Bibliometric Analysis in the Field of Quantum Technology

Abstract

The second quantum technological revolution started around 1980 with the control of single quantum particles and their interaction on an individual basis. These experimental achievements enabled physicists, engineers, and computer scientists to utilize long-known quantum features—especially superposition and entanglement of single quantum states—for a whole range of practical applications. We use a publication set of 54,598 papers from Web of Science, published between 1980 and 2018, to investigate the time development of four main subfields of quantum technology in terms of numbers and shares of publications, as well as the occurrence of topics and their relation to the 25 top contributing countries. Three successive time periods are distinguished in the analyses by their short doubling times in relation to the whole Web of Science. The periods can be characterized by the publication of pioneering works, the exploration of research topics, and the maturing of quantum technology, respectively. Compared to the USA, China’s contribution to the worldwide publication output is overproportionate, but not in the segment of highly cited papers.

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Article
Bibliometric Analysis in the Field of Quantum Technology
Thomas Scheidsteger 1, *,† ,‡ , Robin Haunschild 1,†, ‡ , Lutz Bornmann 2and Christoph Ettl 3


Citation: Scheidsteger, T.;
Haunschild, R.; Bornmann, L.; Ettl, C.
Bibliometric Analysis in the Field of
Quantum Technology. Quantum Rep.
2021,3, 549–575. https://doi.org/
10.3390/quantum3030036
Academic Editor: Lev Vaidman
Received: 8 June 2021
Accepted: 9 August 2021
Published: 15 September 2021
Publisher’s Note: MDPI stays neutral
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iations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany;
r.haunschild@fkf.mpg.de
2Science Policy and Strategy Department, Administrative Headquarters of the Max Planck Society,
Hofgartenstr. 8, 80539 Munich, Germany; lutz.bornmann@gv.mpg.de
3Presidential Division, Administrative Headquarters of the Max Planck Society, Hofgartenstr. 8,
80539 Munich, Germany; christoph.ettl@gv.mpg.de
*Correspondence: t.scheidsteger@fkf.mpg.de
† These authors contributed equally to this work.
‡ The present study is an extended version of a paper presented at the 18th International Conference on
Scientometrics and Informetrics, Leuven (Belgium), 12–15 July 2021.
Abstract:
The second quantum technological revolution started around 1980 with the control of single
quantum particles and their interaction on an individual basis. These experimental achievements
enabled physicists, engineers, and computer scientists to utilize long-known quantum features—
especially superposition and entanglement of single quantum states—for a whole range of practical
applications. We use a publication set of 54,598 papers from Web of Science, published between 1980
and 2018, to investigate the time development of four main subfields of quantum technology in terms
of numbers and shares of publications, as well as the occurrence of topics and their relation to the
25 top contributing countries. Three successive time periods are distinguished in the analyses by their
short doubling times in relation to the whole Web of Science. The periods can be characterized by the
publication of pioneering works, the exploration of research topics, and the maturing of quantum
technology, respectively. Compared to the USA, China’s contribution to the worldwide publication
output is overproportionate, but not in the segment of highly cited papers.
Keywords:
quantum information; quantum metrology; quantum communication; quantum comput-
ing; scientometrics; bibliometrics
1. Introduction
At the end of the 19th century, there was a prevalent opinion that the building of
physics was complete, and nothing new was left to be discovered. However, since approxi-
mately the turn of the 20th century, certain new phenomena that apparently could not be
interpreted in the theoretical frame of classical physics shattered this notion and initiated
an unexpected revolution. The revolution started with Planck’s quantum hypothesis to
derive the correct black body radiation [
1
,
2
] and Einstein’s explanation of the photoelectric
effect [
3
]. Both led to a full-grown quantum theory in the mathematical formulations of the
matrix mechanics of Heisenberg, Born, and Jordan [
4
], as well as of Schrödinger’s wave me-
chanics [
5
]. The primary innovative and non-classical ingredients of the new theory were
the following: (i) a superposition of states was now possible, which had not been thinkable
in the classical framework; (ii) the time evolution of quantum systems was no longer
deterministic and, therefore, required a probabilistic description; (iii) objective properties,
e.g., location and speed at the same time, were no longer existent apart from a determining
measurement; and (iv), most counter-intuitively, particles that are not locally connected
could now correspond via their common wave function—the so-called entanglement.
Quantum theory turned out to be highly consistent with experiment and formed the
basis for the development of solid state physics and for a first quantum technological
revolution. This development led to such applications as lasers, transistors, nuclear power
Quantum Rep. 2021,3, 549–575. https://doi.org/10.3390/quantum3030036 https://www.mdpi.com/journal/quantumrep

Quantum Rep. 2021,3550
plants, solar cells, and superconducting magnets in nuclear magnetic resonance (NMR)
devices and particle accelerators. These applications have in common the exploitation of
quantum behavior—such as the tunneling effect—of great ensembles of particles.
In the late 1970s and early 1980s, scientists learned to prepare and control systems
of single quantum particles, such as atoms, electrons, and photons. The scientists let the
particles interact on an individual basis. This ability sparked a second quantum revolution,
when physicists, engineers, and computer scientists worked together to utilize the long-
known quantum features—especially superposition and entanglement of single quantum
states—for a whole range of practical “next generation” applications. These applications
may be summarized as “quantum engineering” or “quantum technology 2.0” (QT 2.0).
The present study provides a bibliometric analysis of QT 2.0, methodologically fol-
lowing previous studies dealing with research fields such as climate change [
6
], specific
aspects thereof [
7
–
9
], and density functional theory (DFT) [
10
]. The dataset used in this
study has been analyzed in the white paper by Bornmann et al. [
11
], but only for the time
period 2000–2016 and with a focus on Germany. The present study analyzes QT 2.0 over
the time period 1980–2018 with an international perspective on the topic.
2. Quantum Technology 2.0: Foci of Research in Four Subfields
QT 2.0 can be structured in various ways [
12
–
14
]. We prefer to broadly divide them
into four fields, which have substantial overlaps, but do not cover all possible quantum tech-
nologies: (i) quantum information science; (ii) quantum metrology, sensing, imaging, and
control; (iii) quantum communication and cryptography; and (iv) quantum computation.
(i) Quantum information science is the basis for the whole of QT 2.0. It is mainly
the study of the “second-order” effects of quantum theory, firstly recognized by Einstein,
Podolsky, and Rosen [
15
] in their famous EPR gedankenexperiment: quantum systems can
exhibit non-local, entangled correlations unknown in the classical world, which Einstein
opposed as “spooky action at a distance”, insisting that the quantum theory must be
incomplete. Alternatives, e.g., so-called local hidden-variable theories, were proposed; Bell
proved, 30 years later, that these were obliged to fulfill his famous inequality [
16
]. Since
it takes very carefully engineered quantum states to realize and measure these effects, it
took another decade to ascertain their experimental determination [
17
–
19
]; the violation of
Bell’s inequality ruled out local hidden-variables theories. Subsequently, it became feasible
to think of applications of quantum information processing. A milestone year on the path
to exploiting quantum entanglement was 1994: non-local photon correlations over a long
optical fiber could be experimentally demonstrated [
20
] and could, prospectively, after
some improvements, be used for quantum cryptography. Furthermore, an algorithm for a
future quantum computer that could solve a very difficult numerical problem exponentially
faster than all classical computer algorithms known at that time [21] was presented.
A basic prerequisite for quantum information technology is the concept of a qubit or
quantum bit. This is the quantum mechanical generalization of a classical bit, which can be
physically realized as a two-state device (e.g., a ground state level and an excited level of
an ion in an ion-trap, the spin of an electron in a quantum dot, a photon with vertical or
horizontal polarization). According to quantum mechanics, a qubit can stay in a coherent
superposition of both states as long as it is not measured. This would “force” the system
into one of the two states. With several qubits, one can form quantum gates, registers,
and circuits for computational purposes as building blocks for quantum processors. The
engineering challenge is the layout of hardware systems that can handle many qubits,
store them, and keep them stable enough to perform several computation cycles in order
to realize a quantum computer. Not only quantum computing, but also other quantum
technologies are inextricably connected with quantum information science. Especially in
quantum communications, the use of photons (quantum optics) is prevalent because of
their weak interaction with matter and therefore long coherence times. These times are
needed for the transportation of quantum information [16].

Quantum Rep. 2021,3551
(ii) Quantum metrology and sensing offer measurement techniques that provide
higher precision than the same measurement performed in a classical framework. One
well-known example of quantum metrology that had been around for a long time is the
atomic clock [
22
], which uses a characteristic transition frequency in the electromagnetic
spectrum of atoms as a standard. The new generation of quantum logic clocks achieves
a previously unknown accuracy by exploiting the sensitivity of quantum entanglement
against disturbances, measured, e.g., in a single ion [
23
]. Quantum-enabled high-precision
measurements using, e.g., the Josephson effect and the quantum Hall effect, have been es-
sential for the recently completed redefinition of the SI unit system via natural constants [
24
].
Other new devices of quantum sensing are atom interferometry-based gravimeters [
25
] or
magnetic field sensors based on quantum defects in diamonds, which are sensitive enough
to detect changes in single nerve cells [
26
]. Quantum tomography is a mathematical tech-
nique to reconstruct quantum states via a sufficient set of measurements [
27
]. An important
application is the characterization of optical signals, including the signal gain and loss
of optical devices [
28
]. Another relevant application is the reliable determination of the
actual states of the qubits in quantum computing and quantum information theory [
29
,
30
].
Quantum imaging is a new subfield of quantum optics. It exploits quantum correlations
such as quantum entanglement of the electromagnetic field to image objects with a res-
olution or other imaging criteria that are beyond what is possible in classical optics. In
that area, the special technique of “ghost imaging” is using light that has never physically
interacted with the object to be imaged [
31
]. Control of quantum systems is, e.g., achieved
via manipulation of quantum interferences of the wave functions of coherent laser beams.
It is dominantly guided by the so-called quantum optimal control theory [32].
(iii) Quantum communication and cryptography were started with the publication of
the BB84 protocol for quantum key exchange by Bennett and Brassard [
33
]. It is based on
an idea by Wiesner from the early 1970s that had for a long time been unpublished [
34
,
35
]:
Heisenberg’s uncertainty principle would prevent undercover eavesdropping. Later, Ekert
introduced the use of entangled qubits (quantum bits) into quantum key distribution [
36
].
Due to the no-cloning theorem of quantum mechanics, it is not possible—in contrast to the
classical case—to replicate a quantum state exactly [
37
]; it can be done either approximately
or exactly only with a certain probability. Therefore, the information encoded in transferred
qubits cannot be identically copied. Quantum networks consist of quantum processors
as nodes which exchange qubits over quantum communication channels as edges. They
are, therefore, a necessary ingredient of quantum computing. Secure communication in
quantum networks is essential for the long-range transmission of quantum information,
usually by quantum teleportation. This idea was introduced by Bennett et al. [
38
]. Only
four years later, it was experimentally demonstrated by Boschi et al. [
39
] and Bouwmeester
et al. [
40
] independently via entangled photons—the significance of the former being
controversial [
41
]. Another seven years later, entangled photons were used as the basis
for an unbreakable communication code in order to perform a secure money transaction
between two banks in Austria [
42
]. An extended quantum network leads to a quantum
internet, which in addition needs quantum repeaters. They do not work like classical
repeaters due to the no-cloning theorem. They rather build upon entanglement swapping
and distillation and need to store qubits in quantum memory units [
43
]. A recent milestone
was the achievement of all three scientific goals in launching the first quantum satellite,
called Micius, by China: a quantum entanglement distribution over a long distance [
44
], a
satellite-to-ground quantum key distribution between China and Austria by implementing
the BB84 protocol [45], and quantum teleportation [46].
(iv) Quantum computing promises a quantum leap in computational power since pre-
vious speed-ups on the basis of semiconductor technology as described by Moore’s law [
47
]
appear to come to an end [
48
–
50
]. The original idea of quantum computing was expressed
by Feynman [
51
] (this is the transcript of a talk given by Feynman in 1981): Quantum
systems as, e.g., molecules should be simulated by letting a model quantum system evolve
and calculate the system in question. That was a new approach—rather different from

Quantum Rep. 2021,3552
implementing the classical algorithms, e.g., of quantum chemistry. The classical algorithms
consume a high number of computational resources. The first implementation of quan-
tum simulation was the quantum variant of simulated annealing. This is a widely used
Monte Carlo optimization algorithm for finding extrema of multidimensional functions
by mimicking the thermalization dynamics of a system which is slowly cooled. Thermal
excitations allow the system to escape out of local minima. In quantum annealing, this
possibility is much greater by including the tunneling effect [
52
]. In 2011, the Canadian
enterprise D-Wave announced to have built the first commercial quantum annealer [
53
].
The greatest success of this kind of device is the recent (until then not feasible) simulation
of magnetic phase transitions in a 3D lattice of qubits [
54
]. Due to the susceptibility of
quantum computers to decoherence and noise, a substantial performance improvement
can be achieved by the implementation of quantum error correction [55].
Others try to implement a universal model of quantum computation using quantum
logic gates in superconducting electronic circuits. These attempts are most prominent
in popular science presentations and in the media. They are reporting on the efforts
and successes of global players as Google, IBM, and others to reach quantum supremacy.
Quantum supremacy means that a programmable quantum device can solve a problem
that no classical computer can feasibly solve. In 2019, Google claimed to have reached this
goal [
56
] with its quantum processor for a very special problem: this processor checked
the outputs from a quantum random-number generator within minutes, for which the
world’s largest supercomputer would take thousands of years. A third standard model
of quantum computation is the quantum cellular automata, a quantization of classical
cellular automata. They are capable of simulating quantum dynamical systems intractable
by classical means [57].
New algorithms and software are necessary to exploit the advantages of quantum
computing. A quantum algorithm in a narrow sense is an algorithm that exploits quantum
features such as entanglement or superposition, which cannot be ingredients of a classical
algorithm. The very first example of a quantum algorithm provably faster than its classical
counterpart was given by Deutsch in 1985 [
58
], but the most prominent examples with
practical usefulness are Shor’s algorithm for factoring numbers [
21
] and Grover’s algo-
rithm for searching unsorted databases [
59
]. However, there are dozens of other quantum
algorithms [
60
,
61
]. Quantum software comprises the assembling and orchestration of com-
puter instructions to whole programs that can be run on a quantum computer. Even new
high-level programming languages are being developed which especially help to express
the quantum algorithms (see https://en.wikipedia.org/wiki/Quantum_programming#
Quantum_programming_languages for a list of languages—accessed on 29 July 2021).
In recent years, some bibliometric studies have been published on QT. Tolcheev [
62
]
published a bibliometric study on QT including a very broad set of papers (by comprising
all papers that use “quantum” in their title, abstract, or keywords since the year 2000). In
contrast, the present study has a more focused view by including papers from specific
technology-relevant subfields. Tolcheev’s particular attention was directed to the assess-
ment of the publication output of Russian scientists concerning the main WoS Subject
Categories and the degree of international collaboration. Another study on QT by Chen
et al. [
63
] uses the field of quantum information as an application case for a new method.
This method focuses on the scientometric comparison of the Quantum Center of Excellence
of the Chinese Academy of Sciences with three other outstanding international research
units. The authors were interested in the internal team structure, collaborations, and
prospective development. Olijnyk [
64
] was interested in China’s involvement in the area of
quantum cryptography between 2001 and 2017 and witnessed China taking on a leading
role. Dhawan et al. [
65
] focused on the global publication output in quantum computing
research between 2007 and 2016 and reported results concerning the top contributing
countries very similar to ours (see Section 5. Discussion).
Seskir and Aydinoglu [
66
] followed an approach similar to the approach of this study.
They applied an elaborate search query to publications in the WoS until June 2019, informed

Quantum Rep. 2021,3553
by expert knowledge, but different from ours (cf. Section 3.3). Their handling of the topics
was not as detailed as ours, where we tried to suppress noisy terms, and therefore they
arrived at a coarser partitioning of subfields: quantum cryptography and communication,
quantum computing and information theory, and the physical realizations of the respective
concepts. Apart from the identification of a core set of QT 2.0 literature, they also tried
to identify the key players on the level of countries and institutions by analyzing their
collaboration patterns. They identified the same top 25 countries as we did (cf. Section 4.3).
3. Methods and Dataset
3.1. Data Sources
The bibliometric data used in our study are from three sources: (1) the online version
of the Web of Science (WoS) database provided by Clarivate Analytics (Philadelphia, PA,
USA), (2) the bibliometric in-house database of the Max Planck Society (MPG), developed
and maintained in cooperation with the Max Planck Digital Library (MPDL, Munich), and
(3) the bibliometric in-house database of the Competence Centre for Bibliometrics (CCB,
see: http://www.bibliometrie.info/ (accessed on 1 August 2021)). Both in-house databases
were derived from the Science Citation Index Expanded (SCI-E), Social Sciences Citation
Index (SSCI), Arts and Humanities Citation Index (AHCI), Conference Proceedings Citation
Index—Science (CPCI-S), and Conference Proceedings Citation Index—Social Science &
Humanities (CPCI-SSH) prepared by Clarivate Analytics. The availability of bibliometric
data and the syntax the WoS offers, to formulate the complex queries detailed in
Section 3.3
,
have both motivated our choice of database.
3.2. Search Procedure
The analyses considered publications of the document types “Article”, “Conference
Proceeding”, and “Review”. The results are based on 54,598 papers published between 1980
and 2018 in the field of QT 2.0. The search queries that we used for compiling the dataset
including the publications for the different subfields of QT 2.0 are listed and explained in
the following. For each of the different subfields in general, effects of earlier truncation,
usage of quotation marks, and different proximity operators were tested. We carefully
considered the different result sets. Our main goal was to have a sufficient recall and high
precision. The final publication set does not comprise all QT 2.0 publications, nor does it
exclude all irrelevant publications, as will be pointed out in the following. A completely
“valid” publication set is not achievable on such a scale. However, with our carefully
formulated WoS search query, we are confident that we captured most of the relevant
publications regarding QT 2.0 while including only very few irrelevant publications. In
particular, we excluded on purpose the very large literature related to quantum physics
and quantum chemistry that is not linked to the field of QT 2.0.
The searches were done on 25 May 2020 via the online version of WoS and yielded
54,848 publications starting in 1980 until the end of 2018. All WoS internal identifiers
(UTs)—except for 247—could be accessed via the in-house custom database of the MPG in
its version from December 2019.
3.3. Search Queries for Fields of QT 2.0
In the following, we explain the WoS search queries tailored for the different relevant
subfields within the four different fields of QT 2.0 to gain a maximum precision and a high
recall (which is not easily guaranteed). We compared result sets of searching in different
data fields (i.e., topic or title) and with different proximity operators (i.e., quoting search
terms or using the operators AND, NEAR, or SAME). Since we did not find a useful query
for title-only searches, our search queries use the topic field “ts” which comprises title,
abstract, and keywords. The use of proximity operators had to be done differently for each
QT 2.0 subfield. The search formulations are ordered by field. All queries are combined by
the OR operator in the WoS online database.

Quantum Rep. 2021,3554
3.3.1. Quantum Information Science and Quantum Technology in General (Q INFO)
(1)
Quantum information science
ts = (“quantum information*” OR “von Neumann mutual information” OR “quantum
mutual information” OR “quantum fisher information”)
This broad search yields a lot of hits, but successfully excludes non-relevant ones.
In quantum information theory, quantum mutual information, also called von Neumann
mutual information, is a quantum generalization of the Shannon mutual information and
measures the correlation between subsystems of a quantum state [
67
]. Quantum Fisher
information is the quantum analogue of the classical Fisher information of mathematical
statistics and determines the bound for measurement precision. Therefore, it is a matter of
choice whether to assign it to quantum information science or quantum metrology, sensing,
imaging, and control. Use of a narrow proximity operator, e.g., NEAR/1, would result in
many irrelevant publications from other fields that contain compound terms such as, e.g.,
“quantum chemical information” in the context of quantum chemistry or studies using
the “Quantum Geographic Information System” for regional localization of diseases or
geological events.
(2)
Quantum technology in general
ts = (quantum NEAR/2 technolog*)
We added this general search to the first basic and broader topic of quantum informa-
tion science. Only a few of the papers have the concept in their title, but in many of them
QT is explicitly envisaged as a field of application of the physical phenomena described.
The proximity operator is tuned to cover relevant compound terms such as “quantum
optical technologies” or “quantum key distribution technologies” and to exclude irrelevant
hits due to compound terms such as “quantum-inspired classical computing technology”
or “quantum-dot-based display technology”. More than a third of the results of QT contain
the concept of quantum information.
(3)
Quantum theory in connection with qubits
ts = (“quantum theory” SAME (qubit* OR “quantum bit*”))
Quantum theory, the theoretical basis of QT, is a very broad field. Therefore, we
decided to include only publications which contain “quantum theory” and “qubit*” or
“quantum bit*” in the same field of the topic.
3.3.2. Quantum Metrology, Sensing, Imaging, and Control (Q METR)
(1)
Quantum metrology
ts = ((quantum NEAR/10 metrology) OR (quantum NEAR/1 tomograph*) OR “atomic
clock*” OR “ion clock*” OR “quantum clock*” OR “quantum gravimeter*”)
The first proximity operator is also needed to retrieve titles such as “quantum-
enhanced metrology” and especially relevant, but wordier, mentions in the abstract or title
such as, e.g., “A study of quantum Hall devices with different working magnetic fields for
primary resistance metrology”. A greater distance between search terms would reduce the
precision too much. The second proximity operator is also needed to include papers that
contain phrases such as “quantum process tomography” or “quantum state tomography”.
The publication set regarding quantum clocks is a very special case that cannot be retrieved
sufficiently by general search terms. Therefore, the specific search terms “atomic clock*”
and “ion clock*” were included.
(2)
Quantum sensing
ts = ((Quantum NEAR/1 Sensing) OR (Quantum near/1 Sensor*))
Using a quoted search term for this subfield would exclude too many relevant publi-
cations. The chosen proximity operator yields desired results such as “quantum-enhanced
sensing”, “quantum plasmonic sensing”, or even “quanta image sensor”. (When using

Quantum Rep. 2021,3555
proximity operators both plural and singular forms are found.) Using a broader prox-
imity operator such as NEAR/2 would yield too many irrelevant hits such as “quantum
dot-based sensors” or “quantum cascade laser sensor”.
(3)
Quantum imaging
ts = ((“quantum imag*”) OR “ghost imag*”)
For quantum imaging, it was possible to capture most of the relevant publications us-
ing two quoted and truncated strings. Usage of the NEAR/1 operator would have yielded,
e.g., the fear of “images of ghosts” in psychiatric literature or “quantum dot imaging”. This
does not exploit QT 2.0 features but is widely used in biological and chemical research
because of the well-tunable emission spectra of quantum dots. Additionally, imaging
quantum effects in a broader sense are excluded.
(4)
Quantum control
ts = (“quantum control*” OR “control* of quantum” OR “control over quantum”
OR “quantum optimal control” OR “quantum state control” OR “control* quantum” OR
“control* the quantum” OR “quantum coherent control”)
For the subfield quantum control, we decided to use multiple quoted search terms
because “quantum NEAR/1 control” would have led to too many irrelevant hits due to
compound terms such as “quantum path control” in quantum chemistry.
3.3.3. Quantum Communication and Cryptography (Q COMM)
(1)
Quantum communication and networking
ts = (“quantum communication*” OR “quantum network*” OR “quantum optical
communication” OR “quantum state transmission*” OR ((“quantum memor*” OR “quan-
tum storage*”) NEAR/5 photon*) OR “quantum repeater*” OR “quantum internet” OR
(“quantum teleport*” AND (“qubit*” OR “quantum bit*” OR “entangle*”)))
The quoted strings “quantum communication*”, “quantum network*”, “quantum op-
tical communication”, and “quantum state transmission*” yield a rather accurate basis for
this subfield. However, network-related publications regarding optical storage are missing.
We included them by requiring that the term “photon*” appears within five words of the
search terms “quantum memory” and “quantum storage”. The optical storage is especially
important for quantum communication. The qualification of quantum teleportation (a
basic procedure in quantum communication) with qubit or entanglement narrows the focus
down to technological applications as opposed to theoretical or experimental work.
(2)
Quantum cryptography
ts = (“quantum crypto*” OR pqcrypto* OR “quantum key distribution” OR “quantum
encrypt*” OR ((“quantum secur*” OR “quantum secre*”) NOT (“quantum secreted” OR
“quantum secretion”)))
The quoted search term “quantum crypto*” provides a good basis for this subfield.
However, additional search terms, e.g., “pqcrypto*” (for post-quantum cryptography),
“quantum key distribution”, and “quantum encrypt*” were necessary for obtaining an
acceptable recall. Further relevant publications containing “quantum secur*” or “quan-
tum secre*” were included while irrelevant publications, mainly from biology, with the
compound terms “quantum secreted” or “quantum secretion” were excluded.
3.3.4. Quantum Computing (Q COMP)
(1)
Quantum computing
ts = (“quantum comput*” OR “quantum supremacy” OR “quantum error correction”
OR “quantum annealer” OR (quantum NEAR/2 (automata OR automaton)) OR “quantum
clon* machine*”)
The term “quantum annealer” in the search formulation points to more actual technical
realizations than the more abstract term “quantum annealing”. The proximity operator

Quantum Rep. 2021,3556
with automata is so chosen as to include, e.g., the generalized concept of quantum-enabled
finite automata and of quantum cellular automata as well as the “Cellular Automaton
Interpretation of Quantum Mechanics” [
68
] or “quantum evolutionary cellular automata”
or “cellular automaton, based on quantum states”. The capabilities of quantum cloning
machines [
69
] are important for the processing of qubits in quantum computers. Therefore,
we added the last search term to cover such literature, too.
(2)
Quantum hardware systems
ts = (“quantum hardware” OR “quantum device*” OR “quantum circuit” OR “quan-
tum processor*” OR “quantum register*”)
In the case of this subfield, we managed to progress with a combination of general
and specific quoted search terms.
(3)
Quantum simulation
ts = (“quantum simulat*” AND (qubit* OR “quantum bit*” OR “quantum comput*”)
OR “quantum simulator*”) OR (ts = “quantum simulat*” AND wc = (quantum science
technology OR computer science theory methods))
The term “quantum simulation” often means the simulation of quantum systems
performed by classical means. Therefore, the term is widely used in various large fields
such as, e.g., quantum chemistry. Thus, we needed to restrict it somehow. We decided to
use search terms and two relevant WoS subject categories emphasizing the quantum nature
of the simulation itself.
(4)
Quantum algorithms
ts = “quantum algorithm*”
We decided to capture the subfield of quantum algorithms with a single quoted search
term. Broader queries, e.g., using NEAR/1, would also capture irrelevant publications due
to compound terms such as “quantum-inspired algorithm”.
(5)
Quantum software
ts = (“quantum software” OR “quantum cod*” OR “quantum program*”)
In the case of “quantum software”, too many irrelevant publications would be in-
cluded if a broader search query was used. There is no significant overlap with quantum
algorithms, but one third of the results are also found in quantum computing.
3.4. Publication Output and Citation Impact Indicators
We analyzed the number of papers (full counting) broken down by year, field of
QT 2.0, and country. Citation impact analyses are based on time- and field-normalized
indicators. We focused on the share of papers belonging to the 10% most frequently cited
papers in the corresponding publication year, document type, and subject area. In case of
more than one paper with a citation count at the required threshold of 10%, these papers are
assigned fractionally to the top 10% publication set. This procedure ensures that there are
exactly 10% top 10% papers in each subject area [
70
]. The top 10% indicator is a standard
field-normalized indicator in bibliometrics [
71
]. The citation window relates to the period
from publication until the end of 2018.
3.5. Mapping of Research Topics
Besides indicators such as publication output and citation count as measures of
scientific activity and impact, techniques of text mining are also used in bibliometric
studies. The analysis of keywords in a corpus of publications can identify important
research topics and reveal their change and development over time. This analysis can be
managed with the software VOSviewer [
72
]. The software produces networks based on
bibliographic coupling. The nodes in these networks are keywords, their size signifies the
number of corresponding publications, and the distance between nodes is proportional to
their relatedness regarding cited references. Keywords of papers citing similar literature are

Quantum Rep. 2021,3557
located closer to each other. The nodes are divided into classes of similarity, displayed by
clusters of different colors. The network can be controlled by some adjustable parameters
such as minimal cluster size or resolution.
4. Results
In this study, we are interested in answering several research questions. (i) How did
QT 2.0 and its subfields grow overall and compared to each other from 1980 to 2018? (ii)
How did their topical foci change over time? (iii) What are the top contributing countries
in QT 2.0 and its subfields since 2000? (iv) How are research topics and author countries
related?
4.1. Respective Shares of Fields
We retrieved 54,598 publications using the search queries. Table 1shows the number
of papers in the four fields and their percentages of the total number of publications. We
applied whole counting and many papers were assigned to more than one field. Therefore,
the percentages add up to more than 100% and the percentage of papers belonging to only
one field is only about 84%. A graph of the mutual overlap of the four fields is given in
Figure 1.
Table 1. Number and percentage of papers in four fields of QT 2.0.
Field of QT 2.0 Number of Papers
Percentage of the
Number of Distinct
Papers
Number and
Percentage of
One-Field-Only
Papers
Q INFO 16,300 29.85% 9706 (59.55%)
Q METR 12,531 22.95% 9766 (77.93%)
Q COMM 13,985 25.61% 9809 (70.14%)
Q COMP 21,786 39.90% 16,545 (75.94%)
Sum of all fields - 118.77% 45,826 (83.93%)
Figure 1. Venn diagram of mutual overlaps of the four fields of QT 2.0.
4.2. Overall Growth and Growth in Terms of Fields
Figure 2shows the annual publication numbers for QT 2.0 and its four fields for
the period from 1990 to 2018. The numbers are collected in Appendix A.2 in Table A1.

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The annual numbers of publications on QT 2.0 before 1990 never exceeded a dozen per
year—most of them about Q METR. This can be explained by the efforts and achievements
in manipulation, controlling, and measuring of single quantum systems. The first decade
is excluded from the following comparative analyses because the number of documents is
small and their thematic focus is nearly exclusively on Q METR.
Figure 2.
Annual numbers of publications of QT 2.0 and its four fields between 1990 and 2018 (in earlier years the
annual numbers of all fields together never exceed 12) compared to the number of articles, reviews, and conference
proceedings in the whole WoS. The numbers for QT 2.0 and the whole WoS are scaled by factors 2 and 1000, respectively, for
better comparison.
An exponential growth of publications per year occurred between 1990 and 2000,
mainly caused by Q METR and Q COMP (see Figure 2and Table A1). Additionally, Q INFO
emerged as a significant research field. The year 1994 is seen by Dowling and Milburn [
16
]
as the birth year of the quantum information revolution. This year is associated with a
significant experimental step towards practical quantum key distribution [
20
] and the
publication of Shor’s quantum algorithm [
21
] with an exponentially better performance
than the then available classical algorithms for integer factorization. Together with the
introduction of teleportation [
38
], they give ample reason for the significant increases
in Q COMP from 1994 to 1995 (more than doubling and overtaking Q METR) and of
Q INFO and Q COMM from 1995 to 1996 (nearly doubling and more than doubling,
respectively). Q INFO and Q COMM continue in nearly linear growth. In the first decade
of the century, Q COMP is clearly the most strongly represented field, with about twice
as many papers as each of the fields Q INFO and Q COMM. Q INFO and Q COMM have
strong interconnections, coming to the fore especially in this decade. The remarkable peak
in 2009 is probably due to the online demonstration by D-Wave Systems of their quantum
simulator at the Supercomputer Conference SC’07 [
73
]. This demonstration sparked hectic
activity in the field but also sceptical reactions which probably are responsible for the
decline in 2010 and 2011 [
62
]. The final years are characterized by a steady linear growth
and nearly constant shares of the four fields.
Figure 3offers a different view on the development of the four fields between 1990
and 2018 by displaying their respective percentages of the total counts of the papers of all
four fields—partly counted multiple times due to overlaps of the four fields (see Figure 1).

Quantum Rep. 2021,3559
During the first years, Q METR, is clearly dominating, but joined by Q COMP fairly soon
thereafter. From about the year 2000 onwards, Q COMP and the strongly related Q INFO
together have an annual share of about 60% of all QT 2.0 papers.
Figure 3. Annual percentages of the four different fields of QT 2.0 from 1990 to 2018.
To study the time evolution, we divide the period into three phases: (1) from 1980
to 1999; (2) from 2000 to 2011; and (3) from 2012 to 2018. The numbers of papers in the
last two periods are in the same order of magnitude, with 24,322 and 28,132. With 2144
papers, the first (pioneering) phase has less than a 10th of the number of papers in the other
periods. This division into three periods seems suitable to us for presenting the output of
publications and the mapping of research topics.
A measure of the growth of the research fields during the three periods is the doubling
time (see Table 2). The four fields have very similar values to the total QT 2.0. The very
short doubling time of two years is characteristic for the first period until 1999, slowing
down to four years in the second period, and slowing down to seven years during the most
recent period. The last doubling time is comparable to the 5–6 years which Haunschild,
Bornmann, and Marx [
10
] found for the climate change literature until 2014. However, this
time is significantly shorter than the 12–13 years for the overall growth of the WoS records.
Bornmann and Mutz [
74
] calculated an even longer doubling time of nearly twenty-four
(24) years for WoS in the period from 1980 to 2012 by applying a non-linear segmented
regression analysis. During the twenty (20) years from 1991 to 2010, the annual number
of publications grew by a factor of 42, compared to a factor of ten for the climate change
corpus and to a factor of about two for the whole WoS.

Quantum Rep. 2021,3560
Table 2.
Doubling times in years for all QT 2.0 papers and four fields for the years 1980–1999,
1980–2011, and 1980–2018 compared to the whole WoS publication record.
Years All QT Papers Q INFO Q METR Q COMM Q COMP WoS
1980–1999 2–3 1–2 3–4 1–2 1–2 7–8
1980–2011 4–5 4–5 4–5 4–5 5–6 11–12
1980–2018 6–7 5–6 6–7 6–7 7–8 12–13
4.3. Contributing Countries
Many countries are contributing research on QT 2.0 by collaborating with each other.
Table 3lists the 25 top publishing countries with at least 500 papers published between
2000 and 2018 in QT 2.0. Multiple authors of a single paper from the same country are
counted only once, but multi-author papers are fully assigned to several countries so that
the total sum exceeds the number of papers in our dataset. The 25 countries in the table
include more than 90% of the authors, the USA and China alone cover one third, and two
thirds are covered by the first eight countries. The last column shows the corresponding
shares of the countries in the whole WoS in the same period.
Table 3.
The 25 top publishing countries with at least 500 papers in QT 2.0 for the years 2000 to 2018.
The table shows the number of papers, their percentage, and the national share of the whole WoS.
Country Country Code #QT 2.0 %QT 2.0 %WoS
USA us 13,489 18.59 24.29
China cn 12,110 16.69 9.79
Germany de 5291 7.29 5.59
UK gb 4639 6.39 6.57
Japan jp 3982 5.49 4.76
Canada ca 3044 4.20 3.45
Italy it 2894 3.99 3.32
France fr 2558 3.53 3.82
Australia au 2413 3.33 2.59
India in 1711 2.36 2.51
Russian
Federation ru 1687 2.32 1.68
Spain es 1580 2.18 2.64
Switzerland ch 1470 2.03 1.39
Austria at 1213 1.67 0.76
Singapore sg 1074 1.48 0.56
Brazil br 1031 1.42 1.70
South Korea kr 998 1.38 2.21
Poland pl 991 1.37 1.16
Netherlands nl 945 1.30 1.90
Israel il 813 1.12 0.72
Iran ir 719 0.99 0.91
Denmark dk 623 0.86 0.77
Sweden se 585 0.81 1.24
Taiwan tw 535 0.74 1.25
Czech Republic cz 530 0.73 0.62
Analogous evaluations have been made for the four fields of QT 2.0 separately (the
results are not shown). They give a similar picture with nearly the same countries domi-
nating. The same 22 countries are among the top 25 countries in QT 2.0 as a whole and

Quantum Rep. 2021,3561
all four fields, even when we focus on the top 10% most cited papers in QT 2.0 and its
four fields. For both cases (either all papers or only top 10% papers), we calculated two
numbers, indicative of the relative publication output of these countries, measured against
an “expectation value” based on the countries’ overall WoS shares. The first number is the
difference in the last two columns in Table 3(%QT-%WoS); positive and negative signs
indicate more or less publication activity than expected, repectively. The second one is the
corresponding quotient (%QT/%WoS). The quotient is identical to the so-called activity
index (AI), introduced by Frame [
75
], which in turn is a variant of the revealed comparative
advantage (RCA) used in economics [
76
]. AIs greater than 1.0 indicate national publication
outputs higher than expected (from the whole WoS). Both indicators are presented as radar
charts in Figure 4. For each indicator, there is one plot including all papers (on the left) and
one including only the top 10% papers (on the right). In each radar chart, the 22 common
countries are denoted by their respective country codes, starting at the top with the country
with the most publications in QT 2.0 (USA) and descending clockwise. In each radar chart,
the dividing values between under and over achievement are marked by a gray dashed
line at the value 0 for the difference and 1 for the AI.
Figure 4.
Radar charts of the differences (upper graphs) and quotients (activity indices, lower graphs)
of the national shares of papers in QT 2.0 and its four fields from 2000 to 2018. On the left side,
all papers are included; on the right side, only the top 10% most cited papers in the time period
2000–2016 are included. The 22 countries, which are among the top 25 in all fields and the whole QT
2.0, are denoted by their country codes, and ordered clockwise in descending order of the number of
publications. The gray dashed lines at 0 and 1 indicate the expected output of the country.
The most striking insight from these figures is the very different assessment of the two
leading countries with very similar output, the USA and China, in comparison with the
whole WoS: while the USA is less active in QT 2.0 than in other WoS-covered research fields
(QT 2.0: difference =
−
5.7%, AI = 0.77), China is much more active in QT 2.0 than in other
fields (QT 2.0: difference = +6.9%, AI = 1.71). The difference is most pronounced in the field
Q COMM (difference = +15.2%, AI = 2.5). With respect to the top 10% papers, the strong

Quantum Rep. 2021,3562
research focus of China on QT 2.0 is dampened considerably (QT 2.0:
difference = +1.9%,
AI = 1.26; Q COMM: difference = +7.4%, AI = 2.0). Germany has climbed from the third to
the second rank in number of publications. It also shows a higher share of research activity
in QT 2.0 than China (QT 2.0: difference = +2.5%, AI = 1.39). When only highly cited papers
are considered, Germany has comparable strengths in all four fields. Figure 4shows that
Austria, Singapore, and Switzerland contributed rather unexpected high shares of QT 2.0
research in comparison with their research activities as a whole. Austria has an overall AI
of above 2 in QT 2.0 and Q COMP, and of nearly 3 in Q COMM and Q INFO. The AIs even
exceeded this, if focusing on the top 10% papers, leading to values of more than 4. These
high AIs can be explained by the high activities of the groups in Vienna and Innsbruck
concerning quantum teleportation. Singapore has AI values of nearly 3 in the three fields
Q INFO, Q COMM, and Q COMP. Switzerland’s AI value of about 1.6 is mainly caused by
a high value of 2.4 in Q COMM.
4.4. Visualization of the Time Evolution of Research Topics
For the various time periods, we have created keyword maps based on author key-
words and keywords plus assigned by Clarivate Analytics to papers. Usually, we pre-
fer to use author keywords, but in the oldest period there is only a very small percent-
age of papers with author keywords. The number of papers with either author key-
words or keywords plus amounts to about 70% (see Table 4). A common thesaurus
file (https://s.gwdg.de/4DDxsp (accessed on 1 August 2021)) was used to unify singu-
lar/plural forms of words and synonyms as detailed in Table A2 in Appendix A.3. The
minimal number of occurrences of a keyword is chosen such that about 100 keywords
are displayed for each period. We chose default values as VOSviewer parameters for
clustering. For the minimal cluster size, however, we used a value of 5 which resulted in
a well-interpretable network. All VOSviewer maps are provided to the reader as online
versions [
77
] via URLs. They can be used for an interactive inspection, e.g., by zooming in
on the clusters.
Table 4. Occurrences of author keywords and keywords plus in three periods.
Time Period Total Numberof
Papers
Occurrences of
Author
Keywords
(Percentage)
Occurrences of
Keywords Plus
(Percentage)
Occurrences of
Either Type of
Keywords
(Percentage)
1980–1999 2144 445 (20.8%) 1350 (63.0%) 1500 (70.0%)
2000–2011 24,322 10,197 (41.9%) 19,549 (80.4%) 21,888 (90.0%)
2012–2018 28,132 13,766 (48.9%) 22,902 (81.4%) 26,033 (92.5%)
Figure 5displays an overall co-occurrence map of 100 keywords occurring at least
298 times for the period from 1980 to 2018. Maps with about 100 keywords usually are a
good compromise between maintaining readability of the map and displaying most of the
content. In the figure, the four fields of QT 2.0 are nicely discernible by the keywords in four
clusters, whose colors are kept consistent in all networks. In the following explanations of
Figures 5–9, those keywords that are also found in the respective co-occurrence maps are
written in italics:
•
Red (Q METR): The manipulation of single atoms, molecules, and even (electron) spins
as in quantum dots and the quantum control using light fields of coherence (lasers)
lead to the realization of single qubits and of very high-precision quantum clocks.
•
Brown (Q COMP): Quantum computing and computers build on quantum circuits
with logic gates realized as trapped ions, anyons, or in NMR devices. On this hardware,
quantum algorithms have been implemented that are much in need of quantum
error correction.

Quantum Rep. 2021,3563
•
Blue (Q INFO): Quantum entanglement of states is a major subject of quantum infor-
mation science, also investigating entropy and channel capacity in a generalization of
Shannon’s information theory.
•
Green (Q COMM): Quantum communication is built upon the quantum teleportation
of pairs of entangled states, often realized by single photons, as a basis for quantum
cryptography and quantum key distribution.
Figure 5.
Co-occurrence map of the top 100 keywords (author keywords and keywords plus from 1980 to 2018) with four
topical clusters, using the VOSviewer parameters resolution = 1.0 and minimal cluster size = 5. (The two biggest unnamed
green nodes belong to the keywords communication and key distribution, respectively.) For better readability, in compound
keywords quantum is abbreviated to q. Readers interested in an in-depth analysis of our publication set can use VOSviewer
interactively and zoom in on the clusters. An online version is provided at https://s.gwdg.de/1Bg9EB (accessed on 1
August 2021) (cluster colors probably differ).
Figure 6displays the co-occurrence map of 95 keywords occurring at least 12 times in
the period from 1980 to 1999. From the five clusters, two can be associated with the field
Q METR: the red cluster contains keywords such as spectroscopy, atom, quantum clock,
frequency standard, and gravity; the keywords in the violet cluster point in the direction
of quantum control in molecular dynamics, using, e.g., laser pulses. This field dominated
the 1980s. The keyword quantum metrology itself does not occur before 2001, coined as a
named concept only in retrospect. Since 1981, five publications contained the term in their
title but were cited only two times in the first period. The green cluster is focused on Q
COMM. The most frequent keywords quantum cryptography, (quantum) communication,
quantum entanglement, quantum teleportation, and bell theorem point to pioneering
experiments [
8
,
20
] as well as the proposal and realization of quantum communication

Quantum Rep. 2021,3564
protocols and quantum teleportation in the 1980s and 1990s [
33
,
38
,
40
]. The brown cluster
belongs to Q COMP. The most prominent keywords are quantum computing, logic gate,
quantum error correction, and algorithm. The yellow cluster contains keywords that relate
to quantum hardware and methods of its realization. Keywords are quantum device,
quantum dot, quantum cellular automata, and gaas (meaning the semiconductor GaAs,
frequently used in creating quantum dot cellular automata). The keywords in the brown
and yellow clusters indicate the efforts to realize quantum gates and circuits using a variety
of techniques in the earlier years of the period. There was hope for efficient quantum
algorithms in later years, triggered by Shor’s algorithm [
21
]. The field Q INFO is not
explicitly visible (apart from the keyword information in the green cluster). However, it
is implicitly present in the strong connections between the green and brown clusters (Q
COMM and Q COMP) via keywords such as decoherence, quantum entanglement, and
quantum error correction.
Figure 6.
Co-occurrence map of the top 95 keywords (author keywords and keywords plus from 1980 to 1999) with five
topical clusters, using the VOSviewer parameters resolution = 1.0 and minimal cluster size = 5. (The unnamed big brown
node belongs to the keyword computer.) For better readability, in compound keywords the term quantum is abbreviated
to q. Readers interested in an in-depth analysis of our publication set can use VOSviewer interactively and zoom in on
the clusters. An online version is provided at https://s.gwdg.de/SfpspR (accessed on 1 August 2021) (cluster colors
probably differ).
Figure 7displays the co-occurrence map of 98 keywords occurring at least 134 times
in the period from 2000 to 2011. About 60% of the keywords are the same as in Figure 6. In
Figure 7, the keywords are distributed over three clusters (instead of five in Figure 6). The
red cluster in Figure 7can be interpreted as a merging of the two Q METR clusters (red and
violet) from Figure 6. Furthermore, research on quantum hardware is no longer separated

Quantum Rep. 2021,3565
into its own cluster but is incorporated in the brown cluster (Q COMP). For example, the
keywords nmr and trapped-ion point to different approaches for realizing quantum circuits.
The green cluster includes the keywords quantum entanglement, quantum cryptography,
quantum teleportation, and quantum information. The keywords indicate that the cluster
comprises Q COMM and much of Q INFO together. Moreover, quantum key distribution
and questions of security come to the fore in this decade with the first secure money
transaction using QT 2.0 [42].
Figure 7.
Co-occurrence map of the top 98 keywords (author keywords and keywords plus from 2000 to 2011) with three
topical clusters, using the VOSviewer parameters resolution = 1.0 and minimal cluster size = 5. The unnamed big green
node belongs to the keyword quantum entanglement. For better readability, in compound keywords the term quantum is
abbreviated to q. Readers interested in an in-depth analysis of our publication set can use VOSviewer interactively and
zoom in on the clusters. An online version is provided at https://s.gwdg.de/DvXJnY (accessed on 1 August 2021) (cluster
colors probably differ).
Figure 8displays the co-occurrence map of 100 keywords occurring at least 168 times
each between 2012 and 2018. Eighty percent of the keywords are the same as in Figure 7,
but they are distributed over five distinct clusters in Figure 8. The graph for this period
is similar to the overall graph, but with an additional orange cluster. The red cluster can
still be assigned to Q METR, which now also contains the keyword quantum metrology.
The blue cluster located between the brown (Q COMP), green (Q COMM), and red cluster
(Q METR) contains keywords such as quantum entanglement, quantum information, and
entropy. This warrants an assignment of the cluster to Q INFO as a field at the interface
of the other three fields of QT 2.0. The new orange cluster located between the red and

Quantum Rep. 2021,3566
green clusters constitutes an interface area between the two fields Q METR and Q COMM:
quantum optics and quantum memory using photons.
Figure 8.
Co-occurrence map of the top 100 keywords (author keywords and keywords plus from 2012 to 2018) with five
topical clusters, using the VOSviewer parameters resolution = 1.0 and minimal cluster size = 5. For better readability, in
compound keywords the term quantum is abbreviated to q. Readers interested in an in-depth analysis of our publication set
can use VOSviewer interactively and zoom in on the clusters. An online version is provided at https://s.gwdg.de/N5TmUa
(accessed on 1 August 2021) (cluster colors probably differ).
When we inspect the topic maps for the three periods, we see continuity and persis-
tence of clusters as well as a change in the focus and occurrence of keywords. From the first
to the second period, only 60 out of 99 keywords in the maps are identical. From 1980 to
1999 (https://s.gwdg.de/WOaY1F (accessed on 1 August 2021)), the focus had been on the
preparation, manipulation, and control of single quantum systems at the atomic scale and
the pioneering work on building materials, devices, and sensors for quantum metrology.
From the year 2000 to 2011 (https://s.gwdg.de/y668Y5 (accessed on 1 August 2021)), the
focus had, on the one hand, switched to the advanced design of hardware components
for real quantum computers and the development of algorithms utilizing quantum prop-
erties. On the other hand, the exploitation of quantum effects such as entanglement for
secure communication using quantum key distribution had become prominent, favorably
utilizing quantum optics of single photons. From the second to the third period, i.e., the
year 2012 to 2018 (https://s.gwdg.de/nnrm9Y (accessed on 1 August 2021)), nearly 80%
of the keywords remain the same (78 out of 99 and 101 keywords, respectively). There
are only slight changes in the main direction of research, but some keywords moved into
the new clusters of Q INFO and quantum optics. For example, memory and storage are
located in the clusters Q COMP and Q METR in the second period; both keywords are
connected with quantum optics in the third period. This connection probably exists because

Quantum Rep. 2021,3567
of their importance for optical quantum communication networks. The keyword quantum
simulation appears only on the third map in the Q COMP cluster. This coincides with the
enlarged efforts to build a quantum simulator as the fulfillment of Feynman’s vision of a
quantum computer [53,54].
4.5. Visualization of the Geographical Distribution of Research Topics
Figure 9shows a combination of the approaches taken in the previous two sections.
For the period from the year 1980 to 2018, we have produced a co-occurrence map of
countries (denoted by their two-letter country code with a prefixed “@”) with at least
400 occurrences (multiple co-authorships of the same country on a paper are counted
only once) as well as a map of keywords (author keywords and keywords plus assigned
by Clarivate Analytics) with at least 300 occurrences. These thresholds lead to the top
25 countries in Table 3and to 104 keywords, sorted into five topical clusters (by using the
VOSviewer parameters resolution = 1.1 and minimal cluster size = 5). Four clusters in the
figure can be assigned to the four fields of QT 2.0. The fifth cluster comprises quantum
optics, which is also visible in the keyword map of the most recent period (orange cluster in
Figure 8). This last cluster does not contain any of the 25 top countries, but there are many
connections to keywords and countries of the neighboring clusters Q METR, Q COMM,
and Q INFO. These connections confirm its interface function that was also detected for
the third time period in Figure 8. About ten countries are assigned to the clusters Q METR
and Q INFO, respectively. Three countries are assigned to Q COMP and Q COMM each. In
case of Q COMP, India (@in) and Iran (@ir, just above node @in) are mainly connected to
the design of logic gates and circuits.
Figure 9.
Co-occurrence map of (1) the top 25 countries of Table 3(denoted by their two-letter country code with a prefixed
“@”) with at least 400 occurrences and (2) the top 104 keywords (author keywords and keywords plus) with at least
300 occurrences in the total publication set from 1980 to 2018. The map shows five topical clusters, using the VOSviewer
parameters resolution = 1.1 and minimal cluster size = 5. For better readability, in compound keywords the term quantum is
abbreviated to Q. An online version is provided at https://s.gwdg.de/IULOc3 (accessed on 1 August 2021) (cluster colors
probably differ).

Quantum Rep. 2021,3568
We now compare the assessment of the countries in the radar charts for all QT 2.0 pa-
pers in Figure 4with their placement and connections in the co-occurrence map in
Figure 9
.
The large node of China (@cn) in the green cluster (Q COMM) mirrors the dominance
of China with respect to the total number of papers and AI. Germany (@de) with the
third highest publication output and the highest values in Q METR in the radar charts is
consequently located prominently in the red cluster. Germany has significant contributions
to quantum optics and is connected to some other countries in the blue cluster (Q INFO).
Q INFO is the field of Germany’s second highest AI. Germany’s connections to two other
countries in the red cluster may have contributed to their noticeably high AIs of about 2,
which is in contrast to their small share of all QT 2.0 publications: Russia (@ru) with nearly
2.5% also has contributions to quantum optics and to the green cluster of Q COMM with
Russia’s second highest AI; Israel (@il) with just over 1% has the strongest connection to
quantum control and the USA (@us).
We would like to emphasize two other countries. These countries have, with 2.5%,
a small share of all QT 2.0 papers, but a high AI of about 3 in Q INFO and Q COMM.
Singapore (@sg, the unnamed blue node left below “Quantum”) has an especially high
AI of about 3 in Q INFO and Q COMM. In the map, consequently, it can be found in
the blue cluster of Q INFO connected with quantum entanglement and information and
with the UK (@gb). Singapore is additionally connected with the green cluster of Q
COMM and its major contributor China. Austria’s (@at) activities, especially in Innsbruck
and Vienna, are mirrored by its placement in the blue cluster Q INFO which is strongly
connected to quantum entanglement. It is also connected to the green Q COMM keywords
communication and pairs of photons (quantum optics, orange cluster).
5. Discussion
This bibliometric study on QT 2.0 identified four main subject fields, namely Q INFO,
Q METR, Q COMM, and Q COMP. For these four fields, we analyzed their respective
share, their respective growth in the QT 2.0 publication set compared to one another and
to the overall growth of QT 2.0, and the main contributing countries by comparing their
actual to their expected publication output based on the countries’ overall WoS shares. We
provided insight into the time evolution and geographical distribution of specific research
topics through several topic maps. We presented visualizations of the co-occurrence of
keywords during the whole period plus the three distinctive partial periods 1980–1999 (the
pioneering years), 2000–2011 (the exploration years), and 2012–2018 (the maturing years),
as well as of keywords and countries combined.
Of the 54,598 publications in our dataset, the four fields have shares from about one
fifth (Q METR) to two fifths (Q COMP) (see Table 1). In the first decade considered here,
less than 100 publications appeared, most of them in the field Q METR with its pioneering
works on preparing and controlling single quantum systems. During the second decade,
the 1990s, Q COMP joined Q METR in driving the exponential growth, leading to the
ongoing dominance of Q COMP in the new millennium (see Figure 2). Between 1980 and
1999, the doubling time of QT 2.0 was between 2 and 3 years as opposed to the doubling
time of the whole WoS of 7 to 8 years. During the periods until 2011 and until 2018,
respectively, the doubling times still were about half as long as in the whole WoS, with
4 to 5 years and 6 to 7 years, respectively (see Table 2). Tolcheev [
62
] found for a much
broader publication set of all publications containing “quantum” in their title, abstract, or
keywords and for the top 15 countries a doubling time of over 17 years from the year 2000
to 2016. In the most recent decade, QT 2.0 therefore seems to be a very active research area
that is steadily evolving at a rapid pace.
We also analyzed the main contributing countries to QT 2.0. We focused on a time
period with a substantial annual number of papers from the year 1990 until 2018. We
looked at the top 25 contributing countries in more detail and compared their publication
output in QT 2.0 and its four fields to the expected output from the whole WoS (see
Figure 4
). Singling out Q COMP, the top ten contributing countries are the same as in

Quantum Rep. 2021,3569
Dhawan, Gupta, and Bhusan [
65
], even if their less detailed search retrieves only a small
part of the field. We visualized the geographical distribution of research topics with a
co-occurrence map of countries and keywords in Figure 9. The main result is the sharp
contrast of the USA and China, which are the greatest contributors to QT 2.0. The USA
shows a much smaller contribution to QT 2.0 than could be expected from their otherwise
leading role in science. China has a far overproportionate contribution, especially in the
field of Q COMM—corroborated by its hub-like function in the topical map and confirming
the findings of Olijnyk [
64
]. Germany can be found on the third rank of contributors with
a much higher than expected share of QT 2.0 publications. By focusing on highly cited
publications, China’s share and AI are significantly diminished. In the high-impact range,
Germany goes up by one rank to the second place, contributing substantially in each of the
four fields, but notably in Q METR. This result is confirmed by the country’s location in the
corresponding cluster in the topic map.
For the small countries Austria and Switzerland, our current study finds a very high
AI, notably in the field Q COMM. The other small country with an extremely high AI,
Singapore, scores especially high in Q INFO and Q COMM. In the topic map, the blue
cluster of Q INFO has stronger connections to the keywords (quantum) computation and
information as well as entanglement.
The other two noticeable countries with high AI values, especially in Q METR, are
Russia (about 2) and Israel (2 to 2.5). In the topic map, Russian research relates to quantum
optics and keywords of the Q INFO and the Q COMM clusters as well as to the countries
Germany and the USA. This seems to be in accordance with a recent collection of papers on
“Quantum technologies in Russia” in the journal Quantum Electronics [
78
]. Here, research
activities of the recent past and the prospective future focus on the development of optical
quantum memory, of single–photon light sources, and of magnetometers based on NV
centers in diamond. Congruently, Fedorov et al. [
79
] list as main focal topics quantum
communication as well as quantum metrology and sensing, besides quantum computing
and simulation. These topics are supposed to receive a development boost by a recent
huge governmental funding plan [
80
]. The strong collaboration of Russian scientists with
scientists from Germany and the USA which we found in this study agrees with the
findings of Tolcheev [
62
]. Israel has a strong connection to quantum control, spin, and
the USA.
Our findings about the top publishing countries in QT 2.0 agree with national funding
initiatives. The journal Quantum Science and Technology reported in 2019 that ten of the
eleven top countries in Table 3had launched high-budget initiatives in order to consolidate
and substantially enhance their efforts and achievements in QT 2.0: the USA [
81
], China [
82
],
Japan [
83
], Australia [
84
], Canada [
85
], the Russian Federation [
79
], the European Union—
represented by some of its member states [86]—and the UK [87].
In this study, we also investigated the time evolution of research topics, visualized by
co-occurrence maps for different time periods. The map for the period from the year 1980
to 2018 shows clearly distinguishable clusters for the four QT 2.0 fields. The maps for the
three partial periods reveal changes in the focal areas over time. The years 1980 to 1999
were the pioneering years with breakthroughs in the manipulation and measurement of
single quantum systems, the design of quantum logic gates, first quantum algorithms, and
the first quantum teleportation. The years 2000 to 2011 were characterized by an emphasis
on security issues in quantum communication and multiple approaches to building the
first quantum computers. The period from the year 2012 to 2018 displays nearly the same
keywords as the previous period, indicating a maturing of QT 2.0 and a steady work on
improving promising approaches.
This study has focused on QT 2.0 and its four fields in their mutual relation and
development over time, and their occurrence in the main contributing countries as well as
the geographical distribution and the time development of research topics. As many other
similarly designed bibliometric studies, this study has some limitations. (1) The precision
of the search queries is affected by ambivalent meanings of terms that are not qualified by

Quantum Rep. 2021,3570
the term quantum, such as information, computing, etc. (2) For the topic maps, we used
mixed keyword types in order to get a reasonable coverage. During the first two decades,
however, the share of papers with any keyword is just above 70%. (3) The term “quantum
cellular automata” (QCA) is ambiguous. It might mean the implementation of classical
cellular automata on systems of quantum dots as a replacement for classical computation
using CMOS technology [
88
]. However, it may also denote an abstract model of cellular
automata performing true quantum computations, initially proposed by Feynman [
51
]. In
order to better differentiate between the two meanings, some authors refer to the former as
“quantum dot cellular automata”. There are about 1500 hits in our dataset for this term, but
the two meanings are not clearly distinguishable. Thus, we accepted a substantial number
of false positives to not miss the quantum concept. (4) One of the reviewers pointed out
as a limitation of our study the poorer coverage of computer science in the WoS database,
causing a systematic underestimation of the contribution of computer scientists to QT
2.0, especially to Q COMP and Q COMM. This is a well-known disadvantage, mainly
caused by less coverage of conference papers which are an important publication channel
in computer science. The seminal (conference) paper of quantum cryptography [
33
] is
therefore not contained in our dataset. In our study, with a high degree of aggregation,
this circumstance does not seem to distort the overall picture of the long dominance of Q
COMP and Q COMM, but more detailed bibliometric studies of quantum computation or
quantum communication should consider the use of databases with a higher coverage of
computer science.
Future studies could focus on the further development of QT 2.0 research in the fea-
tured countries, such as the USA, China, Germany, Austria, Singapore, and other countries
that now put great efforts and financial means into quantum science and technology. The
competition between China and the USA and the discrepancy in their expected activity we
reported in Section 4.3 are especially worthy of further attention. A related question is the
transfer of QT 2.0 research into the area of (commercial) applications. This would require
the use of another database that relates to, e.g., patents. From the viewpoint of research
topics, the further growth of a field like quantum optics that came up more clearly in the
maturing period of QT 2.0 could warrant a closer look.
Author Contributions:
Conceptualization: T.S., R.H., L.B., C.E. Data curation: T.S., R.H., L.B., C.E.
Formal analysis: T.S., R.H. Investigation: T.S., R.H. Methodology: T.S., R.H. Validation: T.S., R.H.
Visualization: T.S., R.H. Writing—original draft: T.S., R.H. Writing—review and editing: T.S., R.H.,
L.B., C.E. All authors have read and agreed to the published version of the manuscript.
Funding:
The Competence Centre for Bibliometrics is supported by the German Federal Ministry for
Education and Research (BMBF) within the framework of the funding priority “Science and Higher
Education Research” (Grant No: 01PQ17001).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement:
Access to the data used in this paper requires a subscription to the
WoS Core Collection.
Acknowledgments:
The bibliometric data used in this study are from the bibliometric in-house
databases of the Max Planck Society (MPG) and the Competence Centre for Bibliometrics (CCB, see:
http://www.bibliometrie.info/ (accessed on 1 August 2021)). The MPG’s database is developed and
maintained in cooperation with the Max Planck Digital Library (MPDL, Munich); the CCB’s database
is developed and maintained by the cooperation of various German research organizations. Both
databases are derived from the Science Citation Index Expanded (SCI-E), Social Sciences Citation
Index (SSCI), and Arts and Humanities Citation Index (AHCI) prepared by Clarivate Analytics
(Philadelphia, Pennsylvania, USA). We also thank the anonymous reviewers for their insightful
comments and suggestions. Last, but not least, we thank Eléonore Reinéry (Brock University) for
proofreading our manuscript.
Conflicts of Interest: The authors have no competing interest.

Quantum Rep. 2021,3571
Appendix A
Appendix A.1 Combined Search String
The following search string combines the separate topic searches for the different
fields of QT 2.0:
ts = (“quantum theory” SAME (qubit* OR “quantum bit*”)) OR ts = (“quantum
hardware” OR “quantum device*” OR “quantum circuit” OR “quantum processor*” OR
“quantum register*”) OR ts = (“quantum software” OR “quantum cod*” OR “quantum
program*”) OR ts = (“quantum control*” OR “control* of quantum” OR “control over
quantum” OR “quantum optimal control” OR “quantum state control” OR “control* quan-
tum” OR “control* the quantum” OR “quantum coherent control”) OR ts = ((“quantum
imag*”) OR “ghost imag*”) OR ts = ((quantum NEAR/1 sensing) OR (quantum NEAR/1
sensor*)) OR ts = ((quantum NEAR/10 metrology) OR (quantum NEAR/1 tomograph*)
OR “atomic clock*” OR “ion clock*” OR “quantum clock*” OR “quantum gravimeter*”) OR
ts = (“quantum simulat*” AND (qubit* OR “quantum bit*” OR “quantum comput*”) OR
“quantum simulator*”) OR (ts = “quantum simulat*” AND wc = (“quantum science technol-
ogy” OR “computer science theory methods”)) OR ts = (“quantum information*” OR “von
Neumann mutual information” OR “quantum mutual information” OR “quantum Fisher
information”) OR ts = (“quantum crypto*” OR pqcrypto* OR “quantum key distribution”
OR “quantum encrypt*” OR ((“quantum secur*” OR “quantum secre*”) NOT (“quantum
secreted” OR “quantum secretion”))) OR ts = (“quantum communication*” OR “quantum
network*” OR “quantum optical communication” OR “quantum state transmission*” OR
((“quantum memor*” OR “quantum storage*”) NEAR/5 photon*) OR “quantum repeater*”
OR “quantum internet” OR (“quantum teleport*” AND (“qubit*” OR “quantum bit*” OR
“entangle*”))) OR ts = “quantum algorithm*” OR ts = (“quantum comput*” OR “quan-
tum supremacy” OR “quantum error correction” OR “quantum annealer” OR (quantum
NEAR/2 (automata OR automaton)) OR “quantum clon* machine*”) OR ts = (quantum
NEAR/2 technolog*)
Appendix A.2 Publication Output Data
Table A1.
Annual numbers of publications between 1980 and 2018 in QT 2.0 and its four fields as
well as the whole WoS, restricted to the document types article, review, and conference proceeding
and to the five chosen indices SCI-E, SSCI, AHCI, CPCI-S, and CPCI-SSH. The last line gives the resp.
share of each field of the total publication set.
Publication
Year
All
Papers Q INFO Q METR Q COMM Q COMP Sum of
All Fields WoS
1980 8 1 7 0 0 8 445,586
1981 5 0 4 1 0 5 465,913
1982 6 1 5 0 0 6 486,256
1983 7 0 5 0 2 7 512,149
1984 3 0 0 1 2 3 526,237
1985 10 1 6 0 3 10 533,260
1986 9 0 5 0 4 9 536,412
1987 6 1 3 0 2 6 553,109
1988 12 1 6 3 3 13 578,654
1989 12 0 8 0 4 12 651,038
1990 20 0 13 1 6 20 720,896
1991 66 8 34 3 22 67 756,930
1992 65 6 36 12 12 66 758,014
1993 95 10 50 14 22 96 768,353
1994 112 10 63 15 24 112 802,416
1995 153 15 58 22 63 158 820,972

Quantum Rep. 2021,3572
Table A1. Cont.
Publication
Year
All
Papers Q INFO Q METR Q COMM Q COMP Sum of
All Fields WoS
1996 228 25 65 54 99 243 910,249
1997 317 41 98 58 144 341 924,489
1998 462 66 114 83 257 520 951,072
1999 548 87 135 107 274 603 927,431
2000 841 171 190 168 423 952 973,109
2001 1041 232 190 227 554 1203 967,076
2002 1369 324 229 318 715 1586 987,912
2003 1634 379 297 369 848 1893 1,047,409
2004 1709 440 338 402 808 1988 1,094,068
2005 2039 521 427 506 929 2383 1,158,342
2006 2140 571 408 585 922 2486 1,226,838
2007 2418 663 488 647 1021 2819 1,340,147
2008 2565 687 490 706 1084 2967 1,428,198
2009 2946 752 637 805 1222 3416 1,503,129
2010 2762 858 568 743 1086 3255 1,522,424
2011 2858 918 698 738 1022 3376 1,604,339
2012 3018 955 754 742 1122 3573 1,710,599
2013 3391 1144 830 922 1173 4069 1,790,367
2014 3722 1218 856 1031 1347 4452 1,869,915
2015 3937 1368 928 1006 1416 4718 1,943,877
2016 4481 1561 1115 1162 1627 5465 2,019,730
2017 4552 1578 1139 1181 1656 5554 2,048,110
2018 5031 1687 1234 1353 1868 6142 2,041,007
Total 54,598 16,300 12,531 13,985 21,786 64,602 41,906,032
Share of
total
papers 0.30 0.23 0.26 0.40 1.18
Appendix A.3 Description of VOSviewer Thesaurus File
Table A2.
Description of the common thesaurus file (https://s.gwdg.de/4DDxsp (accessed on 1
August 2021)) applied to the VOSviewer keyword maps Figures 5–8. All terms on the maps are
unified concerning singular/plural writing. Sometimes the plural form has been chosen if its more
common (cellular automata) or if no singular form has been found. One stopword (“cannot”) has
been removed. Hyphens have been removed from cellular-automata, electron-spin, error-correction,
single-photon, and nuclear-magnetic-resonance. Some terms, definitely used in the QT 2.0 context,
have been prefixed by “quantum” (entanglement, teleportation, cryptography). The remaining
changes concerning synonyms and abbreviations are listed in the table.
Unified Term in Keyword Maps Mapped Terms
bb84 protocol bb84
cavity qed cavity quantum electrodynamics
ghost image ghosts
nmr nuclear magnetic resonance
noisy quantum channel noisy channel
q cellular automata (quantum-)dot cellular automata, qca
q clock atomic clock, rubidium atomic clock
q computing quantum computation
q error correction error correcting/correction (code), qec
q key distribution quantum key distribution (qkd), qkd
q tomography quantum state tomography

Quantum Rep. 2021,3573
References
1.
Scheidsteger, T.; Haunschild, R.; Bornmann, L.; Ettl, C. Quantum technology 2.0–topics and contributing countries from 1980 to
2018. In Proceedings of the 18th International Conference on Scientometrics & Informetrics (ISSI2021), Leuven, Belgium, 12–15 July
2021; pp. 1009–1019. Available online: https://www.issi-society.org/proceedings/issi_2021/Proceedings%1020ISSI%202021.pdf
(accessed on 29 July 2021).
2. Planck, M. Ueber das Gesetz der Energieverteilung im Normalspectrum. Ann. der Phys. 1901,309, 553–563. [CrossRef]
3.
Einstein, A. Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Ann. der Phys.
1905,322, 132–148. [CrossRef]
4. Born, M.; Heisenberg, W.; Jordan, P. Zur Quantenmechanik. II. Z. für Phys. 1926,35, 557–615. [CrossRef]
5.
Schrödinger, E. Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen. Ann. der Phys.
1926
,
384, 734–756. [CrossRef]
6.
Haunschild, R.; Bornmann, L.; Marx, W. Climate change research in view of bibliometrics. PLoS ONE
2016
,11, e0160393.
[CrossRef]
7.
Marx, W.; Haunschild, R.; Bornmann, L. Climate change and viticulture—a quantitative analysis of a highly dynamic research
field. Vitis 2017,56, 35–43. [CrossRef]
8.
Marx, W.; Haunschild, R.; Bornmann, L. The role of climate in the collapse of the Maya civilization: A bibliometric analysis of the
scientific discourse. Climate 2017,5, 88. [CrossRef]
9.
Marx, W.; Haunschild, R.; Bornmann, L. Climate and the decline and fall of the Western Roman empire: A bibliometric view on
an interdisciplinary approach to answer a most classic historical question. Climate 2018,6, 90. [CrossRef]
10.
Haunschild, R.; Barth, A.; Marx, W. Evolution of DFT studies in view of a scientometric perspective. J. Cheminformatics
2016
,8, 12.
[CrossRef] [PubMed]
11.
Bornmann, L.; Haunschild, R.; Scheidsteger, T.; Ettl, C. Quantum Technology—A Bibliometric Analysis of a Maturing Research
Field. 2019. Available online: https://doi.org/10.6084/m9.figshare.9731327.v1 (accessed on 1 August 2021).
12.
Dowling, J.P.; Milburn, G.J. Quantum technology: The second quantum revolution. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys.
Eng. Sci. 2003,361, 1655–1674. [CrossRef]
13.
Jaeger, L. The Second Quantum Revolution: From Entanglement to Quantum Computing and Other Super-Technologies; Springer:
Berlin/Heidelberg, Germany, 2018. [CrossRef]
14. Long, G.L.; Mueller, P.; Patterson, J. Introducing Quantum Engineering.Quantum Eng. 2019,1, e6. [CrossRef]
15.
Einstein, A.; Podolsky, B.; Rosen, N. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev.
1935,47, 777–780. [CrossRef]
16. Bell, J.S. On the Einstein Podolsky Rosen paradox. Phys. Phys. Fiz. 1964,1, 195–200. [CrossRef]
17.
Aspect, A.; Grangier, P.; Roger, G. Experimental realization of Einstein-Podolsky-Rosen-Bohm gedankenexperiment: A new
violation of Bell’s inequalities. Phys. Rev. Lett. 1982,49, 91–94. [CrossRef]
18. Clauser, J.F.; Shimony, A. Bell’s theorem. Experimental tests and implications. Rep. Prog. Phys. 1978,41, 1881–1927. [CrossRef]
19. Freedman, S.J.; Clauser, J.F. Experimental test of local hidden-hariable theories. Phys. Rev. Lett. 1972,28, 938–941. [CrossRef]
20.
Tapster, P.R.; Rarity, J.G.; Owens, P.C.M. Violation of Bell’s inequality over 4 km of optical fiber. Phys. Rev. Lett.
1994
,73, 1923–1926.
[CrossRef]
21.
Shor, P.W. Algorithms for Quantum computation: Discrete logarithms and factoring. In Proceedings of the 35th Annual
Symposium on Foundations of Computer Science, Santa Fe, NM, USA, 20–22 November 1994; pp. 124–134.
22. Lyons, H. Atomic clocks. Sci. Am. 1957,196, 71–82. [CrossRef]
23.
Chou, C.W.; Hume, D.B.; Koelemeij, J.C.J.; Wineland, D.J.; Rosenband, T. Frequency comparison of two high-accuracy Al+ optical
clocks. Phys. Rev. Lett. 2010,104, 070802. [CrossRef]
24.
Göbel, E.O.; Siegner, U. The New International System of Units (SI)—Quantum Metrology and Quantum Standards; WILEY-VCH:
Weinheim, Germany, 2019.
25.
Snadden, M.J.; McGuirk, J.M.; Bouyer, P.; Haritos, K.G.; Kasevich, M.A. Measurement of the Earth’s gravity gradient with an
atom interferometer-based gravity gradiometer. Phys. Rev. Lett. 1998,81, 971–974. [CrossRef]
26.
Barry, J.F.; Turner, M.J.; Schloss, J.M.; Glenn, D.R.; Song, Y.; Lukin, M.D.; Park, H.; Walsworth, R.L. Optical magnetic detection of
single-neuron action potentials using quantum defects in diamond. Proc. Natl. Acad. Sci. USA
2016
,113, 14133–14138. [CrossRef]
27.
D’Ariano, G.M.; Paris, M.G.A.; Sacchi, M.F. Quantum tomography. In Advances in Imaging and Electron Physics; Hawkes, P.W., Ed.;
Elsevier Academic Press Inc.: San Diego, CA, USA, 2003; Volume 128, pp. 205–308.
28.
D’Ariano, G.M.; Laurentis, M.D.; Paris, M.G.A.; Porzio, A.; Solimeno, S. Quantum tomography as a tool for the characterization
of optical devices. J. Opt. B Quantum Semiclassical Opt. 2002,4, S127–S132. [CrossRef]
29. Blume-Kohout, R. Optimal, reliable estimation of quantum states. New J. Phys. 2010,12, 043034. [CrossRef]
30.
Lvovsky, A.I.; Raymer, M.G. Continuous-variable optical quantum-state tomography. Rev. Mod. Phys.
2009
,81, 299–332.
[CrossRef]
31.
Padgett, M.J.; Boyd, R.W. An introduction to ghost imaging: Quantum and classical. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
2017,375, 20160233. [CrossRef] [PubMed]
32.
Brif, C.; Chakrabarti, R.; Rabitz, H. Control of quantum phenomena: Past, present and future. New J. Phys.
2010
,12, 075008.
[CrossRef]

Quantum Rep. 2021,3574
33.
Bennett, C.H.; Brassard, G. Quantum cryptography: Public key distribution and coin tossing. In Proceedings of the International
Conference on Computers, Systems and Signal Processing, Bangalore, India, 10–12 December 1984; pp. 175–179.
34. Wiesner, S. Conjugate coding. ACM Sigact News 1983,15, 78–88. [CrossRef]
35.
Brassard, G. Brief history of quantum cryptography: A personal perspective. In Proceedings of the IEEE Information Theory
Workshop on Theory and Practice in Information-Theoretic Security, Awaji, Japan, 16–19 October 2005; pp. 19–23. [CrossRef]
36. Ekert, A.K. Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 1991,67, 661–663. [CrossRef]
37. Wootters, W.K.; Zurek, W.H. A single quantum cannot be cloned. Nature 1982,299, 802–803. [CrossRef]
38.
Bennett, C.H.; Brassard, G.; Crépeau, C.; Jozsa, R.; Peres, A.; Wootters, W.K. Teleporting an unknown quantum state via dual
classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 1993,70, 1895–1899. [CrossRef]
39.
Boschi, D.; Branca, S.; De Martini, F.; Hardy, L.; Popescu, S. Experimental realization of teleporting an unknown pure quantum
state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 1998,80, 1121–1125. [CrossRef]
40.
Bouwmeester, D.; Pan, J.-W.; Mattle, K.; Eibl, M.; Weinfurter, H.; Zeilinger, A. Experimental quantum teleportation. Nature
1997
,
390, 575–579. [CrossRef]
41. Vaidman, L. Teleportation: Dream or reality? AIP Conf. Proc. 1999,461, 172–184. [CrossRef]
42.
Knight, W. Entangled Photons Secure Money Transfer. Available online: https://www.newscientist.com/article/dn4914-
entangled-photons-secure-money-transfer/ (accessed on 29 July 2021).
43.
Briegel, H.-J.; Dür, W.; Cirac, J.I.; Zoller, P. Quantum Repeaters for Communication. 1998. Available online: https://arxiv.org/
abs/quant-ph/9803056 (accessed on 1 August 2021).
44.
Yin, J.; Cao, Y.; Li, Y.-H.; Liao, S.-K.; Zhang, L.; Ren, J.-G.; Cai, W.-Q.; Liu, W.-Y.; Li, B.; Dai, H.; et al. Satellite-based entanglement
distribution over 1200 kilometers. Science 2017,356, 1140–1144. [CrossRef] [PubMed]
45.
Liao, S.-K.; Cai, W.-Q.; Handsteiner, J.; Liu, B.; Yin, J.; Zhang, L.; Rauch, D.; Fink, M.; Ren, J.-G.; Liu, W.-Y.; et al. Satellite-relayed
intercontinental quantum network. Phys. Rev. Lett. 2018,120, 030501. [CrossRef] [PubMed]
46.
Ren, J.-G.; Xu, P.; Yong, H.-L.; Zhang, L.; Liao, S.-K.; Yin, J.; Liu, W.-Y.; Cai, W.-Q.; Yang, M.; Li, L.; et al. Ground-to-satellite
quantum teleportation. Nature 2017,549, 70–73. [CrossRef]
47. Moore, G.E. Lithography and the future of Moore law. In Electron-Beam, X-ray, Euv, and Ion-Beam Submicrometer Lithographies for
Manufacturing V; Warlaumont, J.M., Ed.; Proceedings of SPIE; Spie-Int Soc Optical Engineering: Bellingham, WA, USA, 1995;
Volume 2437, pp. 2–17.
48.
Bilal, B.; Ahmed, S.; Kakkar, V. Quantum dot cellular automata: A new paradigm for digital design. Int. J. Nanoelectron. Mater.
2018,11, 87–98.
49.
Lau, F.L.A.; Fischer, S. Embedding space-constrained quantum-dot cellular automata in three-dimensional tile-based self-assembly
systems. In Proceedings of the 4th ACM International Conference on Nanoscale Computing and Communication Machinery,
New York, NY, USA, 27–29 September 2017; pp. 1–6. [CrossRef]
50.
Liu, Y.; Wu, J.J.; Yi, X. Quantum Boson-Sampling Machine. In Proceedings of the 2015 11th International Conference on Natural
Computation, Zhangjiajie, China, 15–17 August 2015; IEEE: Piscataway, NJ, USA, 2015; pp. 340–398. [CrossRef]
51. Feynman, R.P. Simulating physics with computers. Int. J. Theor. Phys. 1982,21, 467–488. [CrossRef]
52.
Finnila, A.B.; Gomez, M.A.; Sebenik, C.; Stenson, C.; Doll, J.D. Quantum annealing: A new method for minimizing multidimen-
sional functions. Chem. Phys. Lett. 1994,219, 343–348. [CrossRef]
53.
Johnson, M.W.; Amin, M.H.S.; Gildert, S.; Lanting, T.; Hamze, F.; Dickson, N.; Harris, R.; Berkley, A.J.; Johansson, J.;
Bunyk, P.; et al.
Quantum annealing with manufactured spins. Nature 2011,473, 194–198. [CrossRef] [PubMed]
54.
Harris, R.; Sato, Y.; Berkley, A.J.; Reis, M.; Altomare, F.; Amin, M.H.; Boothby, K.; Bunyk, P.; Deng, C.; Enderud, C.; et al. Phase
transitions in a programmable quantum spin glass simulator. Science 2018,361, 162–165. [CrossRef]
55.
Pudenz, K.L.; Albash, T.; Lidar, D.A. Error-corrected quantum annealing with hundreds of qubits. Nat. Commun.
2014
,5, 3243.
[CrossRef] [PubMed]
56.
Arute, F.; Arya, K.; Babbush, R.; Bacon, D.; Bardin, J.C.; Barends, R.; Biswas, R.; Boixo, S.; Brandao, F.G.S.L.; Buell, D.A.; et al.
Quantum supremacy using a programmable superconducting processor. Nature 2019,574, 505–510. [CrossRef]
57.
Wiesner, K. Quantum Cellular Automata. In Encyclopedia of Complexity and Systems Science; Meyers, R.A., Ed.; Springer: New
York, NY, USA, 2009; pp. 7154–7164. [CrossRef]
58.
Deutsch, D. Quantum theory, the Church–Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A Math. Phys.
Sci. 1985,400, 97–117. [CrossRef]
59.
Grover, L.K. A fast quantum mechanical algorithm for database search. In Proceedings of the Twenty-Eighth Annual ACM Symposium
on Theory of Computing (STOC’96), Philadelphia, PA, USA, 22–24 May 1996; Association for Computing Machinery: New York, NY,
USA, 1996; pp. 212–219. [CrossRef]
60.
Abhijith, J.; Adedoyin, A.; Ambrosiano, J.; Anisimov, P.; Bärtschi, A.; Casper, W.; Chennupati, G.; Coffrin, C.; Djidjev, H.; Gunter,
D.; et al. Quantum Algorithm Implementations for Beginners. 2018. Available online: https://arxiv.org/abs/1804.03719 (accessed
on 1 August 2021).
61. Jordan, S. Quantum Algorithm Zoo. Available online: https://quantumalgorithmzoo.org/ (accessed on 29 July 2021).
62.
Tolcheev, V.O. Scientometric analysis of the current state and prospects of the development of quantum technologies. Autom. Doc.
Math. Linguist. 2018,52, 121–133. [CrossRef]

Quantum Rep. 2021,3575
63.
Chen, Y.; Zhang, Z.; Tao, C.; Xu, J.; Tian, Q.; Gulín-González, J.; Liu, Q. Scientometric method for comparing on the performance
of research units in the field of quantum information. In Proceedings of the 17th International Conference on Scientometrics &
Informetrics-ISSI2019-with a Special STI Indicators Conference Track, Rome, Italy, 2–5 September 2019; pp. 399–410.
64.
Olijnyk, N.V. Examination of China’s performance and thematic evolution in quantum cryptography research using quantitative
and computational techniques. PLoS ONE 2018,13, e0190646. [CrossRef]
65.
Dhawan, S.M.; Gupta, B.M.; Bhusan, S. Global publications output in quantum computing research: A scientometric assessment
during 2007-16. Emerg. Sci. J. 2018,2, 228–237. [CrossRef]
66.
Seskir, Z.C.; Aydinoglu, A.U. The landscape of academic literature in quantum technologies. Int. J. Quantum Inf.
2021
,19, 2150012.
[CrossRef]
67.
Preskill, J. Chapter 10. Quantum Shannon Theory. In Quantum Information; California Institute of Technology: Pasadena, CA,
USA, 2016.
68.
t Hooft, G. The Cellular Automaton Interpretation of Quantum Mechanics; Springer International Publishing: Basel, Switzerland, 2016;
Volume 185. [CrossRef]
69.
Fan, H.; Wang, Y.-N.; Jing, L.; Yue, J.-D.; Shi, H.-D.; Zhang, Y.-L.; Mu, L.-Z. Quantum cloning machines and the applications. Phys.
Rep. 2014,544, 241–322. [CrossRef]
70.
Waltman, L.; Schreiber, M. On the calculation of percentile-based bibliometric indicators. J. Am. Soc. Inf. Sci. Tec.
2013
,64, 372–379.
[CrossRef]
71.
Hicks, D.; Wouters, P.; Waltman, L.; de Rijcke, S.; Rafols, I. Bibliometrics: The Leiden Manifesto for research metrics. Nature
2015
,
520, 429–431. [CrossRef]
72.
van Eck, N.J.; Waltman, L. Software survey: VOSviewer, a computer program for bibliometric mapping. Scientometrics
2010
,84,
523–538. [CrossRef] [PubMed]
73.
Burnette, E. D-Wave Demonstrates Latest Quantum Computer Prototype at SC07. Available online: https://www.zdnet.com/
article/d-wave-demonstrates-latest-quantum-computer-prototype-at-sc07/ (accessed on 29 July 2021).
74.
Bornmann, L.; Mutz, R. Growth rates of modern science: A bibliometric analysis based on the number of publications and cited
references. J. Assoc. Inf. Sci. Technol. 2015,66, 2215–2222. [CrossRef]
75. Frame, J.D. Mainstream research in Latin America and the Caribbean. Interciencia 1977,2, 143–148.
76.
Mittermaier, B.; Holzke, C.; Tunger, D.; Meier, A.; Glänzel, W.; Thijs, B.; Chi, P.-S. Erfassung und Analyse Bibliometrischer
Indikatoren für den PFI-Monitoringbericht 2018. 2017. Available online: http://hdl.handle.net/2128/16265 (accessed on 1
August 2021).
77.
van Eck, N.J.; Waltman, L. VOSviewer Goes Online! (Part 1). Available online: https://www.leidenmadtrics.nl/articles/
vosviewer-goes-online-part-1 (accessed on 29 July 2021).
78. Kalachev, A.A. Quantum technologies in Russia. Quantum Electron. 2018,48, 879. [CrossRef]
79.
Fedorov, A.K.; Akimov, A.V.; Biamonte, J.D.; Kavokin, A.V.; Khalili, F.Y.; Kiktenko, E.O.; Kolachevsky, N.N.; Kurochkin, Y.V.;
Lvovsky, A.I.; Rubtsov, A.N.; et al. Quantum technologies in Russia. Quantum Sci. Technol. 2019,4, 040501. [CrossRef]
80. Schiermeier, Q. Russia joins race to make quantum dreams a reality. Nature 2019,577, 14. [CrossRef]
81. Raymer, M.G.; Monroe, C. The US National Quantum Initiative. Quantum Sci. Technol. 2019,4, 020504. [CrossRef]
82.
Zhang, Q.; Xu, F.; Li, L.; Liu, N.-L.; Pan, J.-W. Quantum information research in China. Quantum Sci. Technol.
2019
,4, 040503.
[CrossRef]
83.
Yamamoto, Y.; Sasaki, M.; Takesue, H. Quantum information science and technology in Japan. Quantum Sci. Technol.
2019
,
4, 020502. [CrossRef]
84. Roberson, T.M.; White, A.G. Charting the Australian quantum landscape. Quantum Sci. Technol. 2019,4, 020505. [CrossRef]
85.
Sussman, B.; Corkum, P.; Blais, A.; Cory, D.; Damascelli, A. Quantum Canada. Quantum Sci. Technol.
2019
,4, 020503. [CrossRef]
86.
Riedel, M.; Kovacs, M.; Zoller, P.; Mlynek, J.; Calarco, T. Europe’s Quantum Flagship initiative. Quantum Sci. Technol.
2019
,
4, 020501. [CrossRef]
87. Knight, P.; Walmsley, I. UK national quantum technology programme. Quantum Sci. Technol. 2019,4, 040502. [CrossRef]
88. Lent, C.S.; Tougaw, P.D.; Porod, W.; Bernstein, G.H. Quantum cellular automata. Nanotechnology 1993,4, 49–57. [CrossRef]