Introduction

The second quantum technological revolution [1, 2] or quantum technology 2.0 (QT 2.0) that started around 1980 with the control of single quantum particles and their interaction on an individual basis was object of a bibliometric study by Scheidsteger et al. [3]. The authors looked at a publication set of nearly 55,000 papers from Web of Science, published between 1980 and 2018. They investigated the development over time of four main subfields of QT 2.0 in terms of numbers and shares of publications, as well as the co-occurrence of topics and their relation to the top 25 contributing countries. The present study looks at the QT 2.0 literature of the last 40 years from a different perspective. We perform an explorative and comparative study of the historical roots and seminal papers of the four subfields. As suitable method, we chose the reference publication year spectroscopy (RPYS) introduced by Bornmann and Marx [4] as well as by Marx et al. [5] and already successfully applied in some topical fields such as density functional theory [6]. RPYS is going to be applied to an updated version of the publication set mentioned above.

The concerned four main subfields of QT 2.0 are described in more detail in Scheidsteger et al. [3] and are at this point only recapitulated briefly:

  1. (i)

    Quantum metrology and sensing (Q METR) offers measurement techniques that provide higher precision than the same measurement performed in a classical framework. For example, a new generation of quantum logic clocks achieves a previously unknown accuracy by exploiting the sensitivity of quantum entanglement against disturbances. Quantum tomography is a mathematical technique to reconstruct quantum states via a sufficient set of measurements [7].

  2. (ii)

    Quantum information science (Q INFO) is the theoretical basis for the whole of QT 2.0. It is concerned with concepts as the superposition of states and entanglement–a non-local correlation of quantum particles. A basic prerequisite for quantum information technology is the concept of a qubit or quantum bit as the quantum mechanical generalization of a classical bit.

  3. (iii)

    Quantum communication and cryptography (Q COMM) has as basic ingredients Heisenberg’s uncertainty principle that would prevent undercover eavesdropping during a quantum key exchange, and the use of entangled qubits. These are essential for the long-range transmission of quantum information, usually by quantum teleportation, and therefore for the development of quantum networks.

  4. (iv)

    Quantum computing (Q COMP) exploits the superposition and entanglement of quantum states in arrays of qubits that facilitates data processing and calculations not feasible for classical computers.

In an RPYS analysis, the respective publication set with cited references (CRs) is analyzed with regard to the number of cited references in each reference publication year (RPY). A plot of the RPYs against the number of CRs shows early peaks where historical roots can be found. Different metrics (e.g., in how many citing years were references cited very frequently) can be used to further support the analysis of the RPYS results.

The program CRExplorer (www.crexplorer.net) was introduced by Thor et al. [8] for simplifying and supporting this analysis. Advanced indicators that provide new cited references analysis opportunities were included in the capabilities of CRExplorer later [9].

By carrying out RPYS analyses for each subfield of QT 2.0, we want to answer the following research questions:

  • What are the historical roots and seminal papers of the four subfields of QT 2.0?

  • What is the common core corpus of all four subfields in terms of historical roots and seminal papers?

  • What are the distinctives of the four subfields–in terms of papers and topics?

Dataset and Methods

Data Sources

The bibliometric data used in our study are from two sources: (1) the online version of the Web of Science (WoS) database and (2) the bibliometric custom database of the Competence Centre for Bibliometrics (CCB, see: http://www.bibliometrie.info/ (accessed on 1 August 2021)) derived from the WoS Core Collection, including 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) provided by Clarivate Analytics (Philadelphia, PA, USA). Both, the availability of bibliometric data and the flexibility of the syntax that the WoS offers to formulate the complex queries, have motivated our choice of database.

Search Procedure

The analyses by Scheidsteger et al. [3] considered publications of the document types “Article,” “Conference Proceeding,” and “Review.” The results were based on 54,598 papers published between 1980 and 2018 in the field of QT 2.0. The WoS search queries tailored for the four different subfields of QT 2.0 to gain a maximum precision and a high recall were explained by Scheidsteger et al. [3].

For the present RPYS study, we updated the said dataset by including WoS search results for the years 2019 and 2020 with more than 12,500 additional publications. As WoS records and their accession numbers (UTs) can change over time, we re-ran the search queries from Scheidsteger et al. [3] in WoS on 23 September 2021. We found 67,123 records and downloaded them via the 5 k fast download option. We found 66,521 of the downloaded UTs in CCB’s custom database. This publication set between 1980 and 2020 in the field of QT 2.0 could then be assigned to the four subfields.

Datasets for the Subfields of QT 2.0

Table 1 shows the number of papers in the four subfields and their percentages of the total number of publications for the present expanded dataset. We applied whole counting, and many papers were assigned to more than one subfield. Therefore, the percentages add up to more than 100% and the percentage of papers belonging to only one subfield is about 83%. A graph of the mutual overlap of the four subfields is given in Fig. 1. With a total overlap of papers of about 17%, we suppose a separate analysis of all four subfields is warranted.

Table 1 Number and percentage of papers in four subfields of QT 2.0
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Fig. 1
figure 1

Venn diagram with overlapping subfields of QT 2.0

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Reference Publication Year Spectroscopy (RPYS)

The following steps of the analysis could be dealt with interactively in the CRExplorer’s graphical user interface. For each of the four subfields of QT 2.0, we imported the associated WoS datafile including the CRs. After that, clustering and merging of equivalent CR variants was done with Levenshtein threshold 0.75 for the similarity of first author’s last name and the source title and demanding identical volume and (starting) page numbers if available. Thereby, the number of CR variants is usually reduced significantly, but, in case of the given publication set, by no means comprehensibly. Especially the citations to non-sourced references (those papers not indexed by WoS) and/or to conference papers and books or book chapters have a lot of variants that were not captured by our conservative automatic procedure. These variants were due to, e.g., different selections or orders of the bibliographic data in the CRs, misspellings, or assignments of the publication year to dates of later reprinted editions. The merging of these variants is a tedious manual task, but required in order not to lose or underestimate important CRs during the next step of RPYS, i.e., the removal of all CRs with only a certain small number of citations in order to reduce noise. The suitable threshold had to be tuned so that the peak years are sufficiently pronounced. In our case, we found a minimum number of 25 citations that a CR has to have accumulated over the 41 publication years of citing papers (1980–2020). The peak year structure of the spectrogram after applying this threshold to the original full publication set also gives guidance for searching for CR variants that require manual clustering and merging. Table 2 summarizes the relevant numbers regarding cited references and citing publications of the RPYS analyses of the four subfields.

Table 2 Relevant numbers regarding citing publications and CRs of the four RPYS analyses
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The remaining CRs together with their associated N_CR count and other indicators were exported to a CSV file for further inspection and plotting of the spectrogram. For that purpose, we used R [10] with the R package BibPlots [11]. The function rpys_bl aided in analyzing the spectrogram regarding relevant peaks of the five-year median deviation. Tukey’s fences [12] were used to support the identification of the most important peaks: Important peaks were flagged based on the interquartile range of the median deviations [13]. In our case, we chose the parameter outlier = 1. The relevant CRs under the peaks could be determined by the N_CR and PERC_YR values.

Furthermore, we used the advanced indicator N_TOP10 [9] that counts the number of citing years in which the corresponding CR has been cited very frequently. Following Haunschild, Bornmann [14], we expected that highly influential CRs have values of more than half of the number of different citing years. Finally, we checked the CRs with the highest N_CR values which may not be located in peak years.

All these techniques provided the material for intellectual evaluation and estimation of relevance of the detected papers. RPYS enabled us to unearth historical insights from the diligently selected and compiled corpus by applying proven computational and statistical methods insights presumably complementary–but not contradictory–to results of traditional approaches as archival research of primary sources. RPYS would miss those citing papers that are not indexed in WoS, but this method is robust against small losses. On the other hand, WoS is especially suited for this kind of historical investigation because it also contains citations to non-sourced references, as mentioned above. Measuring the importance and influence of a publication cited in the retrieved corpus by counting the number of cited references is a reasonable assumption, regardless of the specific context and sentiment of a given citation which at the moment could only be detailed by separate inspection. Citations can be seen as the wisdom of the scientific crowd [15].

Results

The results for each subfield of QT 2.0 are given along the following structure. The spectrogram is plotted with those years marked that have been identified as outliers by using Tukey’s fences. Each spectrogram is also available in an interactive web-based version that allows to zoom in to desired time periods and to display the top 5 most cited references in each RPY. Then we present a table with the publications that we identified as mainly responsible for these peaks plus publications fulfilling the alternative criteria of either being very frequently cited (i.e., among the 10% most cited ones in more than half of the citing years) or having N_CR > 1,000. Then we discuss these publications in their context and try to explain their historic importance for the respective subfield.

RPYS of Q METR

As first subfield of QT 2.0, we are looking at Q METR, because this research field sparked the second quantum revolution with its pioneering works in controlling and measuring single quantum systems and therefore dominates the relatively small corpus of QT 2.0 literature from 1980 to about 1990 [3]. Figure 2 shows the spectrogram for this subfield with the number of cited references (grey bars) as well as the five-year median deviations (blue lines) in each RPY. The peaks of the five-year median deviation identified as outliers according to Tukey’s fences are marked with asterisks and labeled with the respective RPYs.

Fig. 2
figure 2

Spectrogram of the RPYS analysis for Q METR. An interactive version is available via https://s.gwdg.de/i6W7VX

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Table 3 lists the peak papers and two additional CRs that have been highly cited in more than half of 41 publication years of citing papers. The numbering of CRs started for Q METR will be continued in the other three subfields, i.e., already mentioned works will keep their CR no. and only new ones will be numbered ascendingly and set in standard face.

Table 3 Historical roots of Q METR
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CR7 is the seminal paper for all of QT 2.0. In the EPR gedankenexperiment, quantum systems can exhibit non-local, entangled correlations unknown in the classical world. This property is exploited in the whole of QT 2.0 and therefore also in Q METR. But the popular naming of this concept as entanglement (in German: “Verschränkung”) did not stem from Einstein but from Schrödinger, the other main theoretical physicist who had thought and published in 1935 on the “current situation of quantum mechanics” (Schrödinger [16] consists of three parts–CR8 is only the first part with the most citations.) and in a direct response to CR7 insisted on the counter-classical interpretation of quantum mechanics that “the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts” (CR9).

But other scientists had thought about measurable quantum effects before. Therefore, the first significant peak can be found three years before EPR in 1932. In this year, it seems appropriate to consider all six CRs with at least 25 citations: In CR1, Wigner introduced a quasi-probability distribution (usually called Wigner function) to study quantum corrections to classical statistical mechanics. His goal was to link the wave function that appears in Schrödinger’s equation to a probability distribution in phase space. The Wigner function is foundational for quantum state tomography and quantum imaging. In CR2 to CR4 and CR6, four theoretical physicists published separately an analytic solution to the equations of motion governing the transition dynamics of a two-state quantum system. The mostly so-called Landau-Zener formula is a basis for quantum control of single quantum systems, e.g., in form of the Landau-Zener-Stückelberg(-Majorana) interference in connection with qubits in quantum dots [17]. This is still a hot topic as the “Advanced Workshop on Landau-Zener Interferometry and Quantum Control in Condensed Matter” (http://indico.ictp.it/event/a14133/, accessed 22 Feb 2022) held by the ICTP in 2014 indicates. CR5 is a classical textbook on the mathematical foundations of quantum mechanics and its calculus and statistics as well as the measurement process.

More than 30 years later, Glauber published CR10 and CR11 on the “Quantum Theory of Optical Coherence” which was awarded the Physics Nobel Prize in 2005. The “Glauber-Sudarshan P representation” of the quasi-probability distribution is often used in quantum optics to describe light in optical phase space.

The year 1972 brought several important contributions. The two mathematical papers CR12 and CR15 on differential equations and the theory of linear transformations provided substantial ingredients for quantum control as explained by D’Alessandro et al. [18]. Hafele and Keating reported in CR13 and CR17 on the predicted and the observed relativistic time gains connected with the first transport of atomic clocks around the world in opposite directions during October 1971. The four cesium beam clocks recorded time differences in good agreement with predictions of special relativity theory. In CR14, Happer comprehensively reviewed the state of the art of optical pumping–the light-induced raising of electrons into energetically higher states in atoms or molecules for achieving population inversion. This technique is needed for building lasers and laser-pumped atomic clocks.

CR14 is also declared to be a milestone by the American Physical Society (https://journals.aps.org/pra/50th, accessed 22 Feb 2022): “Nowadays, atomic coherent states are part of the toolbox explained in any quantum optics textbook. Introduced by Arecchi and collaborators in 1972, such states describe the coherence properties of atoms in a laser cavity and have properties similar to optical coherent states.” It is an important contribution to quantum state tomography and metrology, just as the single peak publication CR16 in 1976: Helström’s standard book on “Quantum Detection and Estimation Theory.” The quantum estimation problem was treated even more thoroughly in 1982 in Holevo’s book on “Probabilistic and Statistical Aspects of Quantum Theory” (CR19), e.g., concerning the quantum noise present when laser light is transmitted through glass fibers. These authors’ names are even connected in the often so-called Holevo-Helstrom theorem [19] giving a lower bound for the distinguishability of quantum states. The other paper in 1982 is more of general interest in QT 2.0, but especially in Q COMP: Feynman’s ideas on simulating physics with computers (CR20).

From the end of the 1980s on, the quantum optimal control theory was put into practice, and particularly two research group leaders dominate our list of seminal CRs. There are four papers alone co-authored by Rabitz (Chemistry Dept. of Princeton University), namely CR42 in 1988, CR43 in 1992, CR28 in 2001, and finally, as CR36 in 2010, a review and outlook on the control of quantum phenomena. The latter called CR21 a pioneering work in the application of quantum optimal control to chemical reactions: Tannor and Rice improved the Tannor-Rice pump-dump scheme [20] for controlling the selectivity of product formation in a chemical reaction. The other two CRs in Table 3 in this subject area were produced by Gerber’s research group at the Institute of Physics at the University of Würzburg, Germany. In 1998, CR25 with the title “Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses” achieved a milestone in quantum optimal control of chemical reactions with a method that “works automatically and finds optimal solutions without previous knowledge of the molecular system and the experimental environment. … The experiments reported here represent a step toward synthesizing chemical substances with higher efficiencies while at the same time reducing unwanted by-products” [21, abstract and last sentence]. Three years later they were able to apply this method for the first time to selectively control mixtures of different molecules (CR30).

Apart from CR23 as an important reference work for the whole of Q METR, the year 1989 saw an influential publication in quantum state tomography. Vogel & Risken, theoreticians from the University of Ulm, Germany, proposed in their rapid communication CR22 a method for obtaining quasi-probability distributions that cannot be measured directly–a major problem in quantum state tomography–by measuring appropriate probability distributions of certain quadrature phases. As next related paper, we have CR27 in 2001 by James et al. dealing with the theory underpinning the measurement of density matrices of qubits. Finally, the review CR31 by Giovannetti et al. in 2004 explains how quantum effects can be exploited to overcome even standard quantum limits of measurement that are not connected with Heisenberg’s uncertainty principle. They also give an overview of applications of quantum state estimation. One area is quantum imaging where they esteem the reconstruction of ghost images to be the most famous experiment which was reported by CR24, the rapid communication on optical imaging by means of two-photon quantum entanglement by Pittman et al. in 1995.

However, about 10 years later, it was proposed and demonstrated in three other CRs that many functionalities of a ghost imaging system could be reproduced in a classical setting [22]: CR32 (2004) and CR34 (2005) from Gatti’s theoretical research group on quantum optics at the Università dell’Insubria in Como, Italy, and CR33 (2005) with the first experimental demonstration of two-photon imaging with a pseudothermal source. Another five years later, CR35 from Gatti’s group proposed differential ghost imaging as an improved method of reconstructing the original image.

In the last CR on quantum optics, Sun et al. (CR40) successfully applied 3D computational imaging with a single-pixel detector thereby removing the need for a spatially resolving detector.

At the turn of the millennium, the by far most highly cited CR was published: CR26 by Nielsen and Chuang is the standard work for QT 2.0 on quantum computation and quantum information and the only CR with (even far) more than 1,000 citations–more than twice the N_CR value of the second most-cited one, CR24. CR26 was first published in 2000, followed by reprint editions in several years including the 10th anniversary edition from 2010, which also only is a reprint with some additional material in several editorials on the past ten years. Therefore, in each subfield, we merged all variants from all years to the 2000 edition.

Just one year later, CR29 proved the high interconnections of quantum optics and quantum computation by proposing a scheme for efficient quantum computation with linear optics. CR38 that was published in the last peak year 2013 is related to Q COMP and the improvement of quantum image processing.

The remaining three CRs from the last peak year 2013 are on one hand harking back to the very first applications of quantum metrology and on the other hand pointing to a new field of applications not represented in the former years.

In CR39, Hinkley et al. were able to report an unprecedented accuracy of time measurements using atomic clocks with wide-ranging opportunities, e.g., for improved tests of general relativity, navigation, secure communication, and even for redefining the SI unit of seconds. The review by Doherty et al. (CR37) on nitrogen-vacancy color centers in diamond (NV) “represents the first time that the key empirical and ab initio results have been extracted from the extensive NV literature and assembled into one consistent picture of the current understanding of the center. As a result, the key unresolved issues concerning the NV centre are identified and the possible avenues for their resolution are examined.” [23, abstract] As last paper in our list, CR41 presents the first instance of nanometer-scale thermometry in a living cell as an exciting new application of NV.

RPYS of Q COMM

In this section, we present our results regarding the application area Q COMM. Figure 3 shows the spectrogram for this subfield with the number of cited references (grey bars) as well as the five-year median deviations (blue lines) in each RPY. The peaks of the five-year median deviation identified as outliers according to Tukey’s fences are marked with asterisks and labeled with the respective RPYs.

Fig. 3
figure 3

Spectrogram of the RPYS analysis for Q COMM. An interactive version is available via https://s.gwdg.de/jATDnm

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Table 4 lists the peak papers from the spectrogram of Q COMM and additional CRs that have been highly cited in more than half of 41 publication years of citing papers plus not yet listed CRs with more than 1,000 citations. Already known CRs are attached with their CR no. and printed in italics. New CRs are numbered consecutively and printed in standard face.

Table 4 Historical roots of Q COMM
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Two observations can immediately be made: at first, only five CRs (7, 8, 9, 20, and 26) have been found before in Q METR, all other 41 CRs are new. At second, one author is dominating this subfield with eight outstanding publications: four of them in 1992 alone; seven of them among the top 10% most cited papers in more than 20 years; six of them with more than 1,000 citations in our Q COMM dataset including the highest cited paper CR49 with more than 4000 citations. We talk about Charles H. Bennett from the IBM Watson Research Center. His contribution to a computer science conference in Bangalore, India, together with Gilles Brassard, is widely named as the founding paper of quantum communication and quantum cryptography by introducing the famous BB84 protocol. Therefore, it seems adequate to start our discussion of the Q COMM CR table with this seminal paper, unearth its roots, follow the realizations and improvements of its proposal and also the network of Bennett’s co-authors.

For this discussion, we profited a lot from three highly cited reviews from the CR list written by a leading European research group, the Group of Applied Physics, University of Geneva, Switzerland: CR82, Gisin et al. [24]; CR62, Gisin, Thew [25]; and CR84, Scarani et al. [26]. Additionally, the review by Zeilinger [27] (Institute for Experimental Physics, University of Vienna & Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Vienna) shed light especially on CRs stemming from his extraordinarily successful research group. For the treatment of quantum entanglement, we refer to the respective review by Horodecki et al. [28] which is going to be CR126 in the Q INFO list.

Quantum communication is in a sense all about cryptography and quantum cryptography is according to Elliott [29] the “final insight to transform cryptography in the 20th century.” According to Shannon’s information theory [30], a key for encoding and decoding a message must be as long or even longer as the message itself. The key must have been exchanged between sender and receiver in advance over a secure channel (“key distribution”) and never be used again. Shannon proved that the so-called “one-time pads” proposed by Vernam (CR44) decades ago fulfill these security requirements optimally. Therefore, we included CR44 as the first CR of Q COMM even if the year 1926 is not detected as an outlier peak year. Moreover, it is the first of only a few references in CR49 which proposed a quantum version of key distribution (QKD). “Unfortunately, in practice, it is difficult to distribute completely secret, completely random, one-time pads needed for Vernam ciphers, so they haven’t been widely adopted.” [29] Instead, fixed length secret keys became popular that were reused several times, like the Advanced Encryption Standard with 128-bit keys. In the 1970s, public-key cryptosystems were developed which based the relative security of their key distribution on the difficulty of factoring large numbers.

Now, Bennett and Brassard proposed a radically different approach in CR49. They did not base the encryption on mathematical complexity but on physical laws (especially Heisenberg’s uncertainty principle) which ensure that their quantum key distribution (QKD) of Vernam ciphers can in principle not be compromised; any eavesdropping by third parties would definitely be detected as unavoidable disturbance. CR49 builds on Wiesner’s idea of quantum money from the early 1970s which was only published ten years later (CR63).

CR49 also makes use of Wootters’ no-cloning theorem from 1982 (CR45) which prohibits exact copying of the information encoded in transferred qubits. This fact was in the same year also suggested by CR46. Moreover, CR49 presupposes with CR48 and CR47 the recent experimental realizations by Aspect and co-workers of the EPR gedankenexperiment with the largest violations of Bell’s inequalities [31] to date–in clear accordance with the predictions of quantum theory.

In CR65 from 1991, Artur Ekert (Oxford University) had arrived independently at an entanglement-based variant of BB84 which included a test of Bell’s inequality using EPR pair states. This in turn motivated a “Bell experiment via entanglement swapping” (title of CR67) together with Zeilinger’s group.

In 1992, Bennett, Brassard, and Mermin (CR52) criticized Ekert’s letter CR65 for an allegedly unnecessary emphasis on Bell’s inequalities, but later in that year, they joined forces in publishing an article together in Scientific American [32]. In the same year, Bennett and Wiesner (CR51) came up with dense coding, which can get over the so-called Holevo bound [33]. The Holevo bound limits the amount of classical information carried by one qubit to only one bit. By dense coding, two classical bits can be sent by one a priori entangled qubit [28]. In 1992, Bennett (CR50) also introduced the simpler B92 protocol which uses only two non-orthogonal states, each one coding for one bit value [26]. This protocol is very economic, but needs some external additions to be secure. In the last peak paper in 1992, Bennett, Brassard, and others (CR53) reported the building of a first hardware implementation of the BB84 protocol–eight years after its proposal.

The next milestone connected with Bennett’s name was the discovery of quantum teleportation in 1993 (CR66) which can be seen as the fully quantum version of Vernam’s one-time pad (CR44) and therefore in a sense as the highest form of quantum cryptography [24]. It took only four years until quantum teleportation could be experimentally realized via entangled photons in Zeilinger’s group including Weinfurter (CR69, CR73) and others (CR76)–even to the point of entanglement swapping (CR67, CR75) by entangling two particles which never interacted [27].

Going beyond teleporting the states of finite-dimensional systems–e.g., the two polarizations of photons–quantum teleportation with continuous variables in infinite dimensional systems–such as optical fields–was first proposed by Vaidman (CR68) and experimentally investigated by Braunstein, Kimble, and co-workers (CR74, CR77). Finally, it is one necessary ingredient for building a quantum internet (CR83). In 1996, improvements concerning quantum teleportation via noisy channels were introduced: on the one hand by Bennett et al. (CR70) and on the other hand by Deutsch et al. (CR72).

With the development of QKD protocols, there arose the need for security proofs even for an imperfect apparatus and noisy channels. Deutsch et al. (CR72) provided the notion of entanglement distillation which was used in experiments by Lo and Chau (CR78) to show the security of the BB84 QKD protocol. In 2000, it could rigorously be proven by Shor and Preskill (CR54), who used the ideas of quantum error correction for protecting quantum information against decoherence and quantum noise [26]. To this same end, Bennett et al. (CR71) had utilized entanglement purification protocols and quantum error-correcting codes already in 1996. In 1999, quantum secret sharing was invented by implementing a multipartite generalization of QKD (CR80). After the turn of the century, photon-number-splitting attacks were discovered as a security issue of laser sources in QKD. Both top CRs under the 2005 peak, CR55 and CR56, refer to applications of the so-called decoy state concept as a solution to this problem. In 2007, CR57 reported significant progress concerning the “device-independent security” of Ekert’s QKD protocol but unconditional security could not yet be proven [26].

Cirac and Zoller with various coworkers in Innsbruck contributed two important theoretical studies (CR79, CR81) on quantum repeaters that are needed for quantum communication over long distances. Three of the remaining CRs in the last outlier peak year 2007 are experimental success stories of QKD over longer distances either over a fiber link (CR61) or over free space (CR58, CR59)–thereby demonstrating the feasibility of global key distribution via satellites.

That Kok’s review (CR60) on linear optical quantum computing is a peak paper for Q COMM in 2007 points to the mediating function of quantum optics between Q COMM and Q COMP. In particular, Kok discussed CR64 reporting the Hong-Ou-Mandel effect, the quantum mechanical interference of two photons, which is an essential ingredient for forming logic gates in linear optical quantum computing.

RPYS of Q COMP

In this section, we present our results regarding the application area Q COMP. Figure 4 shows the spectrogram for this subfield with the number of cited references (grey bars) as well as the five-year median deviations (blue lines) in each RPY. The peaks of the five-year median deviation identified as outliers according to Tukey’s fences are marked with asterisks and labeled with the respective RPYs.

Fig. 4
figure 4

Spectrogram of the RPYS analysis for Q COMP. An interactive version is available via https://s.gwdg.de/E12Pep

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Table 5 lists the peak papers from the spectrogram of Q COMP and additional CRs that have been highly cited in more than half of 41 publication years of citing papers plus not yet listed CRs with more than 1,000 citations. Already known CRs are attached with their CR no. and printed in italics. New CRs are numbered consecutively and printed in normal face.

Table 5 Historical roots of Q COMP
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For the subfield Q COMP, we could identify five outlier peaks in the spectrogram with the respective years 1982, 1997/98, 2000, 2005, and 2012. In these years, we found 19 publications that seem to be mostly responsible for the peaks. But in the case of Q COMP, even more further publications, i.e., 23, need to be considered due to the additional criteria. Of the total of 42 CRs, eight had been mentioned before, two of them in Q METR (CR20, CR26) and six in Q COMM (CR45, CR49, CR51, CR63, CR65, CR66). Therefore, 34 CRs are specific for this subfield.

In this subfield, we do not have one outstanding author but a cluster of nine authors with two to six publications, each, and with at least one publication with more than 1,000 citations in our dataset. They account for more than half (24) of all 42 highly influential CRs. Three of these main contributors were awarded ICTP’s Dirac Medal in 2017 for their outstanding contributions to the foundations of quantum computation and communication (https://www.ictp.it/about-ictp/prizes-awards/the-dirac-medal/the-medallists/dirac-medallists-2017.aspx): physicist and computer scientist Charles H. Bennett (IBM Watson Research Center) with six CRs in our Q COMP dataset, theoretical physicist David Deutsch (Oxford) with three CRs, and mathematician Peter W. Shor (MIT and AT&T Bell Labs) with five CRs. They alone account for 13 distinct CRs. The others are the theoretical physicists David P. DiVincenzo (IBM Watson Research Center) with four CRs, Richard P. Feynman (CalTech) with two CRs, William K. Wootters (Williamstown) with three CRs and Emanuel Knill (LANL) with three CRs, the computer scientist Lov K. Grover (AT&T Bell Labs) with two CRs, and the experimental physicist Harald Weinfurter (Innsbruck and Munich) with two CRs. With Grover and Knill, only two of those had not been mentioned before in the discussions of Q METR and Q COMM. These nine contributors now provide a reasonable approach for discussing the historical roots of Q COMP.

The oldest CR in Table 5 is CR101. Whereas the already mentioned papers by Bennett were first and foremost milestones in Q COMM on QKD and quantum teleportation (CR49, CR51, CR65, CR71), this very early work entitled “Logical Reversibility of Computation” is exclusively dedicated to the theory of computing. Building on Landauer [34], he showed that a universal Turing machine could be made both logically and thermodynamically reversible. In a review on the “thermodynamics of computation,” Bennett [35] compared his approach to reversible computing with the proposal of a “billiard ball computer” by Fredkin and Toffoli from earlier in 1982 (CR85). In 1986, Feynman (CR103) was able to confirm Bennett’s conclusions. To quote Feynman’s last sentence: “At any rate, it seems that the laws of physics present no barrier to reducing the size of computers until bits are the size of atoms, and quantum behavior holds dominant sway” [36]. Later on, reversible circuits including Fredkin gates have attracted interest as components of quantum algorithms.

Four years earlier, Feynman’s seminal ideas on simulating physics with computers (CR20) were responsible for more than 1500 citations (4th rank) in our dataset and therefore for this first year with an outlier peak. One of these citing papers, CR98, is also a peak paper in 2005. It presented an efficient quantum algorithm for quantum chemical simulations of molecular energies–very much in the spirit of Feynman’s original ideas from more than 20 years before. The third CR of 1982 was the no-cloning theorem by Wootters and Zurek (CR45) which together with Ekert’s letter (CR65) on QKD and Bennett’s above mentioned four contributions (two of them co-authored by Wootters: CR66 and CR71) points to the strong interconnections in the basics of Q COMM and Q COMP. The sixth contribution by Bennett is again specific for Q COMP. He was able to show the possibility of assembling a fixed set of one and two-qubit gates to a universal quantum circuit (CR109).

David Deutsch single-handedly laid parts of foundations for Q COMP during the 1980s. He introduced the concepts of a quantum Turing machine, of quantum logic gates and quantum circuits (CR102), and four years later, he extended his universal quantum computer to a network model of computations (CR104). He also proposed the first quantum algorithm that is provably faster than its classical counterpart. He later published as a second one the Deutsch-Josza algorithm together with a colleague from Oxford (CR105).

But these first quantum algorithms had no practical applications. This changed dramatically with Shor’s algorithms for prime factorization of large numbers and for the discrete logarithm in 1994 (CR107) which he proved to be polynomial-time algorithms and therefore exponentially faster than any classical algorithm (CR86). These are the second and fourth most cited references in our dataset with more than 2000 respectively 1500 citations–topped only by CR26 with more than 6000 citations in this subfield. There were three other highly cited contributions in three years’ time (only one got less than 1300 citations) by (then) AT&T-based Peter Shor: (i) He introduced quantum error correcting codes to reduce the effect of decoherence (CR110); (ii) he proved promising results concerning their effectiveness (CR114)–thereby improving earlier work by Steane (CR113) which had drawn a connection to linear error correcting codes of classical information theory–and (iii) he participated in the interdisciplinary and international (USA-Canada-France-Israel-UK) collaboration CR109. The second very practical quantum algorithm introduced in the 1990s was Grover’s algorithm for searching unsorted databases (CR112, CR87).

The importance of quantum error correction that is necessary for preserving coherent states against noise and other unwanted interactions in Q COMP and Q COMM was in 1997 subject of an elaborate “theory of quantum error-correcting codes” by Knill and Laflamme (CR89). Four years later, both authors together with Milburn (CR116) showed that efficient quantum computation is possible using only single-photon sources and detectors, and linear optical circuits, the so-called KLM-scheme–a breakthrough according to CR122 in Q INFO [37]. Another four years later, Knill reported that fault-tolerant quantum computing with noisy gates should be possible under a certain threshold of error probabilities (CR95).

In CR93, DiVincenzo elaborated extensively on seven requirements for the “Physical Implementation of Quantum Computation” (title of CR93)–the famous DiVincenzo criteria. In 1998, he had already proposed “quantum computation with quantum dots” (CR98) after contributing to CR71 and CR109, the latter being a remarkable cooperation of computer scientists, mathematicians, and theoretical and experimental physicists, including Weinfurter from Zeilinger’s group in Innsbruck who had also participated in the two Q COMM milestone experiments CR73 and CR75. Ten years later, already as professor in Munich, he contributed to the experimental realization (CR97) of the “one-way” or “measurement-based quantum computer” proposed by Raussendorf and Briegel (CR117). The name indicates the destruction of a prepared entangled resource state by a single qubit measurement and therefore a very different kind of quantum computation, namely irreversible. The experiment succeeded in running the two-qubit Grover’s algorithm on a 2 × 2 cluster state of photons.

DiVincenzo, as co-author, also proposed quantum computation using the spin states of coupled single-electron quantum dots (CR91) as one possibility of fulfilling his criteria in quantum two-level systems in solid-state devices. Seven years later, the successful “coherent manipulation of coupled electron spins in semiconductor quantum dots” could be reported (CR94). Others had investigated nuclear spins of impurity atoms embedded in a substrate (CR92) or reported the successful coherent control of a special type of superconducting qubits as macroscopic quantum states (CR115). Gershenfeld (CR88) proposed an approach for quantum computation that uses the spin of protons in organic molecules and NMR techniques for manipulating and reading out quantum states in bulk. For this kind of systems, Bennett and DiVincenzo [38] reported several successful demonstrations of the recent quantum algorithms and quantum error-correcting codes. A different approach in the theoretical work CR111 using cold trapped ions interacting with laser beams was even awarded the highly prestigious Wolf Prize in Physics 2013 (https://wolffund.org.il/2018/12/11/peter-zoller/, accessed 22 Feb 2022): “In 1995, Cirac and Zoller proposed a model for a quantum computer, which could be practically implemented with the help of trapped ions. The very concrete nature of their proposal led numerous groups worldwide to successful experiments and has inspired many researchers both theorists and experimentalists.”

In the early years of the new millennium, Kitaev proposed a radically new approach to quantum computation: the use of topologically defined quantum gates to preserve quantum information (CR118). He worked on significantly enlarging the error threshold in fault-tolerant quantum error-correcting schemes (CR96). The prediction by Kitaev [39] of so-called Majorana fermions in topological superconductors and their promising use for forming qubits was supported by the observations of CR100 in 2005. In this context of topological information processing, CR99, a clear and accessible introduction to so-called surface code quantum computing, is the last of our influential papers in the last peak year 2012.

CR90, CR106, and CR108 point to the ambiguity of concept names we mentioned as limitation concerning the retrieval of Q COMP publications in Scheidsteger et al. [3]. In these three CRs, the term “quantum cellular automata” meant the implementation of classical cellular automata on systems of quantum dots and not a device exploiting true QT 2.0 effects.

RPYS of Q INFO

Finally, we present the results for the more foundational subfield Q INFO. Figure 5 shows the spectrogram for this subfield with the number of cited references (grey bars) as well as the five-year median deviations (blue lines) in each RPY. The peaks of the five-year median deviation identified as outliers according to Tukey’s fences are marked with asterisks and labeled with the respective RPYs.

Fig. 5
figure 5

Spectrogram of the RPYS analysis for Q INFO. An interactive version is available via https://s.gwdg.de/3x3UcX

Full size image

Table 6 lists the peak papers from the spectrogram of Q INFO and additional CRs that have been highly cited in more than half of 41 publication years of citing papers plus not yet listed CRs with more than 1,000 citations. Already known CRs are attached with their CR no. and printed in italics. New CRs are numbered consecutively and printed in standard face.

Table 6 Historical roots of Q INFO
Full size table

This subfield contains a medium-sized number of papers but with less than 60% the lowest share of papers that belong only to this subfield. Of the 33 CRs in Table 6, eight were first discussed in Q METR, eleven in Q COMM, and six in Q COMP. Only eight have not yet been discussed. For all of these, quantum entanglement is the dominating theme–in the fundamental questions of EPR and Bell’s theorem as well as in applications in Q COMM and Q COMP. The CRs in this subfield focus on the foundational connection to EPR more pronounced than in the other more applied subfields. The CRs have of course been present in the spectrograms but not in outlier peak years or with a sufficiently high N_TOP10 indicator or citation count, respectively.

In CR7, Einstein and co-workers argued–by elaborating on their famous EPR gedankenexperiment–for the incompleteness of quantum theory because it allows for non-local, entangled correlations unknown in the classical world. Alternatives, e.g., so-called local hidden-variable theories, were proposed. Nearly 30 years later, John Bell, in his groundbreaking work on the Einstein-Podolsky-Rosen paradox (CR119), proved that such theories or, more generally, all those supporting a natural notion of locality need to fulfill a certain inequality. Five years later, Clauser, Horne, Shimony, and Holt (CR120) generalized Bell’s theorem and proposed an extension of an existing experiment that should provide a decisive test. Clauser himself participated in one such experiment [40] and co-authored a review on the history of Bell’s theorem and connected experiments [41], but the clearest violation of Bell’s inequality and thereby exclusion of hidden-variables theories came only in 1982 with CR47 and CR48. Clauser and Aspect received the Wolf Prize in Physics 2010–together with Zeilinger, the most important experimental physicist in Q COMM–for “their fundamental conceptual and experimental contributions to the foundations of quantum physics, specifically an increasingly sophisticated series of tests of Bell’s inequalities or extensions there of using entangled quantum states” (https://wolffund.org.il/2018/12/11/anton-zeilinger/, accessed 22 Feb 2022).

Whereas any violation of Bell’s inequality presupposes an entangled quantum state, CR123 provides a proof that the opposite is not true. There is a class of mixed entangled states that do not imply non-locality, i.e., for any local measurement, these states do not violate Bell’s inequality [42]. The Greenberger-Horne-Zeilinger paradox in CR124 is even going beyond Bell’s theorem. By simple logical contradiction without referring to any inequality, any local model is shown to be incompatible with the predictions of quantum theory [42].

The importance of the concept of entanglement for Q INFO is even more emphasized by two other CRs: CR126 is a comprehensive review from 2009 on quantum entanglement by the Horodeckis. CR121 is among others discussed in CR126 as an important result concerning the degree of entanglement in mixed quantum states suffering from decoherence in real life experiments.

The remaining CRs point to entanglement as essential ingredient for the QT 2.0 subfields Q COMM and Q COMP, respectively: In CR125 from 1997, an international collaboration including Cirac and Zoller proposed a scheme for the reliable distribution of entanglement between spatially distant atomic nodes in a quantum network. The last outlier peak year in Q INFO is 2010 with CR122 as most relevant review on quantum computers that we have already used to discuss the latest developments in the leading physical approaches to building quantum computing units in the Q COMP section. It did not show up in the RPYS analysis of Q COMP because 2010 is not an outlier peak year but CR122 is the by far most cited CR in that year.

Comparison of Subfields of QT 2.0

Our first research question concerned the historically most relevant and influential cited references of each of the four subfields of QT 2.0. We tried to identify them by applying RPYS to the associated publication corpora and discussed their contributions to the respective subfield. Our second and third research questions concerned the relationship of the four subfields in terms of their historical roots. Can there be identified a common core corpus of all four subfields? What is the mutual overlap of the subfields? What are the distinctive features in terms of topics?

By comparing the most influential CRs of all subfields, we try to identify a common core corpus or, at least, the mutual overlap of the subfields. Looking at the listing of those 25 CRs that appear in the CR lists of at least two subfields in Table 7, we are able to make some observations.

Table 7 Common 25 most influential CRs in at least two subfields of QT 2.0
Full size table

One obvious observation is the relative distinctness of the influential CRs in Q METR. There are only seven CRs that also appear in at least one of the other lists. Only two of them are listed with only one other subfield, namely the more foundational Q INFO: the important theoretical contributions to the quantum estimation problem in quantum state tomography Helstrom [43] as CR18 and Holevo [44] as CR19.

The other five CRs with overlap to other subfields (CR7, CR8, CR9, CR20, CR26) can rightly be called the common core corpus of all subfields. Only in Q COMP, the year 1935 with CRs 7 to 9 is not assigned an outlier peak year in our RPYS treatment but is the most pronounced peak year before the 1960s–probably due to its very large corpus of CRs after 1990. CRs 7 to 9 by EPR and Schrödinger are the basis of all QT 2.0 reasoning, introducing the concept of entanglement (“Verschränkung”) and elaborating on its puzzling consequences (“EPR paradox”). Entanglement is the main subject of the reasoning in Q INFO, but is also the prerequisite of Q COMM and Q COMP. In his 1981 talk (CR20), Feynman conveyed the vision of a quantum (physical) computer for the simulation of quantum physics which launched the ongoing quest for realizing quantum computing. And last but not least, the comprehensive textbook on “Quantum Computation and Quantum Information” (title of CR26) collected and taught all the relevant insights up to the year 2000 and won its place among the most cited books in physics as the back cover of its 10th anniversary edition [45] proudly noticed. This edition is only a reprint enhanced by some introductory and historical remarks. That is also witness to its timeless worth.

Apart from these common core papers, Q METR’s CRs are concerned with rather distinctive research and technology areas concerning the measurement, manipulation, and control of single quantum systems building on the theoretical groundwork of the 1932, 1963, and 1972 papers for such diverse applications as atomic clocks, quantum (ghost) imaging, or quantum sensing in living cells.

The interrelationship of the other subfields is much closer. Of the remaining 18 CRs that are not included in the Q METR list, but only shared among the other subfields, six are common for all three subfields: two from the 1980s with the no-cloning theorem (CR45) and the BB84 conference contribution (CR49), and four from the 1990s with Ekert’s alternative quantum cryptography with entangled states (CR65) and three Bennett papers on dense coding using entangled states (CR51), the discovery of quantum teleportation (CR66), and entanglement purification and quantum error-correction (CR71). The only paper attributed to Q COMM and Q COMP, but formally not to Q INFO, should be included here: Wiesner’s ideas from the early 1970s on encoding messages in conjugate observables of quantum systems were not published until 1983 (CR63) but were well known to the authors of the BB84 protocol and essential for its formulation.

The six CRs shared by Q INFO and Q COMP can be divided into two groups. The first one consists of Deutsch’s ground-breaking work on the universal quantum computer (CR102) as well as Shor’s algorithm (CR107) and his introduction of quantum error-correcting codes (CR110). The second group encompasses three different proposals for the practical implementation of quantum computing that have later been realized: either using trapped ions (CR111) or the nuclear spin of atoms (CR92) or quantum dots (CR91).

The remaining five CRs are shared by Q INFO and Q COMM. Four from the late 1990s concern improvements of quantum teleportation (CR70) and its experimental realization (CR73, CR74) as well as sustaining it over longer distances (CR79). The earliest CR from 1987 (CR64) on the quantum mechanical interference of two photons is an example of the strong interconnection between Q COMM and Q COMP.

Discussion

To our knowledge, there are no comprehensive studies about the history of QT 2.0 or its subfields in the literature. Some popular venues published sketch-like accounts of the history of, e.g., quantum computing [46]. There are some more personal recollections of the developments by main contributing scientists in focus areas of QT 2.0, e.g., in quantum cryptography [47] or quantum optics [48]. But the present study is the first comprehensive historical study using a quantitative method to look into and beyond the recent 40 years of QT 2.0 literature in order to identify the historic publications in the four subfields separately and in comparison to each other.

In order to answer our first research question, we identified the historically most relevant and influential cited references of each of the four subfields of QT 2.0 by applying RPYS to the associated publication corpora and discussed their contributions to the respective subfield.

Our second research question concerned the relationship of the four subfields in terms of their historical roots. Is it possible to identify a common core corpus of all four subfields? What is the mutual overlap of the subfields? We were able to peel out a common core set of seminal papers: the three 1935 papers by Einstein (CR7) and Schrödinger (CR8, CR9) introducing entanglement and the EPR paradox as well as Feynman’s vision of quantum computing from 1982 (CR20) and the extreme successful compendium and standard teaching work by Nielsen and Chuang (CR26).

Apart from this common set, it is safe to state a distinctiveness of Q METR against the other three subfields. Only two theoretical works from 1976 (CR18) and 1982 (CR19) have a high importance also in Q INFO, and none other in the other fields. As an unexpected fact, main contributions had already been made in 1932, three years before EPR, and formed the basis for this very much experiment-based subfield of QT 2.0.

On the other hand, Q COMM, Q COMP, and Q INFO are highly connected by sharing many influential papers with one another–apart from the common core set of the five publications CR7-CR9, CR20, and CR26 which immediately come to mind in connection with these subfields. Q INFO is in a sense a bracket between Q COMM and Q COMP. It is holding both together by its foundational theoretical and experimental work on entanglement, the concept introduced by the earliest three common core papers form 1935. Entanglement then becomes instrumental for most of the highly influential papers in Q COMM by proposing and providing a means of secure communication, over long distances, and of quantum teleportation. On the other hand, entanglement is the main reason for the exponential speed-up of quantum algorithms [49] and can be very much controlled in linear quantum optical computing.

The distinctive research subjects of these subfields can be extracted from the specialties in the respective CR lists. Q INFO is uniquely concerned with foundational conceptual implications of entanglement, in particular in interaction with EPR. Q COMP has two unique emphases: one on foundational theoretical papers on quantum computing and on the physical realizations of quantum computer hardware, and the other one on quantum algorithms and the requirements for fault-tolerant quantum computing. Q COMM has also two primary foci: one on quantum cryptography, QKD, and entanglement in connection with communication, in particular quantum teleportation, and the other one on the successful use of quantum optics for secure long-distance communication and quantum networks. Similar to Q METR, the RPYS analysis of Q COMM also found a contribution published several years before EPR, namely CR44 which had been foundational to quantum cryptography.

Limitations of our work concern the retrieval of relevant publications as well as the methodology of the RPYS analysis. Search terms used for the confinement of research areas can suffer from ambiguities that yield false positives that do not belong to the targeted subjects. In case of Q COMP, this even led to three CRs in Table 5 (CR90, CR106, CR108) that are no true QT 2.0 papers (see the last paragraph of section RPYS of Q COMP). The CRs retrieved from the WoS do only contain first author names and therefore in many cases obscure the association with well-known longstanding research groups. In the results’ sections, we tried to indicate these connections but probably not comprehensively. The RPYS method itself is dependent on the answers to some question the researcher has to think about. Should different editions of books be merged together? How tolerant should one be with misspellings of names and dates? Depending on the size of the respective sets of CRs, what should a reasonable threshold for removing less cited CRs be in order to reduce noise? Concerning this last question, we decided to make a compromise and to take N_CR = 25 as a common threshold for all four subfields despite their significantly varying numbers of publications and CRs. In the same manner, the thresholds of N_CR = 1,000 and N_TOP10 > 20 as criteria for identifying highly influential papers can be justified but still are to some extent arbitrary. In this context, one has to be aware of the influence of the threshold for the removal of CRs on the N_TOP10 values: Removing poorly cited CRs increases the threshold for the remaining CRs to be counted among the top 10% most cited papers. Therefore, fewer CRs exceed the chosen minimal N_TOP10 value when more poorly cited CRs are removed. Furthermore, the selection of peaks in the spectrogram can be performed subjectively or via statistical means, as we chose to do here. The interested reader can inspect the smaller peaks via the interactive versions of the RPYS spectrograms.