By exploiting the principles of quantum mechanics to manipulate information at the microscopic scale, quantum computers promise to build an emerging technological approach to computation that, even when fully realized, will be fundamentally different from the one that has prevailed since the first days of the information age, namely deterministic operations on bits, the same building block (0 and 1) of digital computers [1]. In quantum computing, information is encoded using two orthogonal states of a microscopic object known as a quantum bit or qubit. Coherently controlled through the action of other quantum systems, they guarantee the success of computation in strongly correlated quantum states. Each qubit’s state is probabilistically determined by wave-like entities.
1. Introduction to Quantum Computing
A quantum state can exist in a coherent superposition of all states. For example, unlike a normal bit that can only be in the state 0 or in the state 1, a qubit can be in the state |ψ〉= cos θ|0〉 + eiφ sin θ|1〉 for any θ∈ [0,π/2] and ϕ∈ [0, 2π]. Having both 0 and 1 simultaneously as its value is called ‘superposition’. After some time of evolution, it is recovered a classical bit with a definite population of 0 and 1. Complex amplitudes encode relative probabilities of finding the qubit in the state 0 or 1, generating interference among all the possible paths (eigenstates of other observables) available at once to the quantum entity.
In addition to this, qubits have a property known as ‘entanglement’. This particular quantum correlation results from the coherent coupling of particles and cannot be described in terms of classical stochastic variables. In this sense, entanglement goes beyond classical Pearson correlation: changing the state of the first qubit also changes the state of the second, no matter how far apart they are. Remote qubits become only classical if one performs a measurement on one of them. This projection of the quantum state instantly collapses the wavefunction of both particles, providing a probabilistic distribution of eigenstates of the measured observable (e.g., they can only be found in |0〉 or in |1〉). This idiosyncrasy of quantum mechanics gives rise to new technologies involving real-time communication with applications in secure cryptography (where only a trusted party can access the information), teleportation (instantaneous transmission of information) and super-dense coding (where one qubit sends two bits of information). Strongly correlated qubits acquire both digital and analog nature that brings quantum computers tremendous computational power [2].
2. Fundamentals of Quantum Mechanics
The behavior of all particles, such as electrons, photons, and atoms, can be described in a simple but powerful way. The idea is that all particles, whether on the macroscopic level of baseballs or on the inaccessible quantum level of electrons, propagate freely until they are observed, whereupon they materialize in a determinate way. Untold wonders of the theory follow from these simple notions [3]. It is good to have a grasp of the main ideas so that the application and limits of the theory can be better understood. However difficult it is to understand, observe, and control quantum mechanics, it is worth the effort because it is the foundation for future computers serving society. Further, computation is intimately related to the very foundations of quantum mechanics.
Two foundational principles underlie the exotic behavior of matter and its representation in quantum theory: 1) Complementarity states that two mutually exclusive pictures, wave and particle, describe every physical system, whether macroscopic, microscopic, or both. The exact nature of the system is unknowable and not simply one picture or the other. The two pictures yield different results for certain kinds of experiments. There is no test that can make one of the two more “real” than the other. The pictures are usually reconciled conceptually in a three-dimensional setting, focusing on the difference in behavior of microscopic quantum systems and massive classical systems [4]. 2) Quantum state is an attribute of a physical system. The observed trajectory of the system does not completely specify its state, nor does the knowledge of the initial state completely determine behavior. Such lack of knowledge is probabilistic in character. Rather, the system is described in the form of a superposition of states and acts as a whole. All activities of the system up until an observation are accounted for by the superposition and the linear transformations associated with it, and these deterministically determine the probabilities of the different possibilities.
2.1. Superposition and Entanglement
In the macroscopic world, objects can seemingly take on one definite property or state at a time, like location, angular momentum, or colour. However, upon examining objects at a microscopic level—atoms, electrons, photons, etc.—a distinct behaviour emerges. Classical mechanics is replaced with a probabilistic theory embodied in quantum mechanics, where quantum systems can take on many properties, and therefore states, simultaneously. Other exotic phenomena arise, such as entanglement, in which the fate of one involved particle depends on the fate of another, regardless of the separation distance [5].
The first extraordinary consequence of superposition is that quantum objects can exist in multiple states simultaneously. For example, in the macroscopic world, a coin is typically either heads-up or tails up, two mutually exclusive states. However, quantum systems can exist as a linear combination of states. For a coin, this could mean being “in” a configuration where there is a certain probability of arriving at either head or tail on measurement, encoding the probabilities in complex values. The resulting physical state, a superposition of states, is the basis for the astonishing parallel processing power of quantum computers. This ability is embodied in the qubit, the quantum analogue of a classical bit. A classical bit is either a zero or one, while a qubit can be modeled in the Bloch sphere, where the poles represent the two states while states on the surface describe superpositions of the two [4].
2.2. Quantum Gates and Circuits
Universal quantum computation relies on the ability to perform general transformations on a finite-dimensional Hilbert space. CNOT gate and single-qubit gates are the main building blocks of quantum circuits that realize unitary operations on two-level systems. Its generalization to N two-level systems is considered general N-qubit gates, where the unitary operation is applied on N qubits, but other qubit systems such as qutrits and hypercubits are also possible. A recipe to exactly realize general multi-qubit gates in terms of CNOT gates and one-qubit gates on arbitrary five-qubit states. By recursively constructing CNOT networks acting on Y, C, and J states, networks to realize arbitrary n-qubit CNOT gate in terms of CNOT gates and one-qubit gates. The circuit decomposition based on arbitrary single-qubit gates and CNOT gates is proven polynomially efficient. A conjecture on the optimality of the De Val-Carce languages based on prior results on CNOT counts and gate complexity is discussed [6].
Quantum circuits, which serve as quantum versions of classical circuits by employing qubits (i.e., quantum bits). A qubit adopts a two-state device realized in either a quantum optics, solid state, or atomic setup. Processing quantum data is largely performed through the application of quantum gates on quantum states; a quantum gate performs unitary transformations on one or more quantum states. A small collection of one- and two-qubit gates is universal, meaning that, using these building blocks, one can realize any unitary transformation on the qubits within a finite error tolerance [7]. Consequently, any quantum algorithm can be written as a quantum circuit composed of a sequence of these gates, where time is treated as a discrete variable. A quantum circuit consists of a parallel combination of quantum gates acting on a number of qubits, which is the quantum analog to a classical circuit composed of logic gates acting on bits.
3. Differences Between Classical and Quantum Computing
The classical digital computer, as developed by Babbage, Turing, von Neumann, and others, manipulated wires and magnetic bits that could each take on one of two values. In that model, computation was carried out using the principle of series of operations on the bits in a predefined sequence. It follows the deterministic principle where the next step of computation is completely determined by the configuration at the moment of computation. On the other hand, in the modern conception of the quantum computer, light quanta is manipulated in such a manner that superpositions of light states (and hence computations) are considered. In that model, a “shift of state” is launched where any single configuration of the light matter is not adequate for the computation. Therefore, the understanding of “state” must be extended. This postulation is given by the quantum theory of light. The light can be in certain non-intuitive states such as coherence and entanglement, enabling this new computation model.
Quantum computing is expected to solve the intractable problems for classical computing. This means that problems whose solution requires exponential time for classical computors can be solved in polynomial time by quantum systems. Here, the NP-complete problems, such as the factorization problem, the traveling salesman problem, and others, are of prime interest. The race is on trying to build physically scalable systems to implement quantum systems (e.g., ion traps, superconducting qubits, quantum dots) and hence quantum computers or quantum devices.
The relation of complexity theory to physical systems indicates that the efficiency of computation must be modeled as a resource such as time, storage, and material used. What can be computed is determined by physics [4]. Indeed, there may be complex decision problems, agrarian to model as computational questions, so problems whose solution requires resources exponentially exceeding those readily available. Now, it is proven that there exists physically possible systems for which generator of any feasible computation can be constructed. However, these systems would be in principle unsuitable for computation because any detail knowledge of the initial condition could not be in practice arranged. In this context, quantum mechanics is of particular interest. Such systems exist where there are no bounds on the time to carry out the computation [8].
4. Quantum Algorithms
[4]. Any conventional algorithm can be simulated on a quantum computer, but most run exponentially slower than the best-known classical algorithms. Some classically intractable problems may be solvable on a quantum computer [9]. Increasing the probability of getting the right answer on a simple kind of quantum computer requires more clever manipulation of the internal quantum state. Here, some early algorithms developed specifically for quantum computers are described, including an algorithm to estimate the number of solutions to an NP complete problem. Proposed some quantum algorithms, in which quantum Fourier transform is used to perform some sub-exponential steps. These algorithms promise higher efficiency than any known classical algorithms on the same problem. An example is discussed for the quantum algorithm to find the period of a function, which has led to a polynomial-time solution of a major cryptographic problem.
4.1. Shor’s Algorithm
One of the most important quantum algorithms is the polynomial-time algorithm for factoring large numbers discovered by Peter Shor in 1994. The discovery of this algorithm was a watershed moment in the quantum computing field. Shor’s algorithm is significant because it is an example of a situation in which quantum mechanics is believed to give a dramatic advantage over classical computation. Shor’s algorithm demonstrates a striking case in which a problem thought to be intractable can be solved efficiently with a practical quantum computer. Large number factorization is of extreme importance in cryptography and security, and the emergence of a feasible quantum computer could thus prove to be transformational for cryptography and security. Until Shor’s algorithm, the focus of scientists developing quantum computers was on simulating physical systems or on using quantum computers as hands-on devices to perform calculations. While exciting, these lines of work did not equal the revolutionary impact of Shor’s algorithm on the domains of cryptography and security [4].
An instructive quality of Shor’s algorithm, beyond its clear computational power, is that it is based on a clear and well-known mathematical problem involving linear algebra (the period-finding problem) and quantum mechanical implementations of that problem [10]. The crux of the matter is that a classical computer needs exponential time (number polynomial in the size of the input) to solve the problem while a quantum computer executes this in polynomial time (number polynomial in the size of the input). It is a good example of the general theme of finding hidden structures in data, already done in the case of Grover’s search algorithm. It also employs and combines known quantum algorithms and ideas both in quantum gate models and quantum Fourier transform. Understanding Shor’s algorithm provides insights into the transformative power of quantum computation.
4.2. Grover’s Algorithm
The search problem on the weighted toy database is formulated, where each database element is endowed with a certain weight. Given this database, a target is blindly searched. First, by employing uniform randomness in the search strategy, it was shown that one must spend at least O(∑ wi/j) queries in the worst case. Such a quantity is referred to as the search complexity since it depends on all weight values. Next, by using the Grover search algorithm in quantum computing, it was shown that the quantity must be augmented by the toughness factor Rmax/Rmin, i.e., O(Rmax/Rmin(∑wi/j)).
The resulting complexity exhibits a non-monotonic dependence on the Mach numbers, i.e., the speedup only occurs for a certain range of Mach numbers. The debate regarding optimality of Grover algorithms has been ongoing for the past 25 years, with many results on whether the search strategy is optimal having been derived. Motivated by the above explorations, weighted means beyond Grover’s original query bounds are analyzed with regard to the quadratic speedup. Using probability-weighted Erdos-Renyi graphs as underlying structures, it was found that Grover-like constraints could be a necessary condition for achieving the quadratic speedup in search times [11].
5. Quantum Hardware
Hardware refers to physical components and devices, such as computers, atomic and molecular devices, superconducting devices and all the operating systems and every day used items built from these systems. Hardware is distinguished from software, which is typically written using high-level programming languages and runs on some hardware. A quantum computer consists of a large number of quantum registers—each made of some physical hardware (molecules, atoms, ions, photons, superconducting circuits, etc.) that store information in discrete quantum states. Qubits are usually a tiny unit of information storage in a quantum computer, holding a state consisting of two bits (|0⟩ and |1⟩) of a classical binary register, plus some combination of the two called a superposition.
A quantum register stores some number n of qubits, which corresponds to potential addresses for up to 2n different classical registers N. A quantum workstation would consist of quantum hardware, with millions of qubits and a very large number of registers, along with the necessary electronics, magnetic confinement systems, or any other techniques required to manage and debug the interaction with an enormous number of classical computers in charge. As a consequence of rules of quantum mechanics, when two quantum systems (in this case, workstations and computers) interact, the total state of the augmented system is in general completely different from the previous ones; from this point of view, quantum states are not “consumed” in transactions, but are much closer to being copied and instantaneously sent [5].
5.1. Qubits and Quantum Registers
The basic building block of quantum computing is the quantum bit, or qubit for short. A qubit is the basic unit of quantum information, similar to a bit, the basic unit of classical digital information, in a digital computer. However, there are essential differences between bits and qubits. A bit is a binary digit that can take values 0 or 1, whereas a qubit is a quantum two-level system that can exist in a superposition of states |0〉 and |1〉 [3]. A qubit registers a wide range of values between 0 and 1 simultaneously, and the set of qubits that can exist in its own independent quantum state is larger than a classical bit register. A quantum register is a controller of the dynamics and memory of a qubit. A quantum register of n qubits can be formed from any n two-level quantum systems. A qubit like γ−Fe2O3 nanoparticles may be seen as catalytic qubits manifesting a many-body nature, and may be viewed as proper quantum computers: logic gates, memory, and parallel/probabilistic processors of information. As a consequence of qubit additivity, two quantum mechanical subsystems may be considered independently without loss of generality. The concept of quantum registers is thus proposed as the basic unit of quantum information [12].
Qubits can be realized using a variety of physical systems, and the most commonly used systems include superconducting qubits, trapped ion qubits, semiconducting qubits, topological qubits, and photonic qubits. Superconducting qubits are lumped element Josephson circuits. There are various degrees of freedom in the qubit realizations, such as phase, charge, voltage, and flux. On the other hand, photonic qubits should be on polarization, position, time-bin, parity, or even nonlocal degrees of freedom. Here, different types of superconducting qubits, especially charge qubits and phase qubits are discussed. In addition, energy levels and Hamiltonian of various types of qubits are introduced. Semiclassical analysis is emphasized to consider Josephson inductances and capacitances. To give a further insight into the qubit behavior, energy potential diagrams are portrayed, where local minima appear, and it is possible to analyze energy eigenvalues, probability distribution of energy eigenstates, tunneling currents, energy spectra, and Anderson localization phenomena.
5.2. Quantum Processors and Quantum Gates
Hardware elements are the basic components which are used to implement the quantum operations and enable the quantum computation [13]. These are also the building elements of the quantum computers. The quantum processors and quantum gates are the hardware elements which are responsible for performing all the necessary operations and computations in a quantum computer [7].
The quantum processor is the physical chip which implements the quantum gates and runs the quantum algorithms. It consists electrode structures that create an effective electromagnetic potential for quantum system manipulation like ions and atoms. The quantum gates are the basic component that enables the manipulation of the quantum information. These gates act on the qubits to rotate their states or to entangle them with controlled probability. A universal quantum computer requires a set of gates which can implement any unitary transformation.
6. Quantum Error Correction
Quantum error correction describes the ideas underlying the protection of stored quantum information against decoherence in a simple two-qubit example. It addresses the nature and functionality of decoherence-free subspaces, the quantum state that frustrates both information loss and attempted eavesdropping (with a possible explanation of preparing it), and problems associated with trying to robustly perform quantum operations and transfer quantum information. To foreshadow the forthcoming discussion of quantum error mitigation, there is no equivalent classical error correction by which inherently noisy algorithms can be compensated with high fidelity and at low overheads [14]. Subsequently, the types of noise that can be corrected and the requisite overheads (in terms of extra qubits, classical processing, and gates) are investigated. A method for estimating the performance of quantum error correction with imperfect components is outlined, including possible drawbacks such as unequal qubit fidelity.
***1. Introduction***. Under the standard quantum computation model, a quantum computer with a number of qubits operates on quantum mechanical variables and may run noisy algorithms. Undesirable interactions with the environment fundamentally limit the precision with which quantum gates can be executed, thereby introducing the possibility of undetected errors accumulating during runs. This is similar to the problem of noisy classical computation, which can generally be solved by error detection and correction. However, the place of quantum information within quantum physics renders quantum errors different, presenting a non-trivial challenge [15]. It is the aim to provide an overview of quantum error correction as it relates to this challenge.
7. Applications of Quantum Computing
Dynamic pricing in e-commerce is a reality, and although websites such as Amazon.com play the game coyly—for example, by inflating prices on machines and browsers that have been seen before—inside the machine, algorithms perform this task broadly and in real time. Large banks employing numerous strategies, such as holding or releasing trading information, evaluate the reactions of thousands of algorithms making transactions on millionths of seconds time scales. Optimizing against similar events is called the “game of life” (ref: 40acac7b-9ed9-4c5f-9b39-ce4123490c7d).
Hamiverse technology is included in the game, owing to its excellent soundness that guarantees costs for all players, all gains dissipated in energy and temperature penalties. It turns out that ensuring optimal search (on the rainbow) is roughly impossible in the presence of environmental couplings; fluctuations inhibit coherent patterns in the game. As dynamics classes range between classical (Markov chain) and quantum (hypercubic Cayley graph) ones, implications on this space of complexity classes are briefly discussed. Expansion properties of the classical landscape impose constraints on the initial arrangement of the players [16]. For instance, linear customization-coarsening times diverge severely as one transits from global synchrony to a local one. The impossible euphoric dynamics living in the past of “Hunger Games” intensifies as newly winning strategies penalize common patterns. Finally, equilibrium states with entangled classical or quantum states/gains of the players are obtained at mean-field levels of description.
7.1. Cryptography and Security
The field of cybersecurity concerns the prevention of unauthorized access to, or attacks on, computer networks or systems providing information or services. Here the threats to security are categorized into four types. The first type is natural disasters causing physical damage to server farms or other technical infrastructure. This is not specifically a threat to computer networks as it applies to any activity relying on such infrastructure, be it banking, telephone networks, the power grid, or information systems. At a higher technical level, ‘hacker’ attacks consist of unauthorized access to information systems or the disruption of their usual services. These attacks may exploit vulnerabilities in hardware or software components of systems, due for instance to bugs or improper configuration of servers providing the system’s services. This class of attacks traditionally falls under the term computer security. Related to these attacks are different types of malware, malicious software designed to infiltrate information systems, track users’ behavior, gather sensitive information (e.g. passwords, credit cards), or disrupt its normal operation (viruses, worms, etc.) [17].
There do currently exist cryptographic systems that could remain secure against computers with quantum hardware. In general implementations they would take advantage of quantum states of light propagating in optical fibers. The simplest and currently most widespread example is a key distribution protocol called BB84. It allows two remote parties to prepare and share a secret random key for use in encrypting messages. It rests on sending pairs of single photons prepared in states chosen at random from a given set of four possibilities. If the states are manipulated by an outsides eavesdropper, they will be perturbed in such a way that upon subsequent measurements the parties’ prediction will, with a certain probability, not be fulfilled. This can be detected and the key discarded. Because no large computer can compute the set of quantum states transmitted, a copy can be only be guessed at, thus preventing the eavesdropper from being able to use the knowledge gained. In a sense it is therefore more secure than classical schemes, under the initial assumption of controlled conditions [18].
7.2. Optimization and Machine Learning
Exponential speedup for an ensemble of QAOA based approximated quantum optimizers is presented, together with a metric to characterize the harmonicity of a landscape that lead to such speedup. The device-independent evaluation of a quantum machine learning algorithm is demonstrated. Any quantum States witnessed by a Bell inequality can be effectively clustered using a 1-Nearest neighbor classifier, that achieves a better performance than any classical approach, provided that the sample size is larger than the dimension of the Hilbert space. The quantum bra-ket representation of data states and a centralized deployable strategy for generating them is introduced. An innovation algorithm for bosonic devices is reported, and its performance in the torchvision dataset for MNIST classification is assessed [19].
The hithero unstudied slowdown for a Quantum QUBO-QAOA optimizer on a tree-structured landscape characterized by strong competition is also addressed. A quantum learning algorithm is developed, to learn the orbits of unistep (generalized) symmetric complex quantum walks [20]. Any orbit in such a set needs survives witness a deterministically, at an exponentially long time. Bandgap disorder in open Quantum graphs, a spectral analysis and consequences for singular propagation of diffusion is also studied.
8. Challenges and Limitations in Quantum Computing
Today, quantum computing is still struggling with its infancy phase [21]. Albeit sound theoretical bases, no quantum computers could be produced to get rid of the hard problems of classical computers or the commonly used classical algorithms implemented on them. Quantum computers are at least exponentially harder to manage on a large scale than classical computers. In cloud services, expected scalability from the provider side may not be achieved with quantum computers, even if they can be used for practical applications [22].
The still high cost production of quantum computers means high costs for quantum cloud computing (QCC) systems too. Traditional cloud services are cheaper than purchasing and maintenance of on-premise computers, therefore it is expected that the same thing could be done with quantum computers too, however this expectation remains uncertain. Recent economic trends showed that maintaining the same QCC cost with traditional cloud services can be hard to achieve simply because of the provided service pace.
9. Future Prospects and Developments
The anticipated advancement of quantum computing with respect to its hardware and software facets is elaborated upon, starting from the present day and going on for the next decade. Apart from the technology developments, the evolving players in the quantum ecosystem, the business models undertaken, and the potential innovations and applications of quantum computing are also deliberated on, arriving at a conclusion of what may be the expectations of quantum computing by the end of 2030.
Mainly because today’s quantum computers are still in their early stage of development, so-called NISQ computers. Even though these devices have a small number of quantum bits, results of experiments already exhibit noise, inhomogeneity of qubit parameters and decoherence. Conducted experiments mainly use linear optics systems, where the input state is generally a coherent state and the output state is a number Fock state of a predetermined size. Proofs of the ability of devices to produce such outputs as a consequence of nonclassical behavior have been presented. Concerns about the computational complexity of simulating NISQ devices using classical teams have also been raised. Upcoming quantum nanoscience experiments with superconducting qubits are predicted to outperform the best classical computers nu. Quantum speedup could be experimentally demonstrated even for shallow quantum circuits, reaching sizes unreachable with the best classical simulation. Up to the best authors’ knowledge, no implementation is planned at such a level of complexity and dimension, that would provide an evidence against classicality [8].
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