Full quantum tomography study of Google’s Sycamore gate on IBM’s quantum computers

Abstract

The potential of achieving computational hardware with quantum advantage depends heavily on the quality of quantum gate operations. However, the presence of imperfect two-qubit gates poses a significant challenge and acts as a major obstacle in developing scalable quantum information processors. Google’s Quantum AI and collaborators claimed to have conducted a supremacy regime experiment. In this experiment, a new two-qubit universal gate called the Sycamore gate is constructed and employed to generate random quantum circuits (RQCs), using a programmable quantum processor with 53 qubits. These computations were carried out in a computational state space of size 9×10159 \times 10^{15} 9 × 10 15 . Nevertheless, even in strictly-controlled laboratory settings, quantum information on quantum processors is susceptible to various disturbances, including undesired interaction with the surroundings and imperfections in the quantum state. To address this issue, we conduct both quantum state tomography (QST) and quantum process tomography (QPT) experiments on Google’s Sycamore gate using different artificial architectural superconducting quantum computer. Furthermore, to demonstrate how errors affect gate fidelity at the level of quantum circuits, we design and conduct full QST experiments for the five-qubit eight-cycle circuit, which was introduced as an example of the programability of Google’s Sycamore quantum processor. These quantum tomography experiments are conducted in three distinct environments: noise-free, noisy simulation, and on IBM Quantum’s genuine quantum computer. Our results offer valuable insights into the performance of IBM Quantum’s hardware and the robustness of Sycamore gates within this experimental setup. These findings contribute to our understanding of quantum hardware performance and provide valuable information for optimizing quantum algorithms for practical applications.

Full text

AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36
https://doi.org/10.1140/epjqt/s40507-024-00248-8
R E S E A R C H Open Access
Full quantum tomography study of Google’s
Sycamore gate on IBM’s quantum computers
Muhammad AbuGhanem1* and Hichem Eleuch2,3,4
*Correspondence:
gaa1nem@gmail.com
1Faculty of Science, Ain Shams
University, Cairo, 11566, Egypt
Full list of author information is
available at the end of the article
Abstract
The potential of achieving computational hardware with quantum advantage
depends heavily on the quality of quantum gate operations. However, the presence
of imperfect two-qubit gates poses a significant challenge and acts as a major
obstacle in developing scalable quantum information processors. Google’s Quantum
AI and collaborators claimed to have conducted a supremacy regime experiment. In
this experiment, a new two-qubit universal gate called the Sycamore gate is
constructed and employed to generate random quantum circuits (RQCs), using a
programmable quantum processor with 53 qubits. These computations were carried
out in a computational state space of size 9 ×1015. Nevertheless, even in
strictly-controlled laboratory settings, quantum information on quantum processors
is susceptible to various disturbances, including undesired interaction with the
surroundings and imperfections in the quantum state. To address this issue, we
conduct both quantum state tomography (QST) and quantum process tomography
(QPT) experiments on Google’s Sycamore gate using different artificial architectural
superconducting quantum computer. Furthermore, to demonstrate how errors affect
gate fidelity at the level of quantum circuits, we design and conduct full QST
experiments for the five-qubit eight-cycle circuit, which was introduced as an
example of the programability of Google’s Sycamore quantum processor. These
quantum tomography experiments are conducted in three distinct environments:
noise-free, noisy simulation, and on IBM Quantum’s genuine quantum computer. Our
results offer valuable insights into the performance of IBM Quantum’s hardware and
the robustness of Sycamore gates within this experimental setup. These findings
contribute to our understanding of quantum hardware performance and provide
valuable information for optimizing quantum algorithms for practical applications.
Keywords: Google’s quantum AI; Sycamore gate; Quantum supremacy; Quantum
process tomography; Quantum state tomography; IBM’s quantum computers
1 Introduction
Traditional computerslack the capability to effectively simulate quantum mechanical sys-
tems. This is primarily due to the exponential growth in data requirements when attempt-
ing to comprehensively simulate quantum systems. In contrast, quantum computers [1]
utilize the distinctive characteristics of quantum systems on which they are built to effi-
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 2 of 25
ciently process vast amounts of information in a polynomial timeframe [2]. This allows
them to handle exponentially larger quantities of data compared to classical counterparts.
Quantumprocessorsareon the verge of fulfilling their transformative promise in revolu-
tionizing thefieldofcomputing[1,3–7].Remarkableprogress has been achieved in the de-
velopmentofquantumcomputers, particularly those utilizing superconducting qubits [8].
Among the most promising candidates are programmable processors based on supercon-
ducting qubits with Josephson junctions [9–14]. These processors offer the potential for
scalable quantum computation on solid-state platforms, presenting an exciting avenue for
advancement [15,16].
One significant breakthrough lies in the utilization of superconducting transmon
qubits [17–19], enabling Noisy intermediate-scale quantum (NISQ) architectures [20]to
execute computations in a vast Hilbert space with a dimension of approximately 253 [21].
Thisrepresentsanextraordinary leap forward,asquantumcomputersin theNISQ era can
tackle computational challenges deemed impossible for classical counterparts [22]. The
implicationsof this progress areprofound, known asquantum supremacy regime[22–24],
as it opens up new possibilities for tackling complex problems that were previously over-
whelming, surpassing the limitations of classical systems and unlocking unprecedented
computational power [22].
Random quantum circuits (RQCs) are considered challenging to simulate using clas-
sical systems. Achieving the ability to perform such simulations is a key milestone. In
their work [21], Google’s Quantum AI and collaborators revealed that Sycamore, a pro-
grammable quantum processor consisting of 53 superconducting qubits, successfully ex-
ecuted a computation in a Hilbert space of dimension 253. This achievement claimed to
surpass the capabilities of the Summit supercomputer, the most powerful classical super-
computer at that time. Initially estimated to take approximately 10,000 years on a con-
ventional computer, the task purportedly was completed in just 200 seconds on Google’s
quantum computer [21].
Soon after the publication, a debate erupted over the potential overestimation of the
amount of time required to complete the same problem on a supercomputer. However,
subsequent research demonstrated that an equivalent task could be simulated on classical
computers, such as the Summit supercomputer, within a matter of days using classical
simulation algorithms based on tensor networks [25–27]. During this debate, a significant
issue was raised: whether there could be trustworthy techniques to precisely characterize
the genuine quantum computing capabilities.
Today’s NISQ computers are capable of performing impressive algorithms. However,
each quantum processor’s capability is limited by hardware errors. For example, errors
such as drift [28,29], coherent noise [30–32] and crosstalk [33]. These challenging errors
make it difficult to reliably predict a processor’s capability and cause many programs to
fail.
The ability to attain computational hardware with quantum advantage relies greatly on
the quality of quantum gate operations. However, the existence of imperfect two-qubit
gates presents a notable challenge and serves as a significant barrier in the advancement
of scalable quantum information processors.
Quantum computing’s advancement hinges on our ability to accurately characterize
quantum systems. Quantum state tomography and quantum process tomography play
pivotal roles in this endeavor. QST allows us to fully describe the state of a quantum sys-
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 3 of 25
tem, providing crucial insights into its properties and behavior. Meanwhile, QPT enables
us to rigorously assess the performance of quantum gates and operations, shedding light
on their effectiveness and fidelity. Together, these techniques empower researchers to un-
derstand, verify, and optimize quantum systems, driving progress towards realizing the
transformative potential of quantum computing.
In this paper, we utilized a state-of-the-art superconducting quantum computer to con-
duct full quantum tomography experiments on a universal two-qubit gate known as the
Sycamore gate. This gate was constructed and employed to generate random quantum cir-
cuits (RQCs) on 53 qubits, synonymous with a computational state-space of size 9 ×1015.
The Sycamore gate belongs to a class of 2-qubit fermionic simulation gates [34]. More-
over, to demonstrate the impact of errors and their effects on gate fidelity at the level of
quantum circuits, we conducted full QST experiments on the five-qubit eight-cycle cir-
cuit. This specific circuit was selected as an illustrative example of the programmability of
Google’s Sycamore quantum processor. These QST experiments were conducted in three
distinct environments: a noise-free setting, a noisy simulated environment, and on IBM
Quantum’s genuine quantum computer.
Ourfindings shed light on the effectiveness ofdifferentimplementations and the sources
of errors, revealing fidelity results that are comparable to both theoretical (ideal) expecta-
tions and real-world experimental executions.
2Sycamore RQCs
An astounding engineering achievement was introduced in [21], aiming to reach a quan-
tum supremacy regime. A programmable superconducting processor was constructed,
featuring a novel class of RQCs. These circuits consist of alternating layers of 1-qubit and
2-qubit gates. Additionally, fast, high-fidelity quantum gates capable of running simulta-
neously over a 2-dimensional qubit array were developed. The Sycamore processor is dis-
tinguished by its ability to perform high-fidelity one-qubit and two-qubit quantum gates,
not only in isolation, but also while executing feasible computations with simultaneous
gate operations on various qubits [21].AfullcharacterizationofGoogle’sQuantumAI
processor, Sycamore, has been presented in [35].
The Sycamore quantum computer employs transmon qubits [17], which can be consid-
ered as nonlinear superconducting resonators functioning at 5–7 GHz. The quantum bits
are encoded as the resonant circuit’s two lowest quantum eigenstates. Additionally, each
transmon has two controls: a microwave drive for excitation of the qubit and a magnetic
flux control for frequency tuning. To read the qubit state, each qubit is associated with
a linear resonator [12]. Moreover, every qubit is linked to its neighboring qubits via ad-
justablecouplers[36,37], as depicted inFig. 1. Because of this coupler design, the coupling
between qubits can be quickly tuned from totally off to 40 MHz. During the supremacy
experiment,as one qubit wasdefective, the processor had 53-qubitsas well as 86-couplers.
In every random circuit performed on the 53-qubit Sycamore quantum processor [21],
each circuit consists of ncycles. Each “cycle” combines a single-qubit gate layer with a
two-qubit gate layer, as depicted in Fig. 2.IntheseRQCs,threesingle-qubitgatesare
constructed and executed: the √X,√Y,and√Wgates, such that
√X=1
√21–i
–i1,(1)
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 4 of 25
Figure 1 The layout of Google’s Sycamore quantum processor comprises a rectangular array of 54
programmable Superconducting transmon qubits. Each qubit is interconnected with its four nearest
neighbors through couplers. Due to one qubit not operating perfectly, the machine has 53-qubits and
86-couplers
Figure 2 A schematic diagram illustrating an n-cycle circuit for the Sycamore 53-qubit Random Quantum
Circuits (RQCs). Each cycle comprises two layers: one with randomly chosen 1-qubit gates from the set
{√X(in gray), √Y(in cyan),√W(in orange)}, and another with 2-qubit gates labeled A, B, C, or D. For longer
circuits, the layers follow the sequence A; B; C; D – C; D; A; B. Notably, the 1-qubit gates are not repeated
sequentially, and an additional layer of 1-qubit gates precedes the measurements
√Y=1
√21–1
11
,(2)
√W=1
√21–
√i
√–i1.(3)
In each cycle, there are two layers: one layer consists of randomly selected 1-qubit gates
from the set {√X,√Y,√W}, while the other layer consists of 2-qubit gates. The 1-qubit
gates are not repeated in sequential order, and before the measurements, an additional
layer of 1-qubit gates is applied. The 2-qubit gates in Google’s RQCs are not randomized.
Instead, both the qubit pair and the cycle number contribute to determining the 2-qubit
gates within the RQCs. These gates maintain the quantity of lower and excited states of
the qubits.
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 5 of 25
Table 1 Error rates of individual gates and readout. Measurements averaged over both |0and |1
states. Results reproduced from Google’s Sycamore quantum supremacy regime experiment [21].
The term “cycle” refers to a combination of two layers: one layer consists of 1-qubit gates, while the
other layer consists of 2-qubit gates
Average (mean) error Simultaneous Isolated
Single-qubit 0.16% 0.15%
Two-qubit 0.62% 0.36%
Two-qubit, cycle 0.93% 0.65%
Readout 3.8% 3.1%
The 2-qubit gates targeted for implementation in the quantum supremacy experiment
are referred to as the fSim gates:
fSim(£,φ)=e–i£(X⊗X+Y⊗Y)/2e–iφ(I–Z)⊗(I–Z)/4
=⎛
⎜
⎜
⎜
⎝
10 0 0
0cos(£) –isin(£) 0
0–isin(£) cos(£) 0
00 0eiφ
⎞
⎟
⎟
⎟
⎠.(4)
Equation (4) represents a family of two-qubit gates, known as “fSim” gates, which stands
for fermionic simulation gates [34].Itcanbeexpressedastheproductofatwo-qubit
(iSWAP(£)) gate and a non-Clifford controlled phase gate (CPHASE(φ)).The experimen-
talimplementationoftheentirespaceofgatesdescribedinEq.(4)representsalong-term
goal of Google Quantum AI and collaborators. [21]
In the supremacy experiment [21], the selected two-qubit gates are the Sycamore gates,
which are equivalent to the fSim gate with the two-qubit gates tuned up near to a swap an-
gle £ = π/2 and a conditional phase φ=π/6 radians. That is, Sycamore ≡fSim(π/2, π/6).
The fSim gate is defined to use a sign of £ different from that of the iSWAP gate [38].
Further insights into the 2-qubit gate strategy and the selection of implemented 2-qubit
gates by Google Quantum AI and its collaborators can be found in [21]. Table 1displays
error measurements averaged over both |0and |1states [21]. The isolated readout error
is detected to be 3.1%. Whereas, when running all qubit simultaneously, the readout error
increases slightly to 3.8%.
The two-qubit Sycamore gate transforms the basis states as follows: |0A⊗|0B→
|0A⊗|0B,|0A⊗|1B→–i|1A⊗|0B,|1A⊗|0B→–i|0A⊗|1Band |1A⊗|1B→
e(–iπ/6)|1A⊗|1B. Here, the subscripts A and B denote qubit A and qubit B, respectively.
The matrix operator corresponding to Sycamore gate reads,
Sycamore =⎛
⎜
⎜
⎜
⎝
10 0 0
00–i0
0–i00
00 0e(–iπ/6)
⎞
⎟
⎟
⎟
⎠.(5)
In the context of this paper, the two-qubit Sycamore gate is deconstructed into a set of
1-qubit gates combined with three controlled-X (CX) gates, as illustrated in Fig. 3. Addi-
tionally, Fig. 4depicts the location of the Sycamore gate on the “Weyl chamber” [39,40]in
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 6 of 25
Figure 3 Quantum circuit decomposition of Sycamore gate. The two-qubit gate is deconstructed into a set of
1-qubit gates (the rotation about z-axis, RZand the square root of NOT gate, √X) combined with three
Controlled-X gates
Figure 4 The Sycamore gate located on the “Weyl chamber” in the m1m2m3space. A tetrahedron whose
vertices correspond to O(0,0,0),A1(π,0,0),A2(π/2, π/2, 0) and A3(π/2,π/2, π/2). So that, every two qubit
gate is associated with a point in the “Weyl chamber”. In this representation, orbits of locally equivalent
unitary 2–qubit gates cross the Weyl chamber, constructing a tetrahedron in the cube of vectors mk.The
polyhedron PQA2RST represents the region of the perfect entanglers
the m1m2m3space, with respect to other two-qubit gates such as SWAP, iSWAP, and the
two-qubit quantum Fourier transform (QFT2)[6].
It is worth clarifying that the Sycamore gate, in conjunction with single-qubit gates, con-
stitutes a universal set, as is typical for most two-qubit gates [41]. Furthermore, it is perti-
nent to note that thebroader class of fSim gates might achieve universalityfor the subset of
Hamming-weight-preserving gateswhen utilizedalongside Z-gates.However,theclaimof
universality asserted in this paper specifically addresses the Sycamore gate’s role in combi-
nation with single-qubit gates, rather than its standalone universality. The demonstration
ofuniversalityas outlined by GoogleQuantumAIand collaboratorsin[21]isunderpinned
by establishing that the gate set consisting of CZ and SU(2) is universal [42,43].
3 Full quantum tomography study of Sycamore gate
3.1 QPT experiments
Quantum process tomography (QPT) [44,45] serves as a crucial method for validating
quantum gates and discerning deficiencies in circuit configurations and gate designs. The
principal aim of QPT is to derive a comprehensive characterization of the dynamical map,
or quantum process, through a series of meticulously crafted experiments.
Thetheory of QPT is extensively elucidated in several seminal works,including those by
Kraus [46], Chuang and Nielsen [44], Poyatos et al. [45], Mitchell et al. [47], and O’Brien
et al. [48]. Additionally, the concept of stand-alone QPT, which obviates the need for QST,
is expounded upon in [49].
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 7 of 25
Figure 5 The real (Re.(χ)) and imaginary (Im.(χ)) components of the Choi matrices, derived from Quantum
Process Tomography (QPT) experiments conducted on the two-qubit Sycamore gate, are examined. (a) The
theoretical expectations for the ideal Choi matrix are considered. (b) QPT experiments are conducted in a
noiseless environment. (c) QPT experiments are carried out in a simulated noisy environment. (d) QPT
experiments are performed on IBM’s real quantum computer, ibm_oslo. These experiments entail 4000
repetitions in each environment. The resulting process fidelity (FP) values are 0.9625, 0.6048, and 0.8102,
respectively. In the Hinton diagram representation, positive elements are depicted as white squares, while
negative elements are represented by black squares. The size of each square corresponds to the magnitude of
its associated value, indicating its relative importance
In our investigation, we leverage the Choi-Jamiolkowski representation [50,51]topro-
vide a comprehensive characterization of our quantum operations via QPT experiments.
The Choi representation offers a powerful framework wherein a quantum channel can be
entirelydescribed by aunique bipartite matrixknown as the Choimatrix. This representa-
tion is achieved through the Choi-Jamiolkowski isomorphism [50,51]. Further discussion
on alternative representations for quantum channels can be found in the review by Wood
et al. [52]. Nonetheless, the practical realization of full QPT has remained limited to ex-
ceedingly small-scale systems [44,45]. For instance, in scenarios involving a system of N
qubits, resulting in a formidable process matrix dimension of 4N×4N[53].
To effectively visualize the quantum process of the Sycamore gate, we conducted QPT
experiments focusing on the two-qubit gate. These experiments were carried out in three
distinct environments: a noiseless setting, a simulated noisy environment, and on IBM
Quantum’s quantum computer, ibm_oslo. In each environment, we executed the exper-
iments by repeating the QPT process 4000 times for every measurement basis across
all QPT circuits. The acquired experimental data were utilized to reconstruct the cor-
responding Choi matrices for each environment. Figure 5depicts both the real (Re.(χ))
and imaginary (Im.(χ)) components of the Choi matrices for the Sycamore gate.
Toquantitativelyassess theclosenessofthemeasured Choi matrices to the ideal process,
we computed the process fidelity FPof the noisy quantum channels [54]. This allowed
for meaningful comparisons across the different settings. The calculated process fidelities
(FP) for the noiseless, simulated noisy, and real-world ibm_oslo settings were found to be
0.9625, 0.6048, and 0.8102, respectively. These values provide insights into the efficacy of
the Sycamore gate under various conditions.
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 8 of 25
3.2 QST experiments
Quantum state tomography entails reconstructing a system’s quantum state description
from a series of quantum experiments. While a state vector characterizes the state of an
ideal quantum system, a density matrix ρdelineates the state of an open quantum sys-
tem. QST aims to rebuild or approximate this density matrix, through a consecutive set
of repeated measurements.
To achieve this, the quantum state is initially prepared using a state preparation circuit,
followed by a sequence of measurements corresponding to different operators. The re-
sultant measurement data are utilized to approximate or reconstruct the state. Various
techniques [55–59] have been developed to reconstruct a physically valid representation
of the density matrix, reflecting the actual state obtained in the system, from the mea-
sured data. Additional studies of QST in noisy environments [60–63] have also provided
valuable insights.
Here, we conduct QST experiments for the universal two-qubit Sycamore gate. The ex-
periments are performed in three different environments: a noisy simulated environment,
a noise-free, and on a real IBM Quantum’s quantum computer. For our experiments, we
utilized the qasm_simulator [64] configured to simulate the perfect (noiseless) quantum
computer. Additionally, we employed the 27-qubit IBM noisy simulator [64], which incor-
porates decoherence effects from earlier experimental implementations to compare with
real (noisy) quantum computing architectures. Here, the input state of the Sycamore gate
is prepared as the |0⊗|0state. In the context of this paper, the notation xy serves as
shorthand for the |x⊗|ystate, where xand yrepresent the binary values 0 or 1.
In these experiments, measurements of all qubits in the X,Y,andZPauli bases
{X,Y,Z}⊗{X,Y,Z}are added to the appended state preparation circuit of Sycamore gate.
This necessitates the execution of nine measurement circuits for the two-qubit Sycamore
gate. The resultant 36 measurement outcomes across all circuits, as depicted in Fig. 6,
yield a tomographically overcomplete basis, facilitating the reconstruction of the quan-
tum state.
We utilize this data to reconstruct density matrices for the Sycamore gate, as illustrated
in Fig. 7. This figure showcases both the real and imaginary components of the recon-
structed density matrices derived from conducting QST experiments with 4000 repeated
shots for each measurement basis. The state fidelity for two input states (density matrices)
is computed as follows:
Fs(ρtheor.,ρexper.)=Tr[√ρtheor. ρexper. √ρtheor. ]2,(6)
where ρexper. and ρtheor. are experimental and theoretical density matrices, respectively.
Our results demonstrate a state fidelity FSof 94.65%, 98.59%0 and 88.25%respectively,
corresponding to QST experiments for Sycamore gate in a noisy simulated environment,
noise-free, and on the ibm_oslo quantum computer.
Table 2provides details regarding qubit properties as well as the system experimental
parameters during executing the QST experiment of the Sycamore gate on the ibm_oslo
quantum computer. Figure 14 (appears in the Appendix)illustratesthemachine’sperfor-
mance during the execution of the QST experiments, including readout errors and pa-
rameter distributions for each qubit over the ibm_oslo quantum computer.
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 9 of 25
Figure 6 Experimental measurements over all QST circuits for the two-qubit Sycamore gate. The
experimental data were extracted from the nine measurement circuits performed in the QST experiments.
Moreover, the ultimate outcomes encompass the 36 measurement results across all circuits and qubits within
the Pauli bases {XX,XY,XZ},{YX,YY,YZ}and {ZX,ZY,ZZ}are presented. The experiments were run for 4000
repeated shots for each measurement basis, with a total running time of 13.8 s on the real IBM Quantum’s
quantum machine, ibm_oslo
The running time for our QST experiments on the system was 13.8 second. Addition-
ally, the average relaxation times T1andT2 were found to be 103.819 μs and 90.417 μs,
respectively. The qubit readout assignment error (ξ(Syc.)) averaged 16.348 ×10–3.The
average readout length (λ) was measured to be 910.22 ns. Moreover, the average qubit fre-
quency (ω/2π) was determined to be 5.0779 GHz, while the average qubit anharmonicity
(γ/2π) was calculated as –0.343 GHz. Furthermore, the qubit flip probabilities from |0
to |1(η)andfrom|1to |0(ζ) averaged 16.28 ×10–3 and 16.42 ×10–3 ,respectively.
Theprecisionindeterminingthe readout lengthand the gate error rates underscores our
commitment to transparent reporting of experimental details. While we acknowledge the
inherentchallenges in achieving such levelsof precision,itisimportanttoclarify thatthese
reported values serve to demonstrate the comprehensive characterization and meticulous
control of experimental parameters in our study.
4 Full quantum state tomography of the five-qubit eight-cycle circuit
In this section, we aim to illustrate how errors manifest in gate fidelity at the level of quan-
tum circuits. To achieve this, we design and conduct full QST experiments for the eight-
cycles circuit, depicted in Fig. 2. This circuit, introduced in [21], serves as an illustration
of the programmability of Google’s Sycamore quantum processor. The QST experiments
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 10 of 25
Figure 7 Real (Re.(ρ)) and imaginary (Im.(ρ)) parts of the density matrices; reconstructed from QST
experiments of the two-qubit Sycamore gate. (a) The ideal density matrix represents theoretical expectations.
(b) QST experiments conducted in noise-free environment. (c) QST experiments implemented in a noisy
simulated environment. (d) QST experiments executed on an IBM Quantum’s real quantum computer. These
experiments were executed for 4000 repeated shots in each environment. The state fidelity FScorresponding
to the above-mentioned environments are 0.9859, 0.9465, and 0.8824, respectively. The experiments had a
running time of 13.8 seconds
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 11 of 25
Table 2 Device specification during the execution of QST experiments of the 2-qubit Sycamore gate,
executed on the IBM quantum machine, ibm_oslo
Qubits T1(μs)T2(μs)ω/2π(GHz) γ/2π(GHz) ξ(Syc.) ζ(%) η(%) λ(ns)
Q0148.997 122.185 4.92504 –0.34477 0.01719 0.02520 0.00919 910.2222
Q1123.133 28.9460 5.04644 –0.34329 0.01109 0.01120 0.01100 910.2222
Q264.0071 53.2877 4.96200 –0.34593 0.00709 0.00780 0.00640 910.2222
Q362.4993 39.8356 5.10809 –0.34165 0.01689 0.01420 0.01959 910.2222
Q438.7673 167.231 5.01110 –0.34359 0.02380 0.02300 0.02459 910.2222
Q5154.136 34.7004 5.17334 –0.34156 0.00970 0.01000 0.00940 910.2222
Q6135.197 186.731 5.31935 –0.33762 0.02869 0.02359 0.03380 910.2222
Figure 8 Measurement outcomes for the experimental implementation of the 5-qubit 8-cycles circuit.
Results were obtained from executing the circuit for 20,000 shots on a qasm simulator,aswellasontwoof
IBM Quantum’s quantum computers: ibm_oslo and ibmq_manila. The vertical-axis represents the frequency
(count) associated with each of the five-qubit computational basis states, which are labeled on the horizontal
axis of the histogram. The quantum computation times on both ibm_oslo and ibmq_manila were recorded as
9.9 s and 9.4 s, respectively
are carried out in three environments: noise-free, noisy, and on an IBM Superconducting
quantum computer.
It’s important to highlight that throughout these implementations, we configured the
machine to adhere to Google’s layering configuration. This means that we instructed the
quantum machine not to optimize or minimize the number of gates applied to each qubit
in the original circuit. Instead, we allowed it to execute the circuit while maintaining the
same structured layers as in Sycamore’s processor.
In doing so, we first execute the 5-qubit eight-cycle circuit on a qasm simulator
and on two of IBM Quantum’s quantum computers: the five-qubit quantum machine
ibmq_manila and the seven-qubit quantum machine ibm_oslo. Measurement outcomes
from executing this circuit for 20,000 shots on these architectures are shown in Fig. 8.
Subsequently, we perform QST experiments on this 5-qubit 8-cycle quantum circuit
in three different settings: noisy, noise-free, and on a real IBM quantum computer,
ibmq_manila. This quantum computer is built based on a quantum processor of type Fal-
con r5.11L, which comprises five superconducting transmon qubits, produced by IBM
Quantum [64].
In these full QST experiments, we include measurements of all qubits in the 5-qubit
Pauli bases {X,Y,Z}⊗5to the appended state preparation circuit of the eight-cycles circuit.
This results in a total of 243 measurement circuits (35) that must be executed for this five-
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 12 of 25
Figure 9 Experimental results over a random subset of the 35quantum state tomography measurement
circuits. These circuits were randomly selected from a total of 243 measurement circuits executed in the QST
experiments for the five-qubit eight-cycles circuit. These results include the final outcomes for the 7776 (6n)
measurement outcomes across all circuits of all qubits in the five qubit Pauli bases. The QST experiments were
executed for 4000 repeated shots for each measurement basis on the real IBM Quantum’s quantum machine,
ibmq_manila. The quantum computation running time was 4 minutes and 54 seconds. (Fig. 9continued on
the nextpage...)
qubit eight-cycles circuit. Furthermore, the culmination of the 7776 (65)measurement
outcomes across all circuits furnishes a tomographically overcomplete basis, facilitating
the reconstruction of the (targeted) quantum state. A random sample from the outputs of
these quantum circuits is shown in Fig. 9and Fig. 10.
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 13 of 25
Figure 10 (. .. Continued). The vertical-axis represents measurement outcomes associated with each of the
five-qubit Pauli-basis states, that labeled on the horizontal axis of each histogram by numbers from 01 to 32.
These numbers correspond respectively to the states: (00000, 01000, 01001, 01010, 01011, 01100, 01101,
01110, 01111, 00001, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011,
11100, 11101, 11110, 11111, 00010, 00011, 00100, 00101, 00110, 00111)
We use these results to reconstruct density matrices for the five- qubit eight-cycles cir-
cuit, as depicted in Fig. 11. This figure displays both the real and the imaginary parts
of the reconstructed density matrices obtained from running full QST experiments for
4000 repeated shots for each measurement basis. When performing QST in a noisy simu-
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 14 of 25
Figure 11 QST experiments conducted for the five-qubit eight-cycles circuit. Here, the real (Re.(ρ)) and
imaginary (Im.(ρ)) parts of the density matrices are reconstructed from the 243 (35) measurement quantum
circuits executed in the QST experiments for the 5-qubit 8 cycle circuit. These results include the final
outcomes for the 7776 measurement across all circuits. (a) The ideal density matrix (theoretical expectations).
(b) QST experiments performed in qasm_simulator. (c) QST experiments implemented with a noisy simulated
environment. (d) QST experiments executed on an IBM’s real quantum computer ibmq_manila.These
experiments were repeated for 4000 shot in each environment. The state fidelity FScorresponding to the
experimental implementations with qasm_simulator, noisy, and on the real quantum computer, ibmq_manila
are 97.72%, 51.55% and 15.95% respectively
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 15 of 25
Table 3 Device specification during the execution of QST experiments of the 5-qubit 8-cycle
quantum circuit on the IBM quantum machine, ibmq_manila. Were accessed on July 6, 2023
Qubits T1(μs)T2(μs)ω/2π(GHz) γ/2π(GHz) ξ(8-cycle.) ζ(%) η(%) λ(ns)
Q0137.0243 83.57426 4.962287 –0.344625 0.026499 0.041399 0.0116 5351.111
Q1188.4665 74.88844 4.837877 –0.345283 0.022000 0.024000 0.0200 5351.111
Q2105.1689 20.20507 5.037238 –0.342551 0.029399 0.036599 0.0222 5351.111
Q3229.1088 67.30202 4.950952 –0.343578 0.017700 0.0220 0.0134 5351.111
Q499.18768 39.62336 5.065125 –0.342108 0.015199 0.0254 0.0050 5351.111
Table 4 The process fidelity (FP) and the state fidelity (FS) are evaluated through quantum
tomography experiments conducted on both Google’s Sycamore gate and the 5-qubit 8-cycle circuit,
across various settings
Fidelity QST Google’s Sycamore gate QPT Google’s Sycamore gate QST Google’s 8-cycle circuit
FS(noiseless) FSa(noisy) FS(IBM)bFP(noiseless) FP(noisy) FP(IBM)bFS(noiseless) FS(noisy) FS(IBM)c
Value 98.59% 94.65% 88.25% 96.25% 60.48% 81.02% 97.72% 51.55% 15.95%
aThe state fidelity from running QST with noise.
bBoth QST and QPT have been conducted on the real quantum computer, ibm_oslo.
cQST on the real quantum computer, ibmq_manila.
lated environment, noise-free, and on the real IBM quantum computer, ibmq_manila,we
demonstrate a state fidelity FSof 0.5155, 0.9772, and 0.1595 respectively.
The discrepancy observed between noiseless simulations and real-world quantum de-
vice executions underscores the importance of understanding the impact of noise and
imperfections in practical quantum computing environments. While noise-free simula-
tions may provide valuable insights into theoretical performance, our experiments on real
quantum hardware highlight the challenges and opportunities in optimizing quantum al-
gorithms for real-world applications.
By conducting both QST and QPT analyses, we were able to gain insights into the per-
formanceof IBM Quantum’shardware and therobustness of Sycamore gates, when imple-
mented in this setup. These findings contribute to our understanding of quantum hard-
ware performance and provide valuable information for optimizing quantum algorithms
for practical applications.
Table 3presents hardware performance, qubit properties as well as the system exper-
imental parameters used during the execution of the full QST experiments of the eight-
cycles circuit on the ibmq_manila quantum machine. The quantum computation time of
running our QST experiments on the quantum computer ibmq_manila was 4 m 54 s. The
average relaxation times T1andT2 are respectively 151.791 μs and 57.119 μs. The qubit
readout assignment error (ξ(8 – cycle.)) has an average of 0.02215. The average readout
length(λ) is5351.11 ns. Theaveragequbit frequency (ω/2π) is4.971 GHz, andthe average
qubitanharmonicity (γ/2π) is –0.3436 GHz. The qubit flip probabilities from |0to |1(η)
and from |1to |0(ζ) have an average of 0.0145 and 0.0299, respectively. This study could
extended further considering the per-round scrambling rate for the multi-round 8-cycle
circuit tests.
Table 4compares the state fidelities (FS) and the process fidelities (FP)obtainedfrom
conducting quantum experiments in different environments (noise-free, noisy-simulated,
and on a physical quantum processor). While the fidelity observed in our noiseless sim-
ulations may be slightly below 1 compared to the ideal gate, this discrepancy is within
an acceptable range considering the finite sampling and factors inherent to the simulation
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 16 of 25
methodology, such as numerical precision errors and algorithmic approximations. We ac-
knowledge that this deviation may impact the precision of other reported numbers and
have taken this into consideration in our analysis.
The discrepancy observed in the performance between the QPT and QST experiments
for the Sycamore gate is notable. It is remarkable that the simulated noisy environment
performs better than the real IBM device in the QST experiments but fares poorly in the
QPT experiment. While we currently lack a clear intuition regarding this observation,
several factors may contribute to these differences. These factors include experimental
variability, measurement sensitivity, and the complexity of QPT.
We acknowledge that employing bootstrapping methods to generate estimates of the
uncertainty of the reported fidelities could further enhance the robustness of our analysis.
This approach could provide a more comprehensive understanding of the precision to
which the fidelities were determined, and we consider it as a valuable direction for future
research.
5Conclusion
The potential to attain computational hardware with quantum advantage relies signifi-
cantly on the quality of quantum gate operations. However, the existence of imperfect
two-qubit gates presents a notable challenge and serves as a significant barrier in the ad-
vancement of scalable quantum information processors.
In this study, we utilized state-of-the-art superconducting quantum computers to per-
form comprehensive quantum tomography experiments of a universal two-qubit gate.
This gate falls within the category of 2-qubit fermionic simulation gates, specifically
Google’s Sycamore gate, which played a crucial role in executing the random quantum
circuits (RQCs) that led to the achievement of quantum supremacy regime using Google’s
53-qubits Sycamore Quantum AI.
Additionally, we aimed to demonstrate the influence of undesired operational inter-
ference on the fidelity of the Sycamore gate at the level of quantum circuits. To achieve
this objective, we performed full QST experiments, specifically targeting the five-qubit
eight-cycle circuit. This circuit was selected as an illustrative example to highlight the
programmability of Google’s Sycamore quantum processor. These QST experiments were
carried out in three different environments: a noise-free setting, a noisy simulated envi-
ronment, and on IBM’s Quantum genuine quantum computers.
In our QPT experiments, the process fidelities (FP) for the noiseless, simulated noisy,
and real-world ibm_oslo settings were found to be 96.25%, 60.48%, and 81.02%, respec-
tively. Moreover, in our QST experiments, results revealed that the Sycamore gate could
maintain a relatively high state fidelity in these environments, with state fidelities 98.59%,
94.65%and 88.25%, respectively, when running our QST experiments for 4000 repeated
shots for each measurement basis. However, at the level of quantum circuits, errors
and other sources of unwanted operational interference significantly impact gate fidelity.
We observed a state fidelity of 97.72%, 51.55%and 15.95%when running QST for the
Sycamore gate within the 5-qubit 8-cycle circuit.
Thesefindings highlight the Sycamore gate’srobustness and reliabilityas a component of
a quantum computing system, suggesting its potential for diverse applications. However,
it’s essential to acknowledge that the gate’s fidelity within a quantum circuit was notably
lower in noisy and quantum computer environments compared to noise-free conditions.
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 17 of 25
This indicates a need for continued efforts to improve the gate’s resilience to noise and
external perturbations.
Future research could delve into identifying and mitigating the sources of these imper-
fections, aiming to enhance the overall performance of the Sycamore gate and other quan-
tum computing elements.
Appendix: Hardware performance
In this appendix, we offer comprehensive insights into the hardware characteristics of the
IBM Quantum’s quantum computers utilized in our study, encompassing both the seven-
qubit ibm_oslo and the five-qubit ibmq_manila systems. Figures 12 and 13 present the
readout error map and layout of the ibm_oslo and ibmq_manila quantum computers, re-
spectively. Detailed general specifications and coupling maps for both systems are pro-
vided in Tables 5,6,7,and8.
The quantum computer ibmq_manila have a minimum (maximum) qubit frequency of
4.838 (5.065) GHz, and a minimum (maximum) readout assignment error of 1.690 ×10–2
(3.920×10–2),with a CNOTaverageerrorrateof 0.817%with, and an average H error rate
of 0.036%, with median CNOT gate time 344.889 ns. This quantum computer can execute
up to 100 quantum circuits, with 20,000 maximum shots. The basis gates of this device
encompass the identity (I), the square root of NOT gate (√Xor SX), the NOT gate(X), the
rotation gate around the z-axis (RZ), and the 2-qubit controlled-NOT gate (CX). The cou-
pling maps between its qubits are delineated as follows: CNOT:[Q0;Q1], CNOT:[Q1;Q0],
CNOT:[Q1;Q2], CNOT:[Q2;Q1], CNOT:[Q2;Q3], CNOT:[Q3;Q2], CNOT:[Q3;Q4]and
CNOT:[Q4;Q3].
Figure 12 Thelayoutandtheerrormapoftheibm_oslo quantum machine, presented by IBM Quantum. It
comprises seven superconducting transmon qubits, with a minimum (maximum) qubit frequency of 4.925
(5.319) GHz, and a minimum (maximum) readout assignment error of 0.710 ×10–2 (2.870 ×10–2). The
processor can execute up to 100 quantum circuits, with a maximum of 20,000 shots
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 18 of 25
Figure 13 Readout error map and layout of the ibmq_manila quantum machine. This quantum machine is
based on a Falcon r5.11L quantum processor, produced by IBM Quantum. It consists of five superconducting
transmon qubits, with a minimum (maximum) qubit frequency of 4.838 (5.065) GHz, and a minimum
(maximum) readout assignment error of 1.690 ×10–2 (3.920 ×10–2), with an average CNOT error rate of
0.817% and an average H error rate of 0.036%, with a median CNOT gate time of 344.889 ns. The processor
can execute up to 100 quantum circuits, with a maximum of 20,000 shots
Table 5 General specifications of the seven qubits IBM Quantum’s real machine, ibm_oslo, including
error rates of individual gates and readout. These experimental parameters were accessed on
September 11, 2022
Parameters The ibm_oslo quantum machine
Q0 Q1 Q2 Q3 Q4 Q5 Q6
T1 (μs) 165.51 137.93 152.04 98.280 174.68 126.02 105.89
T2 (μs) 110.68 39.84 28.47 38.53 191.83 34.84 246.23
Frequency (GHz) 4.925 5.046 4.962 5.108 5.011 5.173 5.319
Anharmonicity (GHz) –0.34448 –0.34293 –0.34565 –0.34186 –0.34389 –0.34234 –0.33883
Readout assignment error (10–2) 0.98 1.17 0.71 1.71 2.2 1.16 2.87
Prob. meas|0prep|10.0124 0.014 0.0084 0.0124 0.021 0.0112 0.0314
Prob. meas|1prep|00.0072 0.0094 0.0058 0.0218 0.023 0.012 0.026
Readout length (ns) 910.222 910.222 910.222 910.222 910.222 910.222 910.222
Single-qubit gate error (10–4) 2.978 1.952 2.166 4.017 1.985 3.516 2.141
Table 6 Coupling map between different qubits of the ibm_oslo, indicating the CX error rates
between qubits Qiand their corresponding gate times. Data accessed on September 11, 2022
Qubits Q0Q1Q2Q3Q4Q5Q6
Couplinga[Q0–Q1][Q1–Q2][Q1–Q3][Q1–Q0][Q2–Q1][Q3–Q1][Q3–Q5][Q4–Q5][Q5–Q4][Q5–Q3][Q5–Q6][Q6–Q5]
Gate timeb341.333 248.889 263.111 412.444 213.333 334.222 238.222 305.333 341.333 167.111 405.333 334.222
Gate errorc7.126 5.801 7.571 7.126 5.801 7.571 5.110 9.309 9.309 5.110 7.232 7.232
aCX [Qi–Qj].
bGate time in ns.
cCX error rates (×10–3).
The ibmq_manila quantum machine has the following median error rates and qubit
parameters: Median CNOT error of 7.296×10–3, medianSXerrorof3.270×10–4,median
readout error of 2.640 ×10–2, median T1 of 134.7628 μs, and median T2 of 67.3 μs.
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 19 of 25
Table 7 General specifications and qubit properties of the five-qubit IBM Quantum computer,
ibmq_manila, illustrating error rates of individual gates and readout. This quantum machine has a
median CNOT error: 7.296 ×10–3 , a median SX error: 3.270 ×10–4 , a median readout error:
2.640 ×10–2, a median T1: 134.76284 μs, and a median T2: 67.3 μs. This machine can execute up to
20,000 shots with a maximum of 100 experiments. Accessed July 5, 2023
Parameters System characteristics of ibmq_manila
Q0 Q1 Q2 Q3 Q4 Median
T1 (μs) 134.76284 254.232 107.03353 247.80769 95.08469 134.76284
T2 (μs) 83.57426 74.88844 20.20508 67.30202 39.62336 67.30
Frequency (GHz) 4.96229 4.83787 5.03724 4.95095 5.06512 4.962
Anharmonicity (GHz) –0.344625 –0.345283 –0.342551 –0.343578 –0.342107 –0.34358
Readout assignment error 0.0392 0.0169 0.0346 0.0264 0.0192 2.640 ×10–2
Prob. meas|0prep|10.0604 0.0256 0.0414 0.0362 0.029 0.0362
Prob. meas|1prep|00.018 0.0082 0.0278 0.0166 0.0094 0.0166
Readout length (ns) 5351.111 5351.111 5351.111 5351.111 5351.111 5351.111
ID errora0.000214 0.000326 0.000608 0.000200 0.000455 3.270 ×10–4
Single-qubit Pauli-X gate error 0.000214 0.000326 0.000608 0.000200 0.000455 3.270 ×10–4
Single-qubit SX gate error 0.000214 0.000326 0.000608 0.000200 0.000455 3.270 ×10–4
aIdentity gate.
Table 8 Coupling map between different qubits of the ibmq_manila quantum computer, indicating
CNOT error rates between qubits Qiand corresponding gate times. Accessed July 5, 2023
Qubits Q0Q1Q2Q3Q4
Coupling mapa[Q0;Q1][Q1;Q0][Q1;Q2][Q2;Q1][Q2;Q3][Q3;Q2][Q3;Q4][Q4;Q3]
Gate timeb277.3333 312.8888 469.3333 504.8888 355.5555 391.1111 334.2222 298.6666
CNOT Gate errorc5.469529 5.469529 12.61270 12.62701 8.414589 8.414589 6.176700 6.176700
aQubits coupled by the CNOT gate.
bGate time in (ns).
cCNOT error rates (×10–3).
For each qubit of the ibm_oslo and ibmq_manila quantum computers during QST ex-
periments, Figs. 14 and 15 showcase processor performance, parameter distribution, and
readout errors. Finally, Figs. 16 and 17 represent the real (Re.(ρ)) and imaginary (Im.(ρ))
parts of the density matrices resulting from running QST for the five-qubit 8-cycle circuit,
displayed as 2D (Hinton) diagrams.
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 20 of 25
Figure 14 Qubit properties, parameter distributions, and readout errors for each qubit on the ibm_oslo
quantum processor during the execution of QST experiments for the two-qubit Sycamore gate
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 21 of 25
Figure 15 Parameters distribution, processor performance, and readout errors for each qubit of the IBM
Quantum’s machine, imbq_manila during the execution of QST experiments for the five-qubit eight-cycles
circuit
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 22 of 25
Figure 16 The real (Re.(ρ)) and imaginary (Im.(ρ)) parts of the density matrices resulting from running QST
for the five-qubit 8-cycle circuit are represented as Hinton diagrams. (a) The ideal density matrix representing
theoretical expectations. (b) Quantum State Tomography (QST) experiments conducted on the qasm
simulator.(Fig.16 continuedonthenextpage ...)
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 23 of 25
Figure 17 (... Continued) Thereal(Re.(ρ)) and imaginary (Im.(ρ))partsofthedensitymatricesresultingfrom
running QST for the five-qubit 8-cycle circuit are represented as a Hinton diagram. (c) QST experiments
conducted in a noisy simulated environment. (d) QST experiments carried out on IBM Quantum’s real
quantum computer, ibmq_manila
Abbreviations
RQCs, Random quantum circuits; QST, Quantum state tomography; QPT, Quantum process tomography; NISQ, Noisy
intermediate-scale quantum; fSim, Fermionic simulation; £, A swap angle; RZ, The rotation about z-axis; CPHASE,
Controlled phase gate; QFT, Quantum Fourier transform; Falcon r5.11L, A quantum processor produced by IBM Quantum;
FS, The state fidelity; ρ,Densitymatrix;ρexper., Experimental density matrix; ρtheor., Theoretical density matrix; qasm,
Quantum assembly language; Qi,Theith qubit; CX, The controlled-X gate; Syc., Sycamore;T1, Relaxation time; T2,
Dephasing time; ω/2π, Qubit frequency; γ/2π, Qubit anharmonicity ; ξ, The qubit readout assignment error; ζ,The
qubit flip probabilities from |1to |0;η, The qubit flip probabilities from |0to |1;λ, Readout length; Re., Real part; Im.,
Imaginary Part; GH z, Giga hertz 1 GHz = 1.0 ×109Her tz; MHz, Mega hertz 1 MHz = 1.0 ×106Hertz; μs, Micro seconds,
1μs=1.0×10–6 second; ns, Nano seconds, 1 ns =1.0×10–9 second.
Acknowledgements
We sincerely thank the anonymous reviewers for their insightful feedback and constructive criticism on our paper. Their
expertise and suggestions have undoubtedly improved the quality and clarity of our work. We acknowledge the use of
IBM’s Superconducting quantum computers in this work. However, the views and conclusions expressed in this
manuscript are solely those of the authors and do not necessarily represent the views of IBM Quantum or Google’s
Quantum AI.
Author contributions
M. AbuGhanem: Conceptualization, Methodology, Software, experimental implementations on IBM Quantum’s quantum
computers, Data curation, Formal analysis, Visualization, Investigation, Validation, Writing, Reviewing and Editing.
H. Eleuch: Investigation, Validation, Reviewing and Editing. All authors have approved the final manuscript.
Funding
The authors declare that no funding, grants, or other forms of support were received at any point throughout this
research work.
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AbuGhanem and Eleuch EPJ Quantum Technology (2024) 11:36 Page 24 of 25
Data availability
The datasets generated during and/or analyzed during the current study are included within this article.
Declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
All authors have approved the publication. This research did not involve any human, animal or other participants.
Competing interests
The authors declare no competing interests.
Author details
1Faculty of Science, Ain Shams University, Cairo, 11566, Egypt. 2Department of Applied Physics and Astronomy, University
of Sharjah, 27272 Sharjah, UAE. 3College of Arts and Sciences, Abu Dhabi University, Abu Dhabi 59911, UAE. 4Institute for
Quantum Science and Engineering, Texas A&M University, College Station, TX 77843, USA.
Received: 12 July 2023 Accepted: 13 May 2024
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