem A. ([15, Theorem 1.2]),[1, Theorem 1.1]) Suppose that $f=P[F]$ is a
harmonic mapping in $\mathbb{D}$ and $\dot{F}\in L^{p}(\mathbb{T})$, where $F$
is an absolutely continuous function.

  1. (1)

If $p\in[1,\infty)$, then both $f_{z}$ and $\overline{f_{\overline{z}}}$ are
in $L^{p}(\mathbb{D}).$

  2. (2)

If $p=\infty$, then there exists a harmonic mapping $f=P[F]$, with $\dot{F}\in
L^{\infty}(\mathbb{T})$, such that neither $f_{z}$ nor
$\overline{f_{\overline{z}}}$ is in $L^{\infty}(\mathbb{D}).$

Furthermore, under some additional conditions of $f$, they proved that the
partial derivatives are in $H^{p}(\mathbb{D})$, for $p\in[1,\infty]$.

Theorem B. ([15, Theorem 1.3],[1, Theorem 1.2]) Suppose that $p\in[1,\infty]$
and $f=P[F]$ is a $(K,K^{\prime})$-elliptic mapping in $\mathbb{D}$ with
$\dot{F}\in L^{p}(\mathbb{T})$, where $F$ is an absolute continuous function,
$K\geq 1$ and $K^{\prime}\geq 0$. Then both $f_{z}$ and
$\overline{f_{\overline{z}}}$ are in $H^{p}(\mathbb{D}).$

In [1], the authors showed that Theorem B also holds true for harmonic
elliptic mappings, which are more general than harmonic quasiregular mapping.

In [9], we provide a refinement of the two previous theorems, we prove that
for $1<p<\infty$, both $f_{z}$ and $\overline{f_{\overline{z}}}$ are in
$H^{p}(\mathbb{D})$ without any extra conditions on $f$.

Theorem C. [9] Suppose that $F$ is an absolute continuous function on
$\mathbb{U}$ and $f=P[F]$ is a harmonic mapping in $\mathbb{D}$ and
$\dot{F}\in L^{p}(\mathbb{T})$.

  1. (1)

If $p\in(1,\infty)$, then both $f_{z}$ and $\overline{f_{\overline{z}}}$ are
in $\mathcal{H}^{p}(\mathbb{D}).$ Moreover, there exists a constant $A_{p}$
such that

$\max(\|f_{z}\|_{p},\|\overline{f_{\overline{z}}}\|_{p})\leq
A_{p}\|\dot{F}\|_{p}.$

  2. (2)

If $p=1$ or $p=\infty$, then both $f_{z}$ and $\overline{f_{\overline{z}}}$
are in $\mathcal{H}^{p}(\mathbb{D})$ if and only if $H(\dot{F})\in
L^{p}(\mathbb{U})$. Moreover,

$2izf_{z}(z)=P[\dot{F}+iH(\dot{F})](z),$

where $H(\dot{F})$ denotes the Hilbert transformation of $\dot{F}$.

##  2\. Main results

Assume $z=re^{i\theta}\in\mathbb{D}$, the polar partial derivative of $f$ with
respect to $\theta$ is given by :

(2.1) $\partial_{\theta}f=i(zf_{z}-\bar{z}f_{\bar{z}}).$

###  2.1. Estimate of the angular derivative $\partial_{\theta}f$, $f$ is
$\alpha$-harmonic.

######  Theorem 2.1.

Suppose $\alpha>-1$ and $1\leq p\leq\infty$. Let $f=P_{\alpha}[F]$ be an
$\alpha$-harmonic mapping on $\mathbb{D}$ with the boundary $F$ being absolute
continuous with $\dot{F}\in L^{p}(0,2\pi)$. Then

$\partial_{\theta}f=\mathcal{P}_{\alpha}[{\dot{F}}]\in
h_{\alpha}^{p}(\mathbb{D})$

and

$\|\partial_{\theta}f\|_{p}\leq c_{\alpha}\|\dot{F}\|_{p},$

where $c_{\alpha}$ is defined in (1.6).

Proof. Assume that $f$ is $\alpha$-harmonic function and

$f=\mathcal{P}_{\alpha}[F],$

with $F$ is absolute continuous and $\dot{F}\in L^{p}(0,2\pi)$, with $p\geq
1$.

$f(z)=\mathcal{P}_{\alpha}[F](z)=\frac{1}{2\pi}\int_{\mathbb{U}}P_{\alpha}(ze^{-it})F(e^{it})\,dt.$

For $z=re^{i\theta}$, we have

$\partial_{\theta}P_{\alpha}(ze^{-it})=-\partial_{t}P_{\alpha}(ze^{-it}).$

By integration by parts, we deduce

(2.2) $\partial_{\theta}f=\mathcal{P}_{\alpha}[{\dot{F}}].$

Using Jensen’s inequality, we have

$|\partial_{\theta}f(re^{i\theta})|^{p}\leq
M^{p-1}_{\alpha}(r)\frac{1}{2\pi}\int_{0}^{2\pi}|P_{\alpha}(ze^{-it})||\dot{F}(e^{it})|^{p}\,dt.$

Using Fubini theorem, we obtain

$\frac{1}{2\pi}\int_{0}^{2\pi}|\partial_{\theta}f(re^{i\theta})|^{p}d\theta\leq
M^{p}_{\alpha}(r)\|\dot{F}\|_{L^{p}}^{p}.$

Thus,

$\sup_{0<r<1}\frac{1}{2\pi}\int_{0}^{2\pi}|\partial_{\theta}f(re^{i\theta})|^{p}d\theta\leq
c^{p}_{\alpha}\|\dot{F}\|_{L^{p}}^{p}.$

For $p=\infty$, it is clear that

$|\partial_{\theta}f(re^{i\theta})|\leq M_{\alpha}(r)\|\dot{F}\|_{\infty}\leq
c_{\alpha}\|\dot{F}\|_{\infty}.$

Therefore, we obtain the desired result. ∎

######  Lemma 2.1.

Let $\alpha>-1$ and $f=\mathcal{P}_{\alpha}[F]$ be an $\alpha$-harmonic
mapping on $\mathbb{D}$ with the boundary function $F$ being absolute
continuous. Then

(2.3) ${\overline{z}}\overline{\partial}f(z)=-\frac{1}{2\pi
i}\int_{0}^{2\pi}\frac{(1-|z|^{2})^{\alpha}}{(1-{\overline{z}}e^{it})^{\alpha+1}}\,\dot{F}(e^{it})\,dt.$

Proof. Let $\alpha>-1$, an easy computation shows that

$\overline{\partial}P_{\alpha}(z)=\frac{\alpha+1}{(1-{\overline{z}})^{2}}\left(\frac{1-|z|^{2}}{1-{\overline{z}}}\right)^{\alpha}=(\alpha+1)\frac{(1-|z|^{2})^{\alpha}}{(1-{\overline{z}})^{\alpha+2}}.$

As $f(z)=\frac{1}{2\pi}\int_{0}^{2\pi}P_{\alpha}(ze^{-it})F(e^{it})\,dt,$ we
obtain

$\overline{\partial}f(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\overline{\partial}[P_{\alpha}(ze^{-it})]F(e^{it})\,dt=\frac{1}{2\pi}\int_{0}^{2\pi}\overline{\partial}P_{\alpha}(ze^{-it})e^{it}F(e^{it})\,dt.$

Hence,

$\overline{\partial}f(z)=(\alpha+1)(1-|z|^{2})^{\alpha}\frac{1}{2\pi}\int_{0}^{2\pi}\frac{e^{it}}{(1-{\overline{z}}e^{it})^{\alpha+2}}\,F(e^{it})\,dt.$

Using integration by parts, we obtain

(2.4) $\overline{\partial}f(z)=-\frac{1}{{\overline{z}}}\frac{1}{2\pi
i}\int_{0}^{2\pi}\frac{(1-|z|^{2})^{\alpha}}{(1-{\overline{z}}e^{it})^{\alpha+1}}\dot{F}(e^{it})\,dt.$

∎

Denote

$I_{\alpha}(r)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{(1-|z|^{2})^{\alpha}}{|1-{\overline{z}}e^{it}|^{\alpha+1}}dt,$

where $r=|z|.$

######  Lemma 2.2.

[8, Proposition 1.1] Let $r\in[0,1)$.

  1. (1)

If $\displaystyle\alpha>0$, then

$I_{\alpha}(r)\leq\frac{\Gamma(\alpha)}{\Gamma^{2}(\alpha/2+1/2)}.$

  2. (2)

If $\displaystyle-1<\alpha<0$, then

$I_{\alpha}(r)\leq\frac{\Gamma(-\alpha)}{\Gamma^{2}(1/2-\alpha/2)}(1-r^{2})^{\alpha}.$

######  Proposition 2.1.

Suppose $\alpha>-1$ and $1\leq p\leq\infty$. Let $f=\mathcal{P}_{\alpha}[F]$
be an $\alpha$-harmonic mapping on $\mathbb{D}$ with $F$ is absolute
continuous and satisfies $\dot{F}\in L^{p}(0,2\pi)$. Then

$M_{p}(r,{\overline{z}}\overline{\partial}f(z))\leq
I_{\alpha}(|z|)\|\dot{F}\|_{p}.$

We use the equation (2.4) and apply Jensen’s inequality to estimate the mean
of the function ${\overline{z}}\overline{\partial}f(z)$.

###### Proof.

Using (2.4), we have

(2.5)
$|{\overline{z}}\overline{\partial}f(z)|\leq\frac{1}{2\pi}\int_{0}^{2\pi}\frac{(1-|z|^{2})^{\alpha}}{|1-{\overline{z}}e^{it}|^{\alpha+1}}\,|\dot{F}(e^{it})|\,dt.$

Using Jensen’s inequality, we get

$|{\overline{z}}\overline{\partial}f(z)|^{p}\leq
I_{\alpha}^{p-1}(r)\frac{1}{2\pi}\int_{0}^{2\pi}\frac{(1-|z|^{2})^{\alpha}}{|1-{\overline{z}}e^{it}|^{\alpha+1}}\,|\dot{F}(e^{it})|^{p}\,dt.$

Applying Fubini’s theorem, we obtain

(2.6) $M_{p}^{p}(r,{\overline{z}}\overline{\partial}f(z))\leq
I_{\alpha}^{p}(|z|)\|\dot{F}\|^{p}_{p}.$

∎

######  Corollary 1.

If $\alpha>0$ and $f=\mathcal{P}_{\alpha}[F]$ and $F$ is absolute continuous
such that $\dot{F}\in L^{p}$ with $1\leq p\leq\infty$. Then
${\overline{z}}\overline{\partial}f\in H_{g}^{p}(\mathbb{D})$ and

$\|{\overline{z}}\overline{\partial}f(z)\|_{p}\leq\frac{\Gamma(\alpha)}{\Gamma^{2}(\alpha/2+1/2)}\|\dot{F}\|_{p}.$

###### Proof.

Let $\alpha>0$, by Proposition 2.1 and Lemma 2.2, we deduce that

$M_{p}(r,{\overline{z}}\overline{\partial}f(z))\leq
I_{\alpha}(r)\|\dot{F}\|_{p}\leq\frac{\Gamma(\alpha)}{\Gamma^{2}(\alpha/2+1/2)}\|\dot{F}\|_{p}.$

Therefore

$\|{\overline{z}}\overline{\partial}f(z)\|_{p}\leq\frac{\Gamma(\alpha)}{\Gamma^{2}(\alpha/2+1/2)}\|\dot{F}\|_{p}.$

∎

Combining the previous results with the identity

(2.7) $iz\partial=\partial_{\theta}+i{\overline{z}}\bar{\partial},$

we obtain our first main result

######  Theorem 2.2.

Let $\alpha\in(-1,\infty)$ with $\alpha\not=0$ and let $f=P_{\alpha}[F]$ and
$F$ is absolute continuous such that $\dot{F}\in L^{p}$ with $1\leq
p\leq\infty$.

  1. (1)

If $\alpha>0$, then $\overline{\partial}f$ and $\partial f$ are in
$H_{\mathcal{G}}^{p}(\mathbb{D})\subset L^{p}(\mathbb{D})$.

  2. (2)

If $\alpha\in(-1,0)$, then $\partial f$ and $\bar{\partial}f$ are in
$L^{p}(\mathbb{D})$ for $p<-\frac{1}{\alpha}$.

  3. (3)

For $\alpha\in(-1,0)$ and $p\geq-\frac{1}{\alpha}$ there exits $f$ an
$\alpha$-harmonic function such that $\partial f$ and $\bar{\partial}f$
$\not\in L^{p}(\mathbb{D})$, moreover, $\partial f$ and $\bar{\partial}f$
$\not\in H_{\mathcal{G}}^{1}(\mathbb{D})$.

###### Proof.

(1) and (2) are direct consequences of the previous discussion and Proposition
2.1 and Lemma 2.2.
(3) Let $\alpha\in(-1,0)$ and consider $H(e^{i\theta})=e^{-i\theta}$. We have
$e_{\alpha,-1}(z)=P_{\alpha}[H](z)$ and

$\overline{\partial}e_{\alpha,-1}(z)=(\alpha+1)(1-|z|^{2})^{\alpha}.$

Hence $\overline{\partial}e_{\alpha,-1}\not\in L^{p}(\mathbb{D})$ for
$p\geq-\frac{1}{\alpha}$ and $\overline{\partial}e_{\alpha,-1}\not\in
H^{1}_{\mathcal{G}}(\mathbb{D})$. ∎

Our second main result is the following

######  Theorem 2.3.

Let $\alpha\in(-1,0)$ and let $f=\mathcal{P}_{\alpha}[F]$ and $F$ is absolute
continuous such that $\dot{F}\in L^{p}$ with $1\leq p\leq\infty$. Then
$\overline{\partial}f$ or $\partial f$ is in
$H_{\mathcal{G}}^{p}(\mathbb{D})$, if and only if, $f$ is analytic.

###### Proof.

Consider the series expansion of $f$

$f(z)=\sum_{n=1}^{\infty}a_{-n}e_{\alpha,-n}(z)+\sum_{n=0}^{\infty}a_{n}z^{n}.$

For $n\geq 1$, let

$f_{n}(z):=\frac{1}{2\pi}\int_{0}^{2\pi}f(ze^{it})e^{int}dt,\quad
z\in\mathbb{D}.$

It yields

$f_{n}(z)=a_{-n}e_{\alpha,-n}(z).$

Hence

$\bar{\partial}f_{n}(z)=a_{-n}\bar{\partial}e_{\alpha,-n}(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\bar{\partial
f}(ze^{it})e^{i(n-1)t}dt.$

First, assume that $\overline{\partial}f\in H_{\mathcal{G}}^{p}(\mathbb{D})$,
then by (1.5), it yields $a_{-n}(1-|z|^{2})^{\alpha}$ is bounded, this implies
that $a_{-n}=0$ for $n\geq 1$ and $f$ is analytic.

In the case $\partial{f}\in H_{\mathcal{G}}^{p}(\mathbb{D})$, we deduce that
$\overline{\partial}f\in H_{\mathcal{G}}^{p}(\mathbb{D})$. Indeed, by (2.7)
and Theorem 2.1, we obtain

$i{\overline{z}}\overline{\partial}f(z)=iz\partial
f(z)-\partial_{\theta}f(z)\in H_{\mathcal{G}}^{p}(\mathbb{D}).$

Conversely, assume that $f$ is analytic, then $f=P_{0}[F]$ and
$f^{\prime}(z)=-\frac{i}{z}\partial_{\theta}f(z)=-\frac{i}{z}P[\dot{F}]\in
H^{p}(\mathbb{D}).$ ∎

##  3\. Schwartz Lemma for $\alpha$-harmonic functions

The Schwarz lemma for analytic functions plays a vital role in complex
analysis and has been generalized to various spaces of functions.

Heinz [6] generalized it to the class of complex-valued harmonic functions,
that is, if $f$ is a complex-valued harmonic function from $\mathbb{D}$ into
itself with $f(0)=0$, then for $z\in\mathbb{D}$,

$\ f(z)|\leq\frac{4}{\pi}\arctan|z|.$

Hethcote [7] improved Heinz’s result, by removing the assumption $f(0)=0$, i
.e., let $f$ be a harmonic function from $\mathbb{D}$ to $\mathbb{D}$, then

$\left|f(z)-\frac{1-|z|^{2}}{1+|z|^{2}}f(0)\right|\leq\frac{4}{\pi}\arctan|z|.$

In [11], Li et al. stated a Schwarz type lemma for solutions of the
$\alpha$-harmonic equation $\Delta_{\alpha}(f)=g$, with the condition
$\alpha\geq 0$. For $g=0$, they obtain

Theorem D. [11, Theorem 2.1] Let $\alpha\geq 0$ and
$f^{*}\in\mathcal{C}(\mathbb{U})$ and $f=P_{\alpha}[f^{*}]$ an
$\alpha$-harmonic function on $\mathbb{D}$ such that $f(0)=0$. Then

$|f(z)|\leq 2^{\alpha}\frac{4}{\pi}|f^{*}|_{\infty}\arctan|z|.$

In the proof, the authors used the assumption $P_{\alpha}[|f^{*}|](0)=0$
instead of $P_{\alpha}[f^{*}](0)=0$. Clearly the assumption
$P_{\alpha}[|f^{*}|](0)=0$ implies that $f^{*}=0$ and thus $f=0$. In order to
fill this gap, in [12], the authors proved the following

Theorem E. [12, Theorem 1.1] Suppose that $f^{*}\in\mathcal{C}(\mathbb{U})$
and $f=P_{\alpha}[f^{*}]$.

  1. (a)

If $\alpha\geq 0$, then

$|f(z)|\leq
2^{\alpha+1}\frac{|f|_{\infty}}{\pi}\arctan\left(\frac{1+|z|}{1-|z|}\tan\frac{c\pi}{2}\right).$

  2. (b)

If $\alpha<0$, then

$|f(z)|\leq
2^{1-\alpha}\frac{|f|_{\infty}}{\pi}(1-|z|^{2})^{\alpha}\arctan\left(\frac{1+|z|}{1-|z|}\tan\frac{c\pi}{2}\right),$

where $\displaystyle c=\frac{P_{\alpha}[|f^{*}|](0)}{|f^{*}|_{\infty}}$.

In our opinion, it is more natural to provide Schwarz type lemmas for
$\alpha$-harmonic functions involving $f(0)$ instead of
$P_{\alpha}[|f^{*}|](0)$.

The aim of this section is to prove a Schwarz type lemma for $\alpha$-harmonic
functions from the unit disc to itself in the spirit of Hethcote [7].

Let

$g_{\alpha}(z):=\left(\frac{1-|z|^{2}}{1-\overline{z}}\right)^{\alpha},$

so

(3.1)
$P_{\alpha}(z)=\left(\frac{1-|z|^{2}}{1-\overline{z}}\right)^{\alpha}\frac{1-|z|^{2}}{|1-z|^{2}}=g_{\alpha}(z)P(z),$

where

$P(z)=\frac{1-|z|^{2}}{|1-z|^{2}},$

is the classical Poisson kernel.

Let

$|g_{\alpha}(r)|_{\infty}=\sup_{0\leq\theta\leq
2\pi}|g_{\alpha}(re^{i\theta})|.$

One can check that

(3.2) $|g_{\alpha}(r)|_{\infty}\leq 2^{\alpha},\quad\mbox{for}\quad\alpha\geq
0,$

and

(3.3)
$|g_{\alpha}(r)|_{\infty}\leq(1-r)^{\alpha},\quad\mbox{for}\quad\alpha<0.$

Let $f:\mathbb{D}\to\mathbb{D}$ be an $\alpha$-harmonic mapping from the unit
disc to itself. Then, we can write

$f(z)=\frac{1}{2\pi}\int_{\mathbb{U}}P_{\alpha}(ze^{-it})f^{*}(e^{it})\,dt,$

where $f^{*}$ is the boundary function of $f$, with $|f^{*}|_{\infty}\leq 1.$

Let

$G_{\alpha}(z):=g_{\alpha}[f](z)=\frac{1}{2\pi}\int_{\mathbb{U}}g_{\alpha}(ze^{-it})f^{*}(e^{it})\,dt=\frac{(1-|z|^{2})^{\alpha}}{2\pi}\int_{\mathbb{U}}\frac{f^{*}(e^{it})}{(1-{\overline{z}}e^{it})^{\alpha}}\,dt.$

Remark, in the case $\alpha=0$, $G_{0}(z)=f(0)$.

######  Lemma 3.1.

Let $\alpha>-1$ and $f$ be an $\alpha$-harmonic function from the unit disc to
itself. Then

$\left|f(z)-\frac{1-|z|^{2}}{1+|z|^{2}}G_{\alpha}(z)\right|\leq\frac{4}{\pi}|g_{\alpha}(|z|)|_{\infty}\arctan|z|.$

In particular

  1. (1)

If $\alpha\geq 0$, then

$\left|f(z)-\frac{1-|z|^{2}}{1+|z|^{2}}G_{\alpha}(z)\right|\leq\frac{2^{\alpha+2}}{\pi}\arctan|z|.$

  2. (2)

If $\alpha<0$, then

$\left|f(z)-\frac{1-|z|^{2}}{1+|z|^{2}}G_{\alpha}(z)\right|\leq\frac{4}{\pi}(1-|z|)^{\alpha}\arctan|z|.$

###### Proof.

We have

$\left|f(z)-\frac{1-|z|^{2}}{1+|z|^{2}}G_{\alpha}(z)\right|\leq\frac{1}{2\pi}\int_{\mathbb{U}}\left|P(ze^{-it})-\frac{1-|z|^{2}}{1+|z|^{2}}\right|\left|g_{\alpha}(ze^{-it})f^{*}(e^{it})\
\right|\,dt$

$\leq\frac{4}{\pi}|f^{*}|_{\infty}|g_{\alpha}(|z|)_{\infty}\arctan|z|.$

Indeed, by Hethcote [7], it yields
$\displaystyle\frac{1}{2\pi}\int_{\mathbb{U}}\left|P(ze^{-it})-\frac{1-|z|^{2}}{1+|z|^{2}}\right|dt\leq\frac{4}{\pi}\arctan|z|.$
The conclusion follows from the inequalities (3.2) and (3.3). ∎

######  Lemma 3.2.

Let $\alpha>-1$ and $r\in[0,1)$. Then

$\frac{1}{2\pi}\int_{\mathbb{U}}\left|(1-re^{it})^{-\alpha}-1\right|\,dt\leq\left|(1-r)^{-\alpha}-1\right|.$

Proof. Set $u(r,t)=(1-re^{it})^{-\alpha}.$ Since $u_{r}(r,t)=\alpha
e^{it}(1-re^{it})^{-\alpha-1}$, and $-\alpha-1<0$, we have

$|u_{r}(r,t)|\leq|\alpha|(1-r)^{-\alpha-1},\quad r\in[0,1]\mbox{ and
}t\in\mathbb{R}.$

Next, using $u(r,t)-u(0,t)=\int_{0}^{r}u_{x}(x,t)dx$, we obtain

$|u(r,t)-u(0,t)|\leq\int_{0}^{r}|u^{\prime}_{x}(x,t)|dx\leq\int_{0}^{r}|\alpha|(1-x)^{-\alpha-1}dx=\frac{|\alpha|}{\alpha}((1-r)^{-\alpha}-1)$
and therefore

$\frac{1}{2\pi}\int_{\mathbb{U}}\left|(1-re^{it})^{-\alpha}-1\right|\,dt\leq\left|(1-r)^{-\alpha}-1\right|.$

∎

Combining the previous two lemmas, we obtain the main result of this section.

######  Theorem 3.1.

Let $\alpha>-1$ and $f:\mathbb{D}\to\mathbb{D}$ be an $\alpha$-harmonic
function. Then

  1. (1)

If $\alpha\geq 0$, then

$\left|f(z)-\frac{(1-|z|^{2})^{\alpha+1}}{1+|z|^{2}}f(0)\right|\leq\frac{2^{\alpha+2}}{\pi}\arctan|z|+2^{\alpha+1}(1-|z|)\left(1-(1-|z|)^{\alpha}\right).$

  2. (2)

If $\alpha<0$, then

$\left|f(z)-\frac{(1-|z|^{2})^{\alpha+1}}{1+|z|^{2}}f(0)\right|\leq\frac{4}{\pi}(1-|z|)^{\alpha}\arctan|z|+\left((1-|z|)^{\alpha}-1\right).$

###### Proof.

Let

$H_{\alpha}(z):=\frac{G_{\alpha}(z)}{(1-|z|^{2})^{\alpha}}=\frac{1}{2\pi}\int_{\mathbb{U}}\frac{f^{*}(e^{it})}{(1-{\overline{z}}e^{it})^{\alpha}}\,dt.$

and

$K_{\alpha}(z):=H_{\alpha}(z)-f(0).$

Notice that

$f(z)-\frac{(1-|z|^{2})^{\alpha+1}}{1+|z|^{2}}f(0)=\left(f(z)-\frac{1-|z|^{2}}{1+|z|^{2}}G_{\alpha}(z)\right)+\frac{(1-|z|^{2})^{\alpha+1}}{1+|z|^{2}}K_{\alpha}(z).$

Using the previous lemmas, it yields:

(1) if $\alpha\geq 0$, then

$\left|f(z)-\frac{(1-|z|^{2})^{\alpha+1}}{1+|z|^{2}}f(0)\right|\leq\frac{2^{\alpha+2}}{\pi}|f^{*}|_{\infty}\arctan|z|+\frac{(1-|z|^{2})^{\alpha+1}}{1+|z|^{2}}|K_{\alpha}(z)|$

$K_{\alpha}(z)=H_{\alpha}(z)-f(0)=\frac{1}{2\pi}\int_{\mathbb{U}}\left[(1-{\overline{z}}e^{it})^{-\alpha}-1\right]f^{*}(e^{it})\,dt.$

Thus

$|K_{\alpha}(z)|\leq\frac{1}{2\pi}\int_{\mathbb{U}}\left|(1-|z|e^{it})^{-\alpha}-1\right|\,dt\leq(1-r)^{-\alpha}-1.$

(2) First, we remark that

$\frac{(1-|z|^{2})^{\alpha+1}}{1+|z|^{2}}\leq(1-|z|^{2})^{\alpha}\leq(1-|z|)^{\alpha}.$

The case $\alpha<0$ is treated similarly as the previous case. ∎

Using the identity for $p\in(0,1)$ and $x,y\geq 0$

$(x+y)^{p}\leq x^{p}+y^{p}$

and the convexity of the function $(1-x)^{\alpha}$ , we obtain
$(1-x)^{\alpha}\geq 1-\alpha x$ for $\alpha\geq 1$ and $x\in[0,1]$.

######  Corollary 2.

Let $f:\mathbb{D}\to\mathbb{D}$ be an $\alpha$-harmonic function.

  1. (1)

If $0<\alpha\leq 1$, then

$\left|f(z)-\frac{(1-|z|^{2})^{\alpha+1}}{1+|z|^{2}}f(0)\right|\leq
2^{\alpha+1}\left(\frac{2}{\pi}\arctan|z|+|z|^{\alpha}\right).$

  2. (2)

If $\alpha\geq 1$, then

$\left|f(z)-\frac{(1-|z|^{2})^{\alpha+1}}{1+|z|^{2}}f(0)\right|\leq
2^{\alpha+1}\left(\frac{2}{\pi}\arctan|z|+\alpha|z|\right).$

We close the paper by providing the power series expansion of $H_{\alpha}$, as
it is anti-analytic function.

Since

$\frac{1}{(1-{\overline{z}}e^{it})^{\alpha}}=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!}\,{\overline{z}}^{n}e^{int},$

we deduce that

$H_{\alpha}(z)=\frac{G_{\alpha}(z)}{(1-|z|^{2})^{\alpha}}=f(0)+\sum_{n=1}^{\infty}\frac{(\alpha)_{n}}{n!}\widehat{f^{*}}(-n)\,{\overline{z}}^{n},$

where

$\widehat{f^{*}}(-n)=\frac{1}{2\pi}\int_{\mathbb{U}}f^{*}(e^{it})e^{int}\,dt.$

Here we are using that $f$ has the following series expansion

$f(z)=\sum_{n=1}^{\infty}\widehat{f^{*}}(-n)e_{\alpha,-n}(z)+\sum_{n=0}^{\infty}\widehat{f^{*}}(n)z^{n},$

see [14, Sectiuon 5].

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