Quantum Computing Fundamentals: Beyond Classical Bits

Since its inception by Richard Feynman in 1982, quantum computing has provided an intriguing opportunity to advance computational capabilities over classical computing. Classical computers use bits to process information in terms of zeros and ones. Quantum computers use the complex world of quantum mechanics to carry out calculations using qubits (the quantum analog of a classical bit). The qubit can be in a superposition of the zero and one state simultaneously unlike a classical bit. The true power of quantum computing comes from the complexity of entanglement between many qubits. When entanglement is realized, quantum algorithms for problems such as factoring numbers and solving linear algebra problems show exponential speed-up relative to any known classical algorithm. Linear algebra problems are of particular interest in engineering application for solving problems that use finite element and finite difference methods. Here, we explore quantum linear algebra problems where we design and implement a quantum circuit that can be tested on IBM’s quantum computing hardware. A set of quantum gates are assimilated into a circuit and implemented on the IBM Q system to demonstrate its algorithm capabilities and its measurement methodology.

Probabilistic bits, or p‐bits, represent a novel computational element that bridges the gap between deterministic classical bits and quantum bits (qubits) used in quantum computing. Unlike classical bits that maintain a definite state of 0 or 1, a p‐bit fluctuates between these states in a controlled, stochastic manner. This probabilistic behavior allows for the representation and processing of information in a form that leverages inherent randomness. In this study, a unique approach is presented to generating p‐bits using a hybrid conjugated polymer, poly‐4‐((6‐(4H‐dithieno[3,2‐b:2',3'‐d]pyrrol‐4‐yl)hexyl)oxy)‐N,N‐diphenylaniline (pTPADTP), as a memristive material. The polymer's conjugated backbone, combined with pendant triphenylamine groups, enables the creation of p‐bits through random resistance switching. The stochasticity of this polymeric memristor makes it particularly suited for p‐bit applications in stochastic optimization, probabilistic algorithms, and artificial neural networks. The charge transport in the polymer is facilitated by two synergistic percolation mechanisms: one occurring along the polymer backbone and the other through the pendant triphenylamine groups. The study of p‐bits generated from pTPADTP opens new avenues for advancing both the theory and practice of computation, where uncertainty and randomness are harnessed as valuable computational resources.

Fundamental transformations in the basic logic of computing are few and far between. Since the invention of digital computers in the early 1940s, the logic underlying computation has remained the same, even as computing hardware evolved from vacuum tubes to silicon transistors. With the advent of quantum computation, a fundamental transformation is near. Quantum computation is based on a different type of logic: rather than being in one of the two states of a classical bit, a quantum bit or qubit can be in a superposition of two states simultaneously. Operations and measurements on these qubits obey the constraints of quantum mechanics. It is now understood that quantum computers have great power in principle to go beyond classical computers, but that not every application is well suited to implementation on quantum computers. For reasons explained in more detail in the Introduction, scientific problems in chemical and materials sciences are uniquely suited to take advantage of quantum computing in the relatively near future. Indeed, quantum computing offers the best hope to solve many of the most important and difficult problems in this field. For example, quantum materials, such as superconductors and complex magnetic materials, show novel kinds of ordered phases that are natural from the point of view of quantum mechanics but difficult to access via computation on classical computers. Quantum sensors based on solid materials are already widely used but could be greatly improved with insight from quantum computations, as could materials for information technologies. Quantum chemical dynamics is another example of a problem that is intrinsically well suited to studies on quantum computers. Applications of quantum chemical dynamics include catalysis, artificial photosynthesis, and other industrially important processes. Quantum computers exist in the laboratory and are beginning to exceed 50 qubits, which is roughly the size beyond which their behavior cannot be predicted or emulated on present-day classical supercomputers. While a quantum computer of 50 qubits is almost certainly not powerful enough to tackle the major scientific challenges in chemical and materials sciences, some of these major challenges start to become accessible with a few hundred qubits if error rates can be kept small. This roundtable was convened to ask how emerging quantum computers can be applied to major scientific problems in chemical and materials sciences, in light of Basic Energy Sciences’s leading role in these fields and the Department of Energy’s leading role in high-performance scientific computation more generally. The main outcome of the roundtable was a consensus that there are scientific problems of great importance on which emerging quantum computers have the potential for disruptive impact, and where comparable progress is unlikely to occur by other means.

A quantum computer based on the manipulation and detection of coherent states of electrons localized above a helium film is described. Each quantum logic element (qubit) is made of a combination of the ground and first excited states of an electron trapped in the image potential well at the surface. Potentials applied to micro-electrodes located beneath each electron confine the lateral motion of the electron and are used to operate gates. Mechanisms for one- and two-qubit logic operations and a readout of the final state are discussed. The principle decay channel for the excited state is via the emission of one phonon into the bulk liquid. Dechoherence times are calculated to be of order of 100 μs and allow greater than 10⁴ serial operations.

t is remarkable that today's computers, after the tremendous development during the last 50 years, are still essentially described by the mathematical model formulated by AIan Turing in the 1930's. Turing's model describes computers which operate according to the laws ofclassical physics. What would happen ifa computer was operating according to the quantum laws? Physi­ cists and computer scientists have been interested in this question since the early 1980's, but research in quantum computation real­ ly started to flourish after 1994 when Peter Shor discovered a quantum algorithm to find prime factors oflarge integers effi­ ciently, a problem which is intrinsically hard for any classical computer (see (1) for an introduction into quantum computa­ tion). The lack ofan algorithm for efficient factoring on a classical machine is actually the basis ofthe widely used RSA encryption scheme. Phase coherence needs to be maintained for a sufficient­ ly long time in the memory of a quantum computer. This may sound like a harmless requirement, but in fact it is the main rea­ son why the physical implementation of quantum computation is so difficult. Usually, a quantum memory is thought of as a set oftwo-level systems, named quantum bits, or qubits for short. In analogy to the classical bit, two orthogonal computational basis states 10) and 11) are defined. The textbook example of a quan­ tum two-level system is the 1/2 of, say, an electron, where one can identify the spin up state with 10) and the spin down state with \1). While several other two-level systems have been proposed for quantum computing, we will devote the majority of our discussion to the potential use ofelectron spins in nanostruc­ tures (such as quantum dots) as qubits. Shor's fadoring algorithm We return to Shor's algorithm, since it allows us to explain many important concepts. At the heart of it lies the quantum Fourier transform (QFT). Given n qubits with an orthonormal basis 10), ...,12-1), the QFT is a unitary2x2 matrix UQFTsuch that

Quantum computers have been predicted to be exponentially faster than their classical counterparts for some computations, such as the factoring of large numbers. (See the article by Charles Bennett in PHYSICS TODAY, October 1995, page 24.) Whereas classical computers perform operations on information stored as classical bits, which can be in one of two discrete states, quantum computers perform operations on quantum bits, or “qubits,” which can be put into any superposition of two quantum states.

Quantum computers use processes of a quantum nature manifested with atoms, molecules, molecular clusters, etc. The description of such processes is based on the application of complex numbers and complex matrices. As is well known, the basic notion of classical information theory is a bit [1]. A classical bit takes the values 0 or 1 (and no other). A qubit (quantum bit) is the smallest element that executes the information storage function in a quantum computer [2].

We introduce nebit, a classical bit with a signed probability distribution. We study its properties and basic transformations that can be applied to it. Then, we introduce a simple dynamical model – a classical random walk supplemented with nebits. We show that such a model exhibits some counterintuitive non-classical properties and that it can achieve or even exceed the speedup of Grover’s quantum search algorithm. The proposed classical dynamics never reveals negativity of nebits and thus we do not need any operational interpretation of negative probabilities. We argue that nebits can be useful as a measure of non-classicality as well as a tool to find new quantum algorithms.

Due to the great difficulty in scalability, quantum computers are limited in the number of qubits during the early stages of the quantum computing regime. In addition to the required qubits for storing the corresponding eigenvector, suppose we have additional k qubits available. Given such a constraint k, we propose an approach for the phase estimation for an eigenphase of exactly n-bit precision. This approach adopts the standard recursive circuit for quantum Fourier transform (QFT) in [R. Cleve and J. Watrous, Fast parallel circuits for quantum fourier transform, Proc. 41st Annual Symp. on Foundations of Computer Science (2000), pp. 526–536.] and adopts classical bits to implement such a task. Our algorithm has the complexity of O(n log k), instead of O(n2) in the conventional QFT, in terms of the total invocation of rotation gates. We also design a scheme to implement the factorization algorithm by using k available qubits via either the continued fractions approach or the simultaneous Diophantine approximation.

Quantum bit commitment (QBC) is a quantum version of the classical bit commitment security primitive. As other quantum security primitives and protocols, QBC improves on cheating detection over its classical counterpart. The implementation of the QBC protocol below relies on the use of common quantum gates: the Hadamard gate used for orthonormal bases and the CNOT gate used to swap qubits. The protocol was run and tested on IBM quantum experience (IBM QX). IBM QX offers two different quantum environments: as a simulator and as a real quantum machine. In our implementation, honest and dishonest participants were considered. Results of both the simulation and the quantum execution were compared against the theoretical expectations. The IBM QX simulator gives results that match the theoretical model. The IBM QX real computer deviates from the expected behavior by a measurable amount. Using the standard deviation and the Hamming distance, the conclusion is that the quantum computer is usable as the difference to the simulator is within an acceptable margin of error. The QBC protocol of choice is fully secure against cheating by Bob. The only way Alice can cheat is using multi-dimensional entanglement. The cost for Alice to cheat is exponential in the number of qubits used, namely $$ O(2^{6n + 3k + 1} ) $$.

We show that a class of quantum computations that was recently shown to be efficiently simulatable on a classical computer by Valiant [in Proceedings of the 33rd ACM Symposium on the Theory of Computing (2001), p. 114] corresponds to a physical model of noninteracting fermions in one dimension. We give an alternative proof of his result using the language of fermions and extend the result to noninteracting fermions with arbitrary pairwise interactions, where gates can be conditioned on outcomes of complete von Neumann measurements in the computational basis on other fermionic modes in the circuit. This last result is in remarkable contrast with the case of noninteracting bosons where universal quantum computation can be achieved by allowing gates to be conditioned on classical bits [E. Knill, R. Laflamme, and G. Milburn, Nature (London) 409, 46 (2001)].

Machine learning tasks are an exciting application for quantum computers, as it has been proven that they can learn certain problems more efficiently than classical ones. Applying quantum machine learning algorithms to classical data can have many important applications, as qubits allow for dealing with exponentially more data than classical bits. However, preparing the corresponding quantum states usually requires an exponential number of gates and therefore may ruin any potential quantum speedups. Here, we show that classical data with a sufficiently quickly decaying Fourier spectrum after being mapped to a quantum state can be well-approximated by states with a small Schmidt rank (i.e., matrix-product states) and we derive explicit error bounds. These approximated states can, in turn, be prepared on a quantum computer with a linear number of nearest-neighbor two-qubit gates. We confirm our results numerically on a set of 1024×1024-pixel images taken from the `Imagenette' and DIV2K datasets. Additionally, we consider different variational circuit ansätze and demonstrate numerically that one-dimensional sequential circuits achieve the same compression quality as more powerful ansätze.

This paper initiates a systematic study of quantum functions, which are (partial) functions defined in terms of quantum mechanical computations. Of all quantum functions, we focus on resource-bounded quantum functions whose inputs are classical bit strings. We prove complexity-theoretical properties and unique characteristics of these quantum functions by recent techniques developed for the analysis of quantum computations. We also discuss relativized quantum functions that make adaptive and nonadaptive oracle queries.

We propose the first correct special-purpose quantum circuits for preparation of Bell diagonal states (BDS), and implement them on the IBM Quantum computer, characterizing and testing complex aspects of their quantum correlations in the full parameter space. Among the circuits proposed, one involves only two quantum bits but requires adapted quantum tomography routines handling classical bits in parallel. The entire class of Bell diagonal states is generated, and several characteristic indicators, namely entanglement of formation and concurrence, CHSH non-locality, steering and discord, are experimentally evaluated over the full parameter space and compared with theory. As a by-product of this work, we also find a remarkable general inequality between “quantum discord” and “asymmetric relative entropy of discord”: the former never exceeds the latter. We also prove that for all BDS the two coincide.

Photon is established as a strong and promising candidate in all-optical signal processing and superfast computing. Again, it is found that photon has a successful approach to be used as a quantum mechanical particle. For this reason, it is used as a carrier of information in optical systems instead of electrons in electronic systems. All-optical quantum systems can provide very high speed, secured and noise free communication and computation. Over the last few decades, several all-optical logic processors, algebraic processors, logic gates and their integrated systems are developed by many researchers. All-optical quantum logic gates with qubit data are the basic building block of an optical quantum computer. In quantum computation, qubits or quantum bits are the quantum analogue of classical boolean bits and are used to store information. Several popular encoding techniques (like phase encoding, frequency encoding, intensity encoding, polarization encoding etc.) are used to develop the qubits. Here, in this paper, the gate matrix for quantum optical tristate CNOT logic gate is developed for the first time with a frequency encoding technique. Quantum optical CNOT gate is one of the most important logic gates in the logic family that is extensively used for performing information processing task in optical systems. Tristate-based optical logic operations are proved to increase the data handling capacity. Also, the speed of different types of optical logic operations can be increased with tristate-based optical logic systems. Introduction of contradiction state is another advantage of tristate logic systems. Frequency encoding principle is the best way to encode tristate, quaternary state and multivalued logic states. In the article, the authors have developed a proper truth table and gate matrix of frequency encoded tristate CNOT gate. The all-optical scheme for tristate CNOT gate can be implemented with semiconductor optical amplifier (SOA)-based optical switches. At the same time advantages of quantum optical operations are exploited.KeywordsSemiconductor optical amplifierTristate logicFrequency encoding principle