applied sciences Article Variational Quantum Circuits for Machine Learning. An Application for the Detection of Weak Signals Israel Griol-Barres 1,* , Sergio Milla 2, Antonio Cebrián 3 , Yashar Mansoori 4 and José Millet 3 1 IDEAS-UPV, Vice-Rectorate for Entrepreneurship and Employment, Universitat Politècnica de València, 46022 Valencia, Spain 2 FGYM, Vice-Rectorate for Entrepreneurship and Employment, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] 3 Instituto ITACA, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] (A.C.); [email protected] (J.M.) 4 Department of Technology Management and Economics, Chalmers University of Technology, 412 96 Göteborg, Sweden; [email protected] * Correspondence: [email protected]; Tel.: +34-635-82-28-75 Featured Application: Quantum classifier to detect weak signals. Abstract: Quantum computing is a new paradigm for a multitude of computing applications. This study presents the technologies that are currently available for the physical implementation of qubits and quantum gates, establishing their main advantages and disadvantages and the available frameworks for programming and implementing quantum circuits. One of the main applications for quantum computing is the development of new algorithms for machine learning. In this study, an implementation of a quantum circuit based on support vector machines (SVMs) is described for the resolution of classification problems. This circuit is specially designed for the noisy intermediate-scale Citation: Griol-Barres, I.; Milla, S.; quantum (NISQ) computers that are currently available. As an experiment, the circuit is tested on a Cebrián, A.; Mansoori, Y.; Millet, J. real quantum computer based on superconducting qubits for an application to detect weak signals Variational Quantum Circuits for of the future. Weak signals are indicators of incipient changes that will have a future impact. Even Machine Learning. An Application for experts, the detection of these events is complicated since it is too early to predict this impact. for the Detection of Weak Signals. The data obtained with the experiment shows promising results but also confirms that ongoing Appl. Sci. 2021, 11, 6427. https:// doi.org/10.3390/app11146427 technological development is still required to take full advantage of quantum computing. Academic Editor: Mario Piattini Keywords: quantum computing; variational quantum circuits; quantum support vector machines; weak signals of the future; machine learning Received: 30 May 2021 Accepted: 9 July 2021 Published: 12 July 2021 1. Introduction Publisher’s Note: MDPI stays neutral Approximately every two years the number of transistors in a classical microprocessor with regard to jurisdictional claims in doubles [1]. As a result, transistors are becoming smaller and require lower voltages to published maps and institutional affil- operate. However, current classical computers are reaching the limit of computational iations. capacity because when manipulated in circuits of such small size, electrons tend to act unpredictably. They can also pass through the walls of conduction channels in what is known as the ‘tunnel effect’ [2]. One of the alternatives to classical computing is the exploitation of the laws of quantum mechanics in computational environments. Copyright: © 2021 by the authors. Quantum computing is the development of computational technologies that are based Licensee MDPI, Basel, Switzerland. on the laws of quantum mechanics. This type of computing enables the creation of systems This article is an open access article to store, process, and transfer information encoded in quantum media. The first idea distributed under the terms and of a quantum computer was proposed in 1982 [3] and was based on the first classical conditions of the Creative Commons computing machine defined by Alan Turing [4]. The Turing machine manipulated symbols Attribution (CC BY) license (https:// on a strip according to a table of rules that simulated a computational algorithm. The creativecommons.org/licenses/by/ first proposal for quantum computation replaced the strip with a sequence of two-state 4.0/). Appl. Sci. 2021, 11, 6427. https://doi.org/10.3390/app11146427 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, x FOR PEER REVIEW 2 of 21 symbols on a strip according to a table of rules that simulated a computational algorithm. The first proposal for quantum computation replaced the strip with a sequence of two- state quantum systems. The machine evolved in steps. At the end of each step, the tape was always in a fundamental state of ‘1′ or ‘0′. However, during each step, the machine could be in a superposition of spin states. Soon after the first idea for a quantum computer, some scholars [5] introduced the ‘speculative’ idea that a computer based on the laws of quantum mechanics could perform complex calculations faster than conventional binary computers. Conventional computers convert data into a series of binary digits called bits that Appl. Sci. 2021, 11, 6427 constitute the basic units of information. These bits can only have the values of ‘02′ ofand 21 ‘1′ and are encoded in integrated circuits according to electrical signals of different voltages. Technological advancements have enabled the creation of devices that act as bits. Today an integratedquantum circuit systems. contains The machine millions evolved and millions in steps. of At transistors. the end of each step, the tape was In alwayscontrast, in adata fundamental processed state by of ‘1’quantum or ‘0’. However, computers during use each ‘qubits’ step, the that machine can couldhave the value ofbe ‘0 in′ and a superposition ‘1′ at the same of spin time. states. Unlike bits, qubits can be in a superposition of both Soon after the first idea for a quantum computer, some scholars [5] introduced the states [6].‘speculative’ idea that a computer based on the laws of quantum mechanics could perform Thecomplex Bloch calculationssphere [7] fasteris a thangeometrical conventional representation binary computers. of all possible superposition states that a Conventionalqubit can adopt, computers as can convert be seen data in intoFigure a series 1a. Each of binary point digits on the called surface bits that of the sphere correspondsconstitute the basicto a pure units state of information. of the Hilbert These space bits can of only complex have the dimension values of ‘0’2. The and ‘1’point of coordinatesand are (0,0,1) encoded corresponds in integrated to circuits an eigenvector according to with electrical a positive signals eigenvalue of different voltages.of the Pauli Technological advancements have enabled the creation of devices that act as bits. Today an matrix and the point of coordinates (0,0,−1) corresponds to the eigenvector with negative integrated circuit contains millions and millions of transistors. eigenvalue. InBoth contrast, states data are processed expressed by quantum with the computers notation use |0> ‘qubits’ and that|1> can and have correspond the value to spin upof and ‘0’ andspin ‘1’ down. at the same The time. qubit Unlike can bits, be qubitsrepresented can be in as a superpositiona linear combination of both states of [6 ].both states, as shownThe Bloch in the sphere equation, [7] is a geometricalwhere a and representation b are complex of all possiblenumbers, superposition and where states ||2 + ||2 = 1.that a qubit can adopt, as can be seen in Figure1a. Each point on the surface of the sphere corresponds to a pure state of the Hilbert space of complex dimension 2. The point of coordinates (0,0,1) corresponds|⟩ =|0⟩ to+|1 an eigenvector⟩ = with|0⟩ + a positive |1 eigenvalue⟩ of the Pauli (1) matrix and the point of coordinates (0,0,−1) corresponds to the eigenvector with negative Theeigenvalue. Bloch sphere Both makes states are it possible expressed to with visual theize notation the action |0> and of |1>different and correspond quantum togates graphically.spin upFor and example, spin down. one The of qubitthe most can be used represented quantum as a lineargates combinationis the Hadamard of both states,gate that operatesas on shown a single in the qubit. equation, This where gate ais and equi b arevalent complex to the numbers, combination and where of |twoa|2 rotations,+ |b|2 = 1. one of pi about the z-axis followed by a rotation of pi/2 about E the y-axis E of the Bloch sphere as j i = j i + i = ifa + ifb shown in Figure 1b. y a 0 b 1 rae 0 rbe 1 (1) (a) (b) Figure 1.Figure The Block 1. The sphere Block sphere representing representing a qubit a qubit in inthe the state state of: (a)) |0|0>i and and (b ()b after) after applying applying a a HadamardHadamard gate to gate(a). to (a). The Bloch sphere makes it possible to visualize the action of different quantum gates Thegraphically. Hadamard For gate example, is the one‘quantum of the most Fourier used quantumtransform’ gates performed is the Hadamard on a qubit gate [8]. that The Hadamardoperates gate on is ausually single qubit. the first This gatestage is equivalentin a quantum to the circuit combination because of two it places rotations, the one qubit in a stateof pi of about superposition. the z-axis followed However, by a rotation the ofBloch pi/2 aboutsphere the y-axismodel, of thewhile Bloch conceptually sphere as shown in Figure1b. The Hadamard gate is the ‘quantum Fourier transform’ performed on a qubit [8]. The Hadamard gate is usually the first stage in a quantum circuit because it places the qubit in a state of superposition. However, the Bloch sphere model, while conceptually determining the states that a qubit can have, has the major limitation of not being useful Appl. Sci. 2021, 11, 6427 3 of 21 for showing the ‘entanglement’ of several qubits. One of the most important advantages of state superposition is that, while with n bits one can encode a single state out of 2n possible states, with n qubits one could encode 2n states simultaneously.