Quantum Computing for Antennas and Propagation: A gentle introduction

10 IEEE ANTENNAS & PROPAGATION MAGAZINEFEBRUARY 2025 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ ©SHUTTERSTOCK.COM/ALEXANDER56891 A gentle introduction. Quantum Computing for Antennas and Propagation Problems Marco Donald Migliore Quantum computing is a cutting-edge paradigm in data processing that uses the principles of quantum mechanics to perform calculations. It has promising applications in various fields of engineering, including applied electromagnetism. Like any emerging technology, quantum computing has advantages and limitations that require in-depth understanding to make effective use in real-world engineering applications. This article aims to provide a simple and intuitive introduction to the fundamental principles of quantum computing, delving into the strengths and limitations inherent to this innovative technology and suggesting some applications of interest in the antennas and propagation community. INTRODUCTION In the past few decades, there has been significant progress in the field of quantum computing. Beginning with the pioneering demonstration of a two-qubit quantum logic gate by Monroe Digital Object Identifier 10.1109/MAP.2024.3498696 Date of publication 9 December 2024; date of current version 29 January 2025. 11IEEE ANTENNAS & PROPAGATION MAGAZINE FEBRUARY 2025 et al. [1], the field has progressed to today’s advanced quantum processors, capable of handling several hundred qubits, as Google Sycamore (53 qubits) and IBM Osprey (433 qubits) [2]. Quantum computing represents a potent technological paradigm with promising applications in various fields of engineering, including electromagnetic applications. However, like any emerging technology, it has its own set of limitations alongside its advantages, which are crucial to understand (Figure 1). This article aims to provide an intuitive introduction to the fundamental concepts of quantum computing based on quantum gates [3], clarifying the advantages but also the limitations of this new technique, with a brief digression also on quantum annealing [4]. The simple and intuitive approach followed in this article is intended to lower the barriers that currently hinder applied electromagnetic engineers from fully utilizing this novel technology. The ultimate goal is to inspire more researchers to explore applications of quantum computing in the field of applied electromagnetism, broadening the community that is already currently exploring this fascinating field of research. SOME BASIC CONCEPTS This article is focused on quantum computing without entering into the details of many possible physical implementations of quantum computers. However, since the fundamental strengths and constraints of quantum algorithms are determined by the underlying physics of quantum computers, to fully understand the potential of quantum computing it’s essential to introduce and discuss some key characteristics of quantum systems. Consider a particle having mass m moving in a potential V(x, t). According to the Born’s interpretation the probability density of the position of the particle along x is given by ,, ,xt xt xt 2 WW W = ) ^^ ^hh h , where (,)xtW is the wave function solution and ) denotes complex conjugation. The wave function is solution of the Schrödinger equation [5], [6] ,, , mx Vxtxt t xt 2 i 2 2 2 ' 2 2'2 2 W W -+ = ^^ ^ hh h ; E (1) where h is the Planck constant and /h2'r= is the reduced Planck constant. The wave function has a functional representation as a vector in the complex Hilbert space of square-integrable functions equipped with the L2 standard norm. As previously mentioned, |(,)| xt 2 W denotes the probability density for the particle’s position along x at the time t. Accordingly, the wave function is normalized such that () (, ). txtx d1 2 WW == # In the case of a time-independent potential, V(x, t) = V(x), it is possible to solve the Schrödinger equation using the separation-of-variables method by assuming a solution of the form (,)()(),xt xt }z W= obtaining () () () mdx dVx xEx 2 2 2 2 ' }} -+ = ; E (2) () () i dt dt Et 'zz = (3) where E is a separation constant. To explore the physical significance of E, we substitute the wave equation (, )()( )xt Ae xt ()kx ti}z W == ~- u u u into (2), with k being the wavenumber and ~ the angular frequency of the wave.1 The spatial component simplifies to ()/( )km2 22 ' + () .Vx E= Using the wavenumber k expressed in terms of wavelength m as ()/k2rm= and the de Broglie relation for momentum /,ph m = we derive the classical energy equation /,mv VE2 2 += where E represents the total energy, comprising both kinetic and potential energies. Furthermore, from the time-dependent equation, we have that ,Eh~= showing again that E is the total energy of the quantum system, and indicating the role of the energy as a “driving force” behind the temporal changes of quantum systems. Regarding the solution of the Schrödinger equation, it is interesting to note that in electromagnetism and generally in classical physics, a wave equation requires a second derivative. However, the presence of the imaginary unit in the Schrödinger equation allows the equating of the left and right terms with a first-order temporal derivative, as can be easily verified by substituting the wave solution (,)xtW u into the equation. More generally, the evolution of the quantum system can be written in terms of the Hamilton operator, which acts as a generator of time evolution of the quantum system || t Hi' 2 2HHWW = t (4) where the Dirac bra–ket notation has been used.2 The time-independent Schrödinger equation ||HEHH}} = t (5) 1Rigorously speaking, this is not a valid wave function, since it cannot be normalized in the usual sense over an infinite domain. Instead, it requires the use of a generalized delta function. The problem can be overcome using “box normalization,” where the particle is confined to a finite portion of space [7]. 2The quantum state is expressed using the “ket” notation |,H W whereas the dual vector is expressed as the “bra” |;G W the inner product between two vectors |H W and |H U is |.GH UW FIGURE 1. Quantum computing is a cutting-edge paradigm in data processing that uses the principles of quantum mechanics to perform calculations. The challenge is to exploit the unique properties of quantum mechanics to develop solutions that are more time-effective and energyefficient than conventional alternatives. 12 IEEE ANTENNAS & PROPAGATION MAGAZINEFEBRUARY 2025 reduces to an eigenvalue problem. Since the Hamiltonian is a Hermitian operator, its eigenvalues (,,) 12 f cc are real, while the eigenfunctions (| ,| ,), 12 f HH }} form a basis for the representation of the wavefunctions |,H } i.e., || | aa 11 22 gHHH }} }=++ (6) where ,,aa 12 f are the coefficient of |H } on the |,|,.. 12 HH}} basis. The eigenfunctions | k H } are called eigenstates of the quantum system. Equation (6) represents the |H } state as superposition of (energy) eingenstates, while the eigenvalues are the energies of the eigenstates. The complex coefficients | a kk GH}} = are called probability amplitudes. By applying Born’s rule, we obtain that the probability of the quantum system being in the eigenstate | k H } is || ak 2 . More generally, the solution of the time-dependent Schrödinger equation can be written as |()| | ta a ee //iiEE 11 22 12 fHHH}} W=++ ''-- (7) or equivalently in terms of Hamiltonian operator |()| ()t0e /Hti HHWW ='- t (8) where ,e /Hti'- t giving the time evolution of quantum states, is a unitary operator. The fact that the time evolution of the quantum state involves a unitary operator, (),Ut t can be intuitively explained considering that during the evolution, the total probability must be preserved, i.e., ()t1W =, and the state vector ()tW can be subject only to a rotation in the (complex) Hilbert space, without changing its length. This gives an important property of quantum systems: i.e., that the time evolution of quantum states involves unitary operators. The wave function contains all of the information about a system, but it is not directly observable. Instead, specific physical quantities (observables), such as energy and momentum, can be measured. In particular, each observable is represented by a self-adjoint (Hermitian) operator .A t During the measurement process, the system “collapses” into one of the eigenstates of .A t The expectation value of the measurement is given by || . A GHWW t For example, the momentum operator /iAddx '=- t ^h is used when measuring momentum. Conversely, the Hamiltonian is employed for energy measurements. As a specific instance, if the Hamiltonian’s eigenfunctions are | k H } and the system is in the state |,H W then the measurement yields an energy value Ek with a probability ||, ak 2 where ak is the projection of |H W onto |. k H } Now, let us suppose that we observe a different characteristic of the quantum system, obtained by the observable .B t If the two observables A t and B t have the property that AB BA= ttt t (i.e., they “commute”), they have the same eigenfunctions, and it is possible to measure the two properties of the physical systems associated to A and B at the same time, otherwise there exists an uncertainty relation between the two properties [5]. In summary, when two observables do not commute, meaning they do not have common eigenfunctions, the Heisenberg uncertainty principle dictates that accurate simultaneous measurements of the properties associated with these observables are not feasible. The above results can give some ideas of how quantum computation works. Let us suppose that we have a problem whose possible solutions are ,,..oo 12 f We can imagine to map the solutions into different states of a quantum system. For example, we can use energy as quantity to encode the possible solutions, i.e., energy E1 is associate to ,o1 E2 to ,o2 and so on. We prepare a quantum system in a starting state |()0 H W according to the problem to be solved. Then, we force the quantum system to evolve in time3 according to a unitary operator ()Ut t , chosen such that in the instant tl in which we perform the measurement the quantum state |( )()| ()tUt0 HHWW = t collapses with high probability to the state that maps the solution of the problem. In an intuitive geometrical explanation of the computational process, the (unit length) vector ()0W is rotated in the (complex Hilbert) space until it is parallel to the eigenstate that maps the solution of the problem, e.g., 1 W if o1 is the solution. The outcome of the measurement is the energy level E1 with probability as close to one as |() t H Wl is parallel to | 1 H W in the measurement instant .tl It is important to recognize that the characteristics of the operators involved in the evolution of the system, which are constrained to being unitary, have significant implications for how quantum information can be manipulated and processed in quantum computers. One of the most significant consequences is expressed by the no-cloning theorem, which states that it is impossible to create an exact copy of an arbitrary unknown quantum state [8]. The no-cloning theorem and measurement collapse make quantum error correction extremely challenging, since errors must be corrected without direct observation of quantum states, thereby requiring additional qubits for encoding redundancy and fault tolerance. On the other hand, this theorem has important applications in quantum cryptography, preventing potential eavesdroppers from perfectly cloning quantum states and ensuring detectable anomalies if tampered with due to measurement disturbances [9]. Finally, the collapse of the wave function into one of the eigenstates of the quantum system during the measurement process has some profound consequences since, although quantum systems contain a large amount of information due to superposition, only a part of this information is accessible after measurement. Holevo’s theorem [8] establishes a fundamental limit on the amount of classical information that can be accessed. From a computational point of view, n-qubit quantum systems offer an “internal” (nonaccessible) “computational space” that grows exponentially [i.e., as ()]O2n with the number of qubits. This exponential growth potentially allows quantum computers to solve certain problems much faster than classical 3From a physical point of view, controlled temporal evolution of the quantum system requires a careful modification of the energy of the system, that is obtained by microwave or optical frequency electromagnetic waves interacting with the quantum system. 13IEEE ANTENNAS & PROPAGATION MAGAZINE FEBRUARY 2025 computers. However, measurement collapse limits the ability to directly observe or utilize the full computational space created by quantum superposition. Consequently, while a quantum computer can process a vast number of possibilities simultaneously, only a single outcome can be extracted from each computation. Quantum algorithms must be specifically designed to exploit the computational capabilities before measurement collapse. SINGLE QUBIT QUANTUM PROCESSING A qubit is a two-state (or two-level) quantum-computing system. It is the quantum version of the classical binary bit. A qubit’s quantum state can be described as a linear combination of two orthonormal basis states, represented as {| ,| }01HH , called computational basis [8] |||01HHHab W =+ where .1 22 ab += By representing the basis with a 2D complex state vectors of unit amplitude |,|0 1 0 1 0 1 and HH== ;; EE we have the wave function representation: |. 1 0 0 1 Hab a b W=+= ;;; EEE The qubit state |H W has a geometrical representation in the Bloch sphere as |( /) |( /) |, co ss in20 21 e i HH Hii W=+ z wherein i and z are the angles of the spherical coordinate system (see Figure 2). Note that all of the possible states of the qubit are on the surface at a unit distance from the center of the Bloch sphere. Computing operations on the qubits are performed by quantum gates, which are basically unitary matrices U t that modify the qubit state from |H W to |,H U wherein || .U HHUW = t Unitary assures that the physical process is reversible, avoiding loss of information. Examples of quantum gates operating on a single qubit are the X(NOT), Y and Z gates, represented by the Pauli matrices [8]: ,, . XY Z 0 1 1 0 0 0 1 0 0 1i i == - = - ttt ;; ; EEE (9) These matrices induce a rotation of the state by r around the X, Y, and Z axes of the Bloch sphere, respectively. A further important quantum gate is the Hadamard gate H 2 1 1 1 1 1 = - t ; E (10) that creates the following superposition of states from the computational bases |||| || .H 2 01 2 01 HH HH HH ab UW== + + - t (11) The equal superpositions of computational states (/ )| |12 01HH + ^h and (/ )| |12 01HH - ^h are usually denoted as |H + and |.H - As an example, let us consider a simple quantum computing algorithm. We start with the initial state ||.0HH W= Applying a Hadamard gate, the state is transformed to || |H0 HHH U == + t . This state is a superposition of |0H and |1H with equal amplitudes. When measured in the computational basis, which consists of the states |0H and |,1H the state |H + results in |0H with a probability of (/ )/,12 12 2 = which corresponds to a 50% chance of measuring |.0H However, it is interesting to note that if a second Hadamard gate is applied to the state |H + before measurement, we obtain |||. H0 HHH U=+= t The second Hadamard gate, being its own inverse, returns the state to |,0H resulting in a measurement of |0H with unit probability. This outcome demonstrates that the state provides complete information about the system, underscoring a profound connection between the quantum state and information. This relevance extends well beyond quantum computations and merits further exploration. When considering a finite number of states, e.g., two in the case of a qubit, the information content of the respective quantum system is given by the density matrix t [8], a positive semidefinite matrix that encodes all of the statistical properties of the system. The density matrix for a quantum system in a pure state |H } is given by [8]: ||.HG t}}= (12) A pure state is characterized by a unit value of the trace of the squared density matrix, i.e., () .1Tr 2 t= On the Bloch sphere, pure states are represented by points on the surface of the sphere. 0 0 0 1 ψ ψ χ y z +i –i + + – – 1 0 10 1 +i1 2 2 j 2 2 –i1 -j 1 1 2 2 1 –1 2 2 =+1α = cos(θ/2) β = ejϕ sin(θ/2) ϕ αα ββ θ FIGURE 2. The Bloch sphere, having a unit radius, offers a geometric illustration of a qubit. The orthonormal basis states 0|H and 1|H are represented by the sphere’s north and south poles, respectively. For a specific qubit |,HW the expansion coefficients are a = cosθ/2 and b = sini/2. 14 IEEE ANTENNAS & PROPAGATION MAGAZINEFEBRUARY 2025 In the example, we have a “pure state,” which has maximal information content. This means that the state vector provides complete information about the system, eliminating any statistical ambiguity or uncertainty in measurement outcomes, aside from the inherent probabilistic nature of quantum mechanics. Suppose that a quantum system placed in a pure state is not completely isolated, but interacts with “environment,” i.e., the “external world” that is not part of the intended quantum operation, such as through thermal interactions, particle interactions, or electromagnetic fields. The interaction involves exchange of information between the quantum system and the environment, which effectively acts as a measurement apparatus.4 This causes decoherence, putting the qubit into a “mixed state,” i.e., a statistical mixture of different pure states |, i H } each with a probability .p i For a mixed state, the density matrix is [8] ||. pi i ii HGt} }= / (13) The diagonal elements of t represent the probabilities of the system being found in each of the basis states, while the off-diagonal elements describe the quantum superpositions between the basis states. Mixed states are characterized by a value of the trace of the squared density matrix that is less than 1, i.e., () .1Tr 21 t On the Bloch sphere, mixed states are associated to points inside the sphere. The closer a point is to the center, the more “mixed” or less “pure” the state is, indicating a greater loss of information about the state of the quantum system. Keeping the pure state condition is of paramount importance in quantum processing, and decoherence time is one of the most important limitations in quantum computation. On the other hand, interaction with the environment can be exploited to obtain information about the environment itself. This concept is crucial in many applications. For example, in quantum metrology, it forms the basis of new highly sensitive electric and magnetic field sensors, gravitational sensors, pressure sensors, and quantum clocks [11]. Finally, one of the most impressive practical applications of decoherence is undoubtedly magnetic resonance imaging. This technology exploits the influence of the environment on the decoherence of atomic nuclei to differentiate tissue types, identify pathological changes, and provide detailed internal images [12]. MULTIPLE QUBIT GATES Elaboration using multiple qubits can be analyzed in terms of matrices representing states by vectors having 2n elements, where n is the number of qubits [8]. 4The modeling of decoherence, which arises from interactions with environmental factors, can be formulated using the quantum Langevin equation (QLE) [10]. The QLE extends the classical Langevin equation to quantum systems, incorporating terms for both deterministic evolution (governed by the system’s Hamiltonian) and noise (arising from environmental interactions) to account for both dissipation and fluctuation. The study of such open quantum systems is a specialized area of research within quantum mechanics. For example the state |00H can be represented using the tensor product (Kronecker product) as |||. 00 00 1 0 1 0 1 0 0 0 77 HHH=== R T S S S S S ;; V X W W W W W EE (14) In case n = 2, we have 22 = 4 base vectors |, |,|, |. 00 1 0 0 0 01 0 1 0 0 10 0 0 1 0 11 0 0 0 1 HHHH ==== R T S S S S S R T S S S S S R T S S S S S R T S S S S S V X W W W W W V X W W W W W V X W W W W W V X W W W W W (15) An example of a quantum gate operating on two-qubits is the controlled NOT ()CNOT gate CI X0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 NOT== t t t R T S S S S S ; V X W W W W W E (16) that flips the second qubit (the target qubit) if and only if the first qubit (the control qubit) is |.1H A further example of two-qubits gate is the controlled Z gate, that applies a phase “flip” (i.e., a r change in the relative phase) to the target qubit only if the control qubit is in the state |1H . CI Z0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 Z== - t t t R T S S S S S ; V X W W W W W E (17) Since quantum computing is based on unitary transformations, any elaboration modeled by a unitary operator has its equivalent quantum gate. The most famous example is the discrete Fourier transform, which, in the case of two qubits, involves the following quantum gate: . Fi i i i 2 1 1 1 1 1 1 1 1 1 1 1 1 1 =- - - - - - t R T S S S S S V X W W W W W (18) Verification of the unitary constraint in quantum gates may require circuit overhead that includes the use of extra qubits, called ancillae qubits, and garbage outputs, i.e., outputs that are not a useful part of the result but is necessary for the quantum circuit to preserve a one-to-one mapping. ENTANGLEMENT AND BELL STATES Entanglement [8] is a phenomenon in quantum mechanics where two or more particles become correlated in such a way that the state of one particle cannot be described independently of the state of the other. When particles become entangled, measuring the state of one particle instantaneously influences the state of the other, regardless of the distance between them. Einstein called this spooky action at a distance [13]. Let us consider the state || |aa00 01 12 HHH W =++ ||.aa10 11 34 HH + The state can be represented as tensor 15IEEE ANTENNAS & PROPAGATION MAGAZINE FEBRUARY 2025 product of two states, e.g., |(||)01 11 7 HHHab W=+ (| |),01 22 HHab + only if ,aa aa 14 23 = since ,a112 aa= ,,.aaa 2123 12 412 ab ba bb=== Multiple qubits that can be written as tensor product of the individual qubits are said to be separable. Entangled qubits are multiple bits that are not separable. An example is the Bell state basis, also called the EPR (Einstein, Podolsky, Rosen) basis, that forms a basis of maximally entangled states [8] | || ,| || ,| || ,| || . 2 00 11 2 00 11 2 01 10 2 01 10 H HH H HH H HH HHH UUW W = + = - = + = - +-+ - Imagine that Alice and Bob each have one part of an entangled qubit pair, represented by the state / |( || ). 00 11 2 HHH U=+ + Alice’s qubit exists in a state that’s a superposition of both 0 and 1. When Alice measures her qubit using the standard basis {| ,| },01HH she will observe either a 0 or a 1, each with an equal probability of 1/2. Similarly, if Bob measures his qubit, he will observe the same result as Alice. Individually, Alice’s and Bob’s results appear random. However, upon communicating with each other, they would realize that their results, while seeming random on their own, are actually perfectly correlated: Whenever Alice gets a |,0H so does Bob, and whenever Alice gets a |,1H Bob also gets a |,1H both with certainty. Entanglement is an important and surprising feature of quantum systems, which touches the deepest roots of quantum theory and physics and deserves a brief digression. In 1935, in an article titled “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” [13], Einstein, Podolsky, and Rosen argued for the existence of “elements of reality” that were not part of quantum theory, and speculated that it should be possible to construct a theory containing these hidden variables. In the 1960s John Bell published some mathematical expressions that describe the constraints on correlations between the measurements of entangled particles in a classical system [14]. Bell’s theorem shows that certain types of correlations predicted by quantum mechanics cannot be reproduced by classical systems. The fact that quantum mechanics violates Bell’s inequalities indicates that any hidden-variable theory underlying quantum mechanics must be nonlocal. The violation of Bell inequalities has been experimentally confirmed [15], supporting the quantum mechanical view that entanglement involves nonlocal correlations. The fundamental significance of these studies was recognized with the award of the Nobel Prize in Physics in 2022. Entanglement, together with the no-cloning theorem, is at the basis of a large number of possible applications in secure communications [9]. APPLICATION OF QUANTUM COMPUTING IN ANTENNA ANDPROPAGATION PROBLEMS Traditional approaches to solving complex electromagnetic problems often encounter computational bottlenecks due to the large number of variables involved. Quantum computing offers a paradigm shift, promising unprecedented computational power that can revolutionize the analysis. As an example, the ability to process vast amounts of information in parallel enables quantum computers to optimize antenna designs more rapidly than classical computers, potentially leading to the development of more compact, energyefficient, and high-performance antennas [16], [17], [18], [19]. Quantum computing has also been proposed within the framework of reflecting intelligence surfaces [20] and electromagnetic propagation [21]. Many other potentially disruptive applications are possible, as conducting advanced material simulation and synthesis [22], control of the electromagnetic propagation environment [23], [24], [25], developing new electromagnetic devices [26], [27], sensor systems [28], [29], and numerical simulation methods [30], [31], [32]. AN EXAMPLE OF QUANTUM SOLUTION To give a practical illustration and clarify the benefits of quantum computing, let’s consider the problem of searching a large, unordered database. This challenge has been addressed in the context of intelligent control of the electromagnetic environment for 6G applications, as explored in [24], but its relevance extends beyond this specific use case. A notable instance is the real-time inversion of electromagnetic problems involving nonlinearity, a complex issue with significant implications in microwave tomography. Loosely speaking, the set of possible radiated or scattered fields that are distinguishable in the presence of noise is always finite [33]. An exhaustive “lookup table,” i.e., a database that captures all of the input–output relationships, could potentially allow one to obtain the image from the measured field simply by inspecting the “lookup table.” However, the set of distinguishable fields, even if finite, is extremely large, and identifying the solution in an unsorted database with millions of entries is a daunting task using conventional digital computers that require a number O(N) of queries on an unsorted database having N entries. For instance, when N is equal to ,2n where ,n20= the algorithm accesses the database more than half a million times on average before finding the element solution of the problem. An interesting solution is to use Grover’s quantum search algorithm [34], which requires ()ON queries to find the desired entry in an unstructured database having N2n = entries with an almost 100% chance of success. Let us suppose that we have an unsorted database with N2n = entries, and consider a quantum system and an observable having N eigenvalues, each of them associated to a different eigenstate. Let h m be the hth eingenvalue and | xh H the respective eigenstate. We associate the hth eigenstate | xh H to the hth entry of the datatabase. The Gover algorithm uses two operators, the oracle operator, that is a black-box operation that flips the sign of the state if it is associated to the item of the database that is being searched for, and the diffusion operator, that amplifies the probability that the system collapses to the goal state in the measurement process. 16 IEEE ANTENNAS & PROPAGATION MAGAZINEFEBRUARY 2025 To give a simple example of how this controlled dynamical evolution can be implemented in quantum computers, suppose that ,N4= and that the four entries of the database are associated to the four eigenstates |,|,|,|.00 01 10 11HHHH We suppose that the target state associated to the entry we look for is |.11 H In the following, we will call |11H = the superposition of all of the eigenstates but the target eigenstate: |( |||). 11 3 1 00 01 10HH HH =++ = (19) At the beginning of the procedure, the system is prepared in a uniform superposition (Figure 3) [35] |||( ||||). 2 1 00 01 10 11 17HHHH HHH W=+ += +++ (20) The uniform superposition can be obtained by preparing the quantum system in the ||00 0HH W= state and applying a two-qubit Hadamard gate, whose matrix representation is HH7 (Figure 3). With reference to Figure 3, the projection of |1H W onto the goal state |11H is 1/2. The angle /.2i is equal to /.6r The probability that |1H W collapses to |11H is the square of the projection, i.e., /.41 Application of the oracle operator flips the sign of the goal state, i.e., |11H (Figure 3) [35], obtaining the following entangled state: |( |||| ) 2 1 00 01 10 11 2HH HHH W=++-. (21) Finally, the diffusion operator rotates |2H W by .2i Consequently, the angle of |3H U is ///,6232rrr+= and the superposition of states collapses to |11H with 100% probability (Figure 3). In the simple example above (),N4= the wave function collapses in the goal state in only one iteration. More generally, in case of ,N2n = the oracle-diffusion operators rotates the state vector by / (/),N2 - r requiring the iterative application ()ON times, and the access to the database only ()ON times. It turns out that the identification of the desired entry requires only 1,000 queries in a database having N220 = entries. It must be noted, however, that the implementation of Grover’s algorithm on real quantum computers has been limited to a few qubits. A discussion on the implementation of Grover’s algorithm on quantum computers in the case of few bits is reported in [36]. As the number of bits increases, the implementation of Grover’s algorithm becomes rapidly more complex. Furthermore, the oracle must be designed for the specific application, as it encodes the logic of the pattern-matching problem being addressed into Grover’s algorithm. For instance, in the very simple example discussed previously, each of the four possible goal states requires a different oracle. In general, the implementation of the oracle can become a very complex operation. AN INTERFEROMETRIC-BASED SOLUTION Despite challenges in practical implementation, quantum algorithms hold significant potential advantages over nonquantum algorithms, which can provide a substantial edge in solving certain classes of problems. To clarify the advantages, let us examine the same problem using an analog computer capable of exploiting classical interference. The system comprises four input ports, each equipped with a phase shifter, and two interferometers, each with two input ports. Initially, the phase settings of the phase shifters are [, ,,],00rr with the first two serving as inputs for the first interferometer and the last two for the second interferometer, resulting in zero output from both. The element to be identified is marked by altering the phase of its associated phase shifter by .r In our example, in which the target element is the fourth, the phase shifters would be set to [, ,, ].002rr A nonzero output from the second interferometer indicates the subset containing the element. Further interferometric measurement locates the desired element. Extending this method to a list of N elements is straightforward. The method effectively adapts the binary search algorithm for unsorted lists, requiring only O(logN) queries to identify an element and outperforming the quantum approach for large N. However, this O(logN) efficiency in terms of queries comes at an unacceptable cost for large N. In fact, it requires a number of input ports in the interferometer (and, consequently, an amount of energy) that scales as O(N). This makes the solution impractical for lists containing millions of items. θ/2 = n/6 θ/2 = n/6 θ/2 + θ = n/2 1/2 11 11 11 11 11 11 H H H H Z Z H H Oracle Diffuser 0 0 1/2 1 (a) (b) ψ2 ψ3 ψ1 FIGURE 3. Grover’s algorithm for the case where the goal state is 11 .|H (a) Steps of the algorithm. (b) Implementation using quantum gates. H is the Hadamard gate. The controlled-Z gate (C ˆZ gate) is represented by the two small dots connected by a vertical line. Z is the Z ˆ gate. Note that the structure of the oracle depends on the specific goal state. 17IEEE ANTENNAS & PROPAGATION MAGAZINE FEBRUARY 2025 The crucial difference between classical and quantum solutions is that while a classical interferometer device has one physical port associated with each input wave, the computational space of a quantum computer grows exponentially with the number of qubits, resulting in an exponential increase in computational capacity. A QUICK LOOK AT QUANTUM ANNEALING To conclude this brief overview, it is useful to clarify that in addition to the approach based on quantum gates, there is another method known as quantum annealing [4]. This approach could be particularly promising for solving problems within the antennas and propagation community. Quantum annealing is based on the adiabatic theorem of quantum mechanics, which states that a quantum system will remain in its ground state if the Hamiltonian changes slowly enough and there is a sufficient energy gap between the ground state and the first excited state. Consequently, if the system is initially set in the ground state and the Hamiltonian is altered slowly enough, at the end of the time evolution the system will remain in the ground state. This property can be exploited to solve quadratic unconstrained binary optimization (QUBO) problems ()minfx {,}01x n ! (22) where () f xxQx = < is the cost function, x is a binary vector of length n, Q is a symmetric matrix of real coefficients, and the superscript < stands for transpose. Without claiming rigor, we can imagine that f (x) is characterized by a “landscape” with “hills” and “valleys,” containing a large number of local minima. In quantum annealing, the Hamiltonian is carefully constructed so that the “energy landscape” mirrors this landscape. Consequently, if the system is initially set in the ground state and the Hamiltonian is modified slowly enough toward the desired “energy landscape,” by the end of the time evolution, the system will remain in the minimum energy state, thereby solving the QUBO problem. Quantum annealing offers several advantages over traditional annealing algorithms used in nonquantum computers. The minimization process in quantum annealing typically requires only a few microseconds. Although there is no guarantee of achieving the minimum energy solution, due to the relatively fast evolution of the quantum state, several thousand simulations can be conducted in a very short time. This ensures that the resulting solution, if not at the minimum energy, is very close to it. Additionally, quantum annealing can exploit quantum tunneling to navigate across different “valleys” in the energy landscape, a capability not present in annealing algorithms that operate on nonquantum computers. Quantum annealing is primarily used for optimization problems, such as those found in logistics, finance, and materials science. It is less versatile than gatebased quantum computing but can be highly effective for certain applications. While gate-based quantum computing is still at an experimental stage, quantum annealing has seen practical deployment in areas such as optimizing taxi sharing. Many electromagnetic problems can be recast as QUBO minimization, making quantum annealing an intriguing solution. Excellent examples are discussed in [37], where quantum annealing is proposed for optimizing reflective metasurfaces, and in [38], which discusses antenna beamforming using quantum annealing. The articles also describe the experience of implementing this approach on a real quantum computer, highlighting both the challenges and the significant potential of this technology in solving electromagnetic problems. CONCLUSIONS Quantum computing is an important paradigm that allows us to obtain a computational space that increases exponentially with the number of qubits. This feature opens up new perspectives in the solution of particularly complex electromagnetic problems. However, the structure of a quantum program is significantly different from that of conventional programs we are accustomed to. Quantum gates are a nontrivial extension of digital gates and reflect the mechanism of evolution of the quantum states in which information is encoded. The brief and simple introduction to quantum computing presented in this article mainly focused on the similarities between the approach used in microwave engineering and the approach followed in quantum programming, sacrificing mathematical rigor at some points in favor of an intuitive explanation. In particular, in addition to the undeniable advantages, it is important to understand the limits of quantum computing as well. Currently, there is a huge interest in applying quantum computing to practical applications. However, quantum computing is not a panacea for all problems. In simple terms, a quantum computer is essentially a probabilistic Turing machine [39]. Every problem that can be solved using quantum computers can also be solved using digital computers or analog computers that might exploit classical superposition. The challenge for the future of quantum computing lies in identifying problems where the use of entanglement, along with the exponential increase in computational capacity with the number of qubits, proves advantageous over other solutions in terms of time and energy efficiency. This necessitates a reinterpretation of classical problems in quantum terms. In this context, an interesting example is provided in [40], where quantum properties are utilized to simulate the complex dynamics of coupled classical oscillators. This suggests that quantum resources can potentially address computationally intensive problems that are of significant interest in the antenna and propagation community. Entanglement, together with the no-cloning theorem, is at the basis of a large number of possible applications in secure communications. 18 IEEE ANTENNAS & PROPAGATION MAGAZINEFEBRUARY 2025 ACKNOWLEDGMENT This work was supported in part by the European Union’s Horizon Europe Research and Innovation Program (NextGEM) funded by the European Union under Grant 101057527; by the PRIN2022 P20224NMCF, a tomographic ground penetrating radar for humanitary demining applications (TERRAIN); and by the European Union-NextGenerationEU under the project PNRR RESTART (research and innovation on future telecommunications systems and networks, to make Italy more smart) PE_00000001 (CUP D43C22003080001) [MUR Decree n. 34115/03/2022] Cascade Call launched by the SPOKE 3 POLIMI: “SPARKS” project. AUTHOR INFORMATION Marco Donald Migliore ([email protected]) is with the Department of Electrical and Information Engineering “Maurizio Scarano,” University of Cassino and Southern Lazio, 03043 Cassino, Italy. 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