New Directions in Quantum Computing for a Tectonic Shift of Technological Change


 Quantum computer and computing are areas of theoretical and experimental research having a phase of growth that can generate a tectonic shift of the evolution of technology in society. The scientific and technological development in quantum research is due to a range of driving technological trajectories that are growing to solve more and more complex problems. The goal of this study is to detect emerging research fields and technological trajectories of quantum computing to explain and generalize, whenever possible, the technological characteristics of evolutionary dynamics in science and technology. Database of Scopus concerning documents and patents is used for statistical analyses to determine the growth of research fields and technological trajectories having a high potential impact in science and society. Results suggest that quantum research is driven by emerging scientific and technological trajectories given by Qubits, Quantum optics, Quantum circuit, Semiconductor quantum dots and quantum information. Overall, this study explains, whenever possible, emerging research fields and technological technologies in quantum computing that support scientific and technological change directed to future economic and social progress. Finally, technology analysis of this study can help policymakers to support the allocation of resources for all areas of Quantum Science having a high potential of growth and positive impact in science and society.


Introduction
Quantum technology is directed to transmission and manipulation of quantum bits (or qubits) 1 and has a high potential of improving information processing and communication between remote locations (Kozlowski and Wehner, 2019;Long et al., 2019). Quantum technology is at the initial stage of evolution but in future it will greatly support new computational approaches to model and operationalize rules in algorithmic form to solve complex problems in society (Atik and Jeutner, 2021;Carberry et al., 2021). Principal firms, such as Intel and IBM, are increasing their R&D investments in these critical research fields and technologies (Coccia, 2017a(Coccia, , 2017bMöller and Vuik, 2017). Quantum research and technology is developing manifold trajectories to improve the solution of complex problems and/or the satisfaction of needs in society, such as quantum cryptography and communications, entanglement and discord, quantum algorithms, quantum error correction and fault tolerance, quantum computer science, quantum imaging and sensing, experimental platforms for quantum information, etc. (Dahlberg et al., 2019). The evolution of this technology needs time, R&D investments, selected research policy, strategic decisions of firms and nations, etc. to support a complete and functional quantum ecosystem, based on reliable physical infrastructures, human resources high-skilled and appropriate technological systems to deal with and solve societal problems (cf., Coccia, 2017Coccia, , 2017cCoccia, , 2018aHou and Shi, 2021;Granstrand and Holgersson, 2020;Oh et al., 2016). In addition, the evolution of quantum science requires interdisciplinarity approaches, different know-how and converging technologies from physics, chemistry, operating systems, computer science, artificial intelligence, nanotechnology, computer networks, etc. to support scientific and technological development of quantum computing and information (Kozlowski and Wehner, 2019). Current quantum research is proposing new approaches and applications, such as algorithms to accelerate quantum machine learning over classical computer algorithms (Pande and Mulay, 2020;Rao et al., 2020), techniques and tools for drug discovery (Batra et al., 2021), preliminary approaches of cryptography, etc. There is a vast literature in these research fields, however, which research fields and technological trajectories in quantum computing are supporting the evolution of quantum science and technology are hardly known. This study confronts these problems 1 Qubits in the quantum mechanical are the analogue of classical bits. here by developing a statistical analysis based on data of publications and patents to explain the evolution of emerging research and technological trajectories in quantum research. The balance of the paper proceeds as follows. First, we describe a theoretical framework of this study underpinned in the economics of technology. Second, we present data and methodology used to analyze data and detect scientific and technological trajectories that can explain the evolution of quantum science over time. We then show relevant findings; discussions and conclusion show how the evolution of quantum science and technology proceeds to provide theoretical and innovation management implications.

Theoretical framework
The evolution of technologies is driven by science that is often viewed as a self-organizing system based on various scientific and social changes (Börner, 2011;Sun et al., 2013) 2 . Scholars maintain that the evolution of technologies is driven by the interaction between inter-related technological systems and scientific fields that generate co-evolutionary pathways and technological change (Benamar et al., 2020;Coccia and Watts, 2020;Coccia et al., 2021;Jovanovic et al., 2021). Quantum technology has many characteristics of general-purpose technologies (GPTs) because it has the potential to make prior technical knowledge obsolete and sustain technological, industrial, economic and social change (Sahal, 1981;Bresnahan, 2010). The technological aspects of GPT are basic to analyze and explain the quantum technology, and some brief background is useful to understand and clarify this vital concept for technology analysis here. GPTs are enabling technologies that support clusters of new products and processes (Helpman, 1998, p.3;cf., Calabrese et al., 2005). GPTs generate changes of techno-economic paradigms, which affect almost every branch of the economy and sustain the long-run process of economic growth in human society (Freeman and Soete, 1987, pp.56-57;Bresnahan and Trajtenberg, 1995, p.8) 3 . These path-breaking innovations are mainly of transformative nature and generate a destructive creation, which makes prior products/processes and knowledge obsolete (Calvano, 2007). Lipsey et al. (1998, p.43) define a GPT as: "a technology that initially has much scope for improvement and 2 For role of science and technology in economic and social change see: Ardito et al., 2021;Calabrese et al., 2005;Coccia, 2003Coccia, , 2005aCoccia, , 2005bCoccia, 2008Coccia, , 2014Coccia, , 2015Coccia, , 2015aCoccia, , 2016Coccia, , 2017aCoccia, , 2018bCoccia, , 2018cCoccia, , 2018dCoccia, , 2019Coccia, , 2019aCoccia, , 2019bCoccia, , 2019cCoccia, , 2019dCoccia and Cadario, 2014;Finardi, 2012, 2013;Rolfo, 2000, 2008;Coccia and Watts, 2020;Pagliaro andCoccia, 2021. 3 cf., Hall andRosenberg, 2010;Lipsey et al., 1998;Li, 2015;Ruttan, 1997;Schultz and Joutz, 2010;von Hippel, 1988. New directions in quantum computing for a tectonic shift of technological development, National Research Council of Italy 4 eventually comes to be widely used, to have many users and to have many Hicksian and technological complementarities." Moreover, GPTs exert a pervasive impact across firms, industries and permeate the overall economy and society of nations (Coccia, 2020). GPTs are revolutionary innovations that generate several ripples of techno-economic effects that change the structure of firms, industries, and society (Peirce, 1974). Coccia (2005, pp.123-124) claims, referring to revolutionary innovations, such as GPTs that: "The means of human communication are radically changed and a new means of communication, which heavily affects all the economic subjects and objects, is born, forcing all those who use it to change their habits. A new technoeconomic paradigm is born… The propulsive capacity for development offered by seventh-degree innovation is so high that it hauls the entire economy. Thanks to the new methods of communication, there is also greater territorial, social, and human integration. Another characteristic of seventh-degree innovations is the ease of their spread. The mobility of people, goods, capital, and information increases, and the time taken to travel and communicate is reduced." In fact, GPTs generate clusters of innovations in several industries because they support basic processes/components/technical systems for the architecture of various families of products/processes that are made quite differently. In general, GPTs are characterized by: "pervasiveness, inherent potential for technical improvements and 'innovational complementarities', giving rise to increasing returns-to-scale, such as steam engine, electric motor, and semiconductors" (Bresnahan and Trajtenberg, 1995), p.83). Jovanovic andRousseau (2005, p.1185) show the distinguishing characteristics of a GPT: 1 Pervasiveness: GPT should propagate to many sectors 2 Improvement: GPT should reduce costs of its adopters 3 Innovation spawning: GPT should produce new products and processes (cf. also, Bresnahan and Trajtenberg, 1995). Lipsey et al. (1998, p.38ff) describe other similar characteristics of GPTs, such as: the scope for improvement, wide variety, and range of uses and strong complementarities with existing or potential new technologies (cf., Coccia, 2020).
Another feature of GPTs is a long-run period between their initial research, the emergence as invention and final introduction in new products/processes having a societal impact (Lipsey et al., 1998(Lipsey et al., , 2005. Rosegger (1980, p.198) showed that the estimated time interval between invention and major innovation can be about 50 years: e.g., for electric motor is about 58 years, electric arc lights 50 years, telegraph about 44 years, synthetic resins 52 years, etc.  Overall, then, GPTs are complex technologies that support product/process innovations in several sectors for a corporate, industrial, economic, and social change (Coccia, 2015(Coccia, , 2017d(Coccia, , 2017e, 2020. The characteristics of GPTs can help the explanation of scientific and technological development of quantum science and technology. In particular, the technology analysis of new trajectories of quantum technology is essential to predict technological and social change. Nelson (2008, p.489) claims that: scientific understanding underlying a technology tends to be contained in the applications-oriented sciences … a strong body of scientific understanding enables technological progress to be rapid and sustained … the research in the engineering disciplines and applications-oriented sciences aims to develop an understanding of what is going on in the operation of the relevant field of practice, so as to illuminate how to advance it.
The quantum technology as a complex technology can generate "certain meta-evolutionary processes involving a combination of two or more symbiotic technologies whereby the integrated system structure is drastically simplified" (Sahal, 1985, p.70, original emphasis). Nelson (2008) also argues that the evolutionary growth of complex technologies is due to a process of learning based on the ability to identify, control, and replicate practices (von Tunzelmann et al., 2008, p.479;cf. also Nelson, 2008, p.488). Sahal (1985, p.79, original emphasis) maintains that: "the process of technological evolution is characterized not only by specific innovation avenues that concern individual industries …, but generic innovation avenues as well, that cut across several industries ... it is apparent that the emergence of a new innovation avenue through fusion of two or more avenues or through fission of an existing avenue can give rise to sudden changes in the mode and tempo of technical progress". The goal of investigating here technological trajectories of quantum technology and science is important to clarify how the long-run evolution of this technology depends on the behavior and evolution of different inter-related technologies, i.e., the long-run evolution of technology can be due to interlocking and positive feedbacks between technologies. Hence, the detection of emerging research fields and technology in quantum science can provide main characteristics to understand future evolutionary paths in science and society (Deshmukh and Mulay, 2021). In this context, the study here analyzes publications and patents that are a main unit of scientific and technology analysis to show how scientific fields and technological trajectories evolve over time (Boyack et al., 2009;Jaffe and Trajtenberg, 2002). As matter of fact, quantitative approaches based on bibliometric data of journals are useful techniques to capture information earlier in the cycle of technology development, whereas patents, in contrast, trail behind (Cozzens et al., 2010;Ding et al., 2000). Arora et al. (2013) apply a bibliometric analysis of publication metadata to investigate emerging nanotechnology research and innovation (cf., Chen, 2006). Van den Oord and van Witteloostuijn (2018) develop a multi-level model to study the evolution of emerging biotechnology. In this research stream, the study here has the purpose of detecting emerging technology in quantum science. The idea here is to analyze the evolution of quantum science by examining new technological trajectories that are basic in science, technology, and society. Next section presents the methods of this scientific investigation (cf., Coccia, 2018).

 Sources and Sample
The study uses Scopus (2021)  Organization, and the UK Intellectual Property Office). In particular, the window of "Search documents" in the Scopus (2021) database is used to identify scientific documents having in article title, abstract or keywords the term "quantum computing " and "quantum computer". Scientific products (articles, conference papers, conference reviews, book chapters, short surveys, letters, etc.) and patents are the basic units for technology and scientific analyses that explain the evolution of science and technology in the field of quantum research under study here (Ding et al., 2000;Glänzel and Thijs, 2012;Savov et al., 2020).  Specification of the model and data analysis procedure The tool "Search documents" in Scopus (2021, 2021a) provides a time series of document results and keywords with the highest number of documents in quantum research, given by: Each of this keyword is inserted in the window "search documents" to detect the specific time series of related research fields or technologies that are used to detect the growth of research fields and technologies for a comparative analysis within quantum science.
The study applies some models for scientific and technology analysis in quantum science.
Firstly, trends of research field/technology i at t are analyzed with the following model: a is a constant; log has base e= 2.7182818; t=time; ut = error term yt is scientific products or patents Secondly, the evolution of technology of quantum computing and quantum computer is analyzed with the model of technological evolution by Sahal (1981) in which the number of patents (Y) is a function of the number of scientific production (X), i.e., Y=f(X) considering X and Y two basic elements of the technological system in quantum computer

Results
First, data are transformed in logarithmic scale to have normality in the distribution of variables for appropriate parametric analyses applied here.

Figure 1.
Trends of research fields in quantum research using scientific production. Note: to show better the trends the period starts from 1990

Figure 2
Technological trajectories in quantum science using patents. Note: to show better the trends the period starts from 2000   Note: Y = Patents of quantum computing (Y) or quantum computer (Y'); X= publications in quantum computing, (X') publications in quantum computer (explanatory variables); The standard errors of the regression coefficients are given in parentheses. p is the pvalue. R 2 is the coefficient of determination, S the standard error of the estimate. F the ratio of the variance explained by the model to the unexplained variance.

Discussions
The evolution of quantum science over the last decades is unparalleled (Scheidsteger et al., 2021  The explanation of these emerging trajectories can clarify their potential for a revolutionary shift in science and technology:  Qubits. In classical computing the information is encoded in bits that exist in one of two states: a 0 or a 1. In quantum computing the information is encoded in quantum bits, or qubits, which can exist in superposition (it can be 0 and 1 at the same time). Qubits represent atoms, ions, photons or electrons and their respective control devices that work together to act as computer memory and a processor. Quantum computer can contain these multiple states simultaneously and has the potential to be more powerful than current most powerful supercomputers. Qubits have main applications in physically realized photonic, atomic and solid-state systems, quantum memories , etc. (de Brugière et al., 2022).
 Quantum information is the information of the state of a quantum system. Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using principles of quantum mechanics. Quantum information can solve problems and perform data processing using a quantum system as the information carrier, rather than binary '1's and '0's used in conventional computation.
Quantum information is developing the technology and exploring applications on multiple fronts, such as atomic clocks potentially could be used as quantum sensors. These quantum logic clocks are part of a new generation of ultraprecise timekeeping devices that can also act as sensors of gravity (Wineland et al., 2002). Quantum communication is also one of the applications of quantum information. Another main application is quantum teleportation: quantum information gets instantly transferred from one qubit to another (NIST, 2021).
 Quantum entanglement is a physical phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. To put it differently, quantum entanglement is a state where two systems (a system is usually an electron or photon) are so strongly correlated that the obtaining of information about one system's "state" will give immediate information . New directions in quantum computing for a tectonic shift of technological development,

National Research Council of Italy
15 about the other system's "state", though they are apart these systems (Zou et al., 2021). Quantum entanglement has been demonstrated experimentally with photons and neutrinos (Kocher and Commins, 1967), electrons (Hensen et al., 2015); molecules (Arndt et al., 1999;Nairz et al., 2003), small diamonds (Lee et al., 2011) etc. In addition, the utilization of entanglement in communication, computation and quantum radar are very active areas of research and development. Entanglement can enable quantum cryptography and superdense coding to be faster than light speed communication, and even teleportation, though current problems with quantum computers about time and processing power-consuming.
 Quantum circuit, in quantum information theory, is a model for quantum computation; a quantum circuit is a computational routine consisting of coherent quantum operations on quantum data, such as qubits. Any quantum program can be represented by a sequence of quantum circuits and non-concurrent classical computation (cf., Ovalle-Magallanes et al., 2022)  A classical (or non-quantum) algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical computer. Similarly, a quantum algorithm is a step-by-step procedure, where each of the steps can be performed on a quantum computer.
Quantum algorithms use some essential feature of quantum computation, and they can solve some problems faster than classical algorithms by technological aspects of quantum superposition and quantum entanglement (Lanzagorta et al., 2009;Nielsen and Chuang, 2010;Shao et al., 2019).
 Communication is one of the most promising research fields of quantum physics closely related to important works, such as quantum teleportation, quantum information processing and quantum cryptography. The last one turns to be the most interesting application which aims to protect information channels against eavesdropping by means of quantum cryptography. In fact, the most well-known and developed application of quantum cryptography is quantum key distribution: it describes the use of quantum mechanical effects to perform cryptographic tasks or to break cryptographic systems. One important component of virtually all proper encryption schemes is true randomness which can be generated by means of quantum optics (Bennett et al., 1992).
 Quantum cryptography will lead to the final solution for cyberattacks and for this reason many efforts have been dedicated to this growing field of research. The advantage of quantum cryptography lies in the fact that it allows the completion of various cryptographic tasks that are proven or conjectured to be impossible, such as, it is impossible to copy data encoded in a quantum state. In the presence of an attempt to read the encoded data, the quantum state will be changed due to wave function collapse. This approach could be used to detect eavesdropping in quantum key distribution (Bennett et al., 1992;Chen et al., 2015). Cloud computing and ecommerce can be growing areas by applying quantum computation. Current limitations are due to very large and expensive computers capable to transmitting information using quantum cryptography. Moreover, now there is no algorithm efficient enough to do the factoring quickly, but future technological development could eventually find it. In short, quantum cryptography is a true breakthrough in the field, though we need time for creating learning processes for reliable applications of this technique in practical problems in society.
These findings and discussions show that quantum science is evolving with endogenous processes of interaction between different research fields and technologies, just described, that increase the size and complexity of the ecosystem of quantum science driving scientific and technological change.

Conclusions
Quantum science can support the answer to questions in science and society that need a lot of data; as consequence, quantum technology, as a general-purpose technology, can have main applications in manifold fields and as the technology develops, the more uses we will find for it. The evolution of quantum science shows that new research fields and technologies are growing (e.g., qubits, quantum information and algorithms, quantum communication, etc.) to support the emergence of ecosystem based on a complex network of interconnected systems including scholars, academic institutions, collaborations, financial resources, etc. (Coccia, 2019c, 2019d, 2019e, 2019f, 2019g, 2020d, 2020e, 2020f, 2020g, 2020h, 2021i, Coccia and Bellito, 2018Coccia and Benati, 2018;Coccia and Cadario, 2014;Finardi, 2012, 2013;Rolfo, 2000, 2008) 4 . Results also reveal that quantum computing research and technologies are evolving, more and more, with a transition from hardware to software aspects represented by growing trajectories of quantum information and quantum algorithms (cf., Li et al., 2021).
These results bring us to conclusions that are, of course, tentative. Although this study has provided some interesting, albeit preliminary results, as many other bibliometric studies, it has several limitations. First, the precision of the search queries is affected by ambivalent meanings in quantum computing, such as information, computing, computer, etc. To operationalize the technology analysis of quantum computing directed to measure, assess and predict technological trajectories, this study proposes a simple model of technological evolution by Sahal (2021). This model measures the effect that production of publications has on patents' growth within the field of research, considering publication and patents as elements of the same scientific system. This approach was originated to study biological principles of allometry and subsequently applied in economics of technology to analyze the patterns of technological innovation (Sahal, 1981).
The general model is based on following assumptions.
(1) Suppose the simplest possible case of only two elements, X (publications) and Y (patents), forming a system in a specific scientific and technological domain.
(2) Let Y(t) be the extent of advances of a technology Y at the time t and X(t) be the extent of scientific production underlying the advances of a technology Y. Suppose that both X and Y evolve according to some S-shaped pattern, such a pattern can be represented analytically in terms of the differential equation of logistic function.
For X, scientific production, the starting equation is: The equation can be rewritten as: The integral of this equation is: The growth of X(t) can be described respectively as: Mutatis mutandis, for Patents Y(t) the equation is: The logistic curve here is a symmetrical S-shaped curve with a point of inflection at 0.5K with 2 The expression generated is: The model of technological evolution is given by: where ( ) B is the evolutionary coefficient of growth that measures the evolution of technology Y in relation to scientific production X. , whether technology Y evolves at a lower relative rate of change than X; the whole system has a slowed evolution (underdevelopment) over the course of time.
 B has a unit value: 1 = B , then Y and X have proportional change during their evolution. In short, when B=1, the whole system here has a proportional evolution (growth).
, whether Y evolves at greater relative rate of change than X; this pattern denotes disproportionate advances. The whole system of technology Y has an accelerated evolution (development) over the course of time.