Evolution of quantum computer algorithms from reversible operators

One of the current trends in quantum computing is the application of algebraic topological methods in the design of new algorithms and quantum computers, giving rise to topological quantum computing. One of the tools used in it is the Yang-Baxter equation whose solutions are interpreted as universal quantum gates. Lately, more general Yang-Baxter equations have been investigated, making progress as two-spectral equations and Yang-Baxter systems. This paper intends to apply these new findings to the field of topological quantum computation, more specifically, the proposition of the two-spectral Yang-Baxter operators as universal quantum gates for 2 qubits and 2 qutrits systems, obtaining 4x4 and 9x9 matrices respectively, and further elaboration of the corresponding Hamiltonian by the use of computer algebra software Mathematica® and its Qucalc package. In addition, possible physical systems to which the Yang-Baxter operators obtained can be applied are considered. In the present work it is demonstrated the utility of the Yang-Baxter equation to generate universal quantum gates and the power of computer algebra to design them; it is expected that these mathematical studies contribute to the further development of quantum computers

The main purpose of this review is to deal with questions related to the nature of quantum information and particularly with quantum entanglement as an important component of quantum information and computing. We will not discuss here quantum computer algorithms, like the algorithm by Shor, or their advantages and disadvantages. We only cover the material that lies at the foundations of quantum information and computing and epistemological questions. We attempt to connect the famous debate between Einstein and Bohr on quantum entanglement to some of the latest work on qubits and quantum computing and to the mathematical theorems that form the basis of these discussions, particularly the theorem of Bell. We present a critical analysis of this material and hope that this will give the reader a better understanding of what can be said with certainty about the nature of quantum information.

A quantum computer is the computer technology and the micro physical scientists will combine the technology achievements. Quantum computer is the processing of information by using the quantum properties of particle, a new concept quantum laws to control information transmission and processing of computer. Light has the advantages in space time parallel and high-frequency radiation, which makes the optical become the main target of research model the future of computers. This did not prevent the three valued logic calculus is still an important part of the three values of optical computer. It is precisely because the three value of optical code decoder and the three value, importance, ALU logic optical so, in three valued optical computer all the key parts, we firstly research the value of three logic optical light coding, decoder and a value of three.

Quantum computers are analog devices; thus they are highly susceptible to accumulative errors arising from classical control electronics. Fast operation—as necessitated by decoherence—makes gating errors very likely. In most current designs for scalable quantum computers, it is not possible to satisfy both the requirements of low decoherence errors and low gating errors. Here, we introduce a hardware-based technique for pseudo-digital gate operation. We perform self-consistent simulations of semiconductor quantum dots, finding that pseudo-digital techniques reduce operational error rates by more than two orders of magnitude, thus facilitating fast operation.

Compact realizations of reversible logic functions are of interest in the design of quantum computers. Such reversible functions are realized as a cascade of Toffoli gates. In this paper, we present the first exact synthesis algorithm for reversible functions using generalized Toffoligates. Our iterative algorithm formulates the synthesis problem with d Toffoli gates as a sequence of Boolean Satisfiability (SAT) instances. Such an instance is satisfiable if there exists a network representation with d gates. Thus, we can guarantee minimality. In addition to fully specified reversible functions, the algorithm can be applied to incompletely specified functions. For a set of benchmarks experimental results are given.

This paper explores the impact of quantum computing on modern information security technologies, especially the threat to traditional encryption methods. The parallel processing capability of quantum computing and Shor algorithm make traditional encryption methods face the risk of being cracked. To address this challenge, this paper introduces novel cryptographic methods based on quantum computing, including quantum walking (QW) and quantum entanglement based cryptographic schemes, as well as quantum attack-resistant cryptographic algorithms (PQC). The computing principles, advantages and disadvantages of NTRUEncrypt and Lattice-based encryption algorithms are discussed in detail, and the application and prospect of these new encryption methods in information security are analyzed.

The deterministic quantum computing with one qubit (DQC1) is a mixed-state quantum computation algorithm that evaluates the normalized trace of a unitary matrix and is more powerful than the classical counterpart. We find that the normalized trace of the unitary matrix can be directly described by the entangling power of the quantum circuit of the DQC1, so the nontrivial DQC1 is always accompanied with the non-vanishing entangling power. In addition, it is shown that the entangling power also determines the intrinsic complexity of this quantum computation algorithm, i.e., the larger entangling power corresponds to higher complexity. Besides, it is also shown that the non-vanishing entangling power does always exist in other similar tasks of DQC1.

A quantum computation algorithm is demonstrated using single photons and linear optics. Quantum registers are realized by modes and polarization of photons, and the unitary transformations for them are implemented with linear optics. For each quantum computation, the answer to the Deutsch Jozsa problem for any 4 bit digit is given by a single-photon detection signal with a small error rate. The origin of the error is also studied and categorized according to the sources: systematic errors of quantum gates, phase relaxation, amplitude dumping, and misreadout. It is suspected that a demonstration of 11 qubits using linear optics anda single photon with less than a 20 % error rate is achievable by the technique used in the experiment. As a further objective, an idea to construct controlled-NOT gate using quantum phase gate (QPG) given by Chuang et. al., is introduced and experiments on the QPGs are also reviewed.

SUMMARY Quantum computing has attracted considerable attention in recent years because it promises speedups that conventional supercomputers cannot offer, at least for some applications. Though existing quantum computers (QCs) are, in most cases, still too small to solve significant problems, their future impact on domain sciences is already being explored now. Within this context, we present a quantum computing concept for 1-D elastic wave propagation in heterogeneous media with two components: a theoretical formulation and an implementation on a real QC. The method rests on a finite-difference approximation, followed by a sparsity-preserving transformation of the discrete elastic wave equation to a Schrödinger equation, which can be simulated directly on a gate-based QC. An implementation on an error-free quantum simulator verifies our approach and forms the basis of numerical experiments with small problems on the real QC IBM Brisbane. The latter produce simulation results that qualitatively agree with the error-free version but are contaminated by quantum decoherence and noise effects. Complementing the discrete transformation to the Schrödinger equation by a continuous version allows the replacement of finite differences by other spatial discretization schemes, such as the spectral-element method. Anticipating the emergence of error-corrected quantum chips, we analyse the computational complexity of the best quantum simulation algorithms for future QCs. This analysis suggests that our quantum computing approach may lead to wavefield simulations that run exponentially faster than simulations on classical computers.

We introduce a fault-tolerant construction to implement a composite quantum operation of four overlapping Toffoli gates. The same construction can produce two independent Toffoli gates. This result lowers resource overheads in designs for quantum computers by more than an order of magnitude. The procedure uses Clifford operations and 64 copies of the non-Clifford gate $T=\mathrm{exp}[i\ensuremath{\pi}(I\ensuremath{-}{\ensuremath{\sigma}}^{z})/8]$. Quantum codes detect errors in the circuit. When the dominant source of error is $T$-gate failure with probability $p$, then the composite Toffoli circuit has a postselected failure rate of $3072{p}^{4}$ to lowest order.

Certain algorithms for quantum computers are able to outperform their classical counterparts. In 1994, Peter Shor came up with a quantum algorithm that calculates the prime factors of a large number vastly more efficiently than a classical computer. For general scalability of such algorithms, hardware, quantum error correction, and the algorithmic realization itself need to be extensible. Here we present the realization of a scalable Shor algorithm, as proposed by Kitaev. We factor the number 15 by effectively employing and controlling seven qubits and four "cache qubits" and by implementing generalized arithmetic operations, known as modular multipliers. This algorithm has been realized scalably within an ion-trap quantum computer and returns the correct factors with a confidence level exceeding 99%.

Unravelling the source of quantum computing power has been a major goal in the field of quantum information science. In recent years, the quantum resource theory (QRT) has been established to characterize various quantum resources, yet their roles in quantum computing tasks still require investigation. The so-called universal quantum computing model (UQCM), e.g. the circuit model, has been the main framework to guide the design of quantum algorithms, creation of real quantum computers etc. In this work, we combine the study of UQCM together with QRT. We find, on one hand, using QRT can provide a resource-theoretic characterization of a UQCM, the relation among models and inspire new ones, and on the other hand, using UQCM offers a framework to apply resources, study relation among these resources and classify them. We develop the theory of universal resources in the setting of UQCM, and find a rich spectrum of UQCMs and the corresponding universal resources. Depending on a hierarchical structure of resource theories, we find models can be classified into families. In this work, we study three natural families of UQCMs in detail: the amplitude family, the quasi-probability family, and the Hamiltonian family. They include some well known models, like the measurement-based model and adiabatic model, and also inspire new models such as the contextual model that we introduce. Each family contains at least a triplet of models, and such a succinct structure of families of UQCMs offers a unifying picture to investigate resources and design models. It also provides a rigorous framework to resolve puzzles, such as the role of entanglement versus interference, and unravel resource-theoretic features of quantum algorithms.

Quantum Machine Learning (QML) is a new phraseology in the world today. With various upcoming Artificial Intelligence (AI) technologies, everyday data scientists and corporate around the world are trying to harness its power. QML borrows concepts of Quantum Computing and QML algorithms enhance the existing Machine Learning (ML) algorithms by processing datasets more efficiently. Quantum Computers are a powering force driving QML algorithms that boosts computational power and helps analyse data. QML algorithms can be applied in sectors involving number crunching, pattern identification and so on. In the existing world, QML can be a game-changer for organizations that work on identifying product correlations and customer behaviour in a brief time. This paper discusses studies related to principles of Quantum Mechanics, Quantum Computing and QML algorithm. Additionally, it explores opportunities and challenges faced in the field of QML. Lastly, the paper talks about QML applications and compares various available QML toolkits from the market. Future direction and exhaustive survey of papers from various spectrums related to Quantum Computing is presented in this paper.

Modem digital security based on classical cryptographic methods is likely to become obsolete because quantum algorithms such as Shor's and Grover's algorithms will become available. The paper uses research data and experimental findings and methodological analysis to understand new trends in the field. The manuscript examines the path toward post-quantum cryptography together with the necessary tactics to defend future security systems facing quantum-computing technological advancement.

Demonstrations of quantum advantage for certain sampling problems have generated considerable excitement for quantum computing and have further spurred the development of circuit-model quantum computers, which represent quantum programs as a sequence of quantum gates acting on a finite number of qubits. Amongst this excitement, analog quantum computation has become less prominent, with the expectation that circuit-model quantum computers will eventually be sufficient for emulating analog quantum computation and thus rendering analog quantum computation obsolete. In this work we explore the basic requirements for emulating a specific analog quantum computation in the circuit model: the preparation of a biased superposition of degenerate ground states of an Ising Hamiltonian using an adiabatic evolution. We show that the overhead of emulation is substantial even for this simple problem. This supports using analog quantum computation for solving time-dependent Hamiltonian dynamics in the short term and midterm, assuming analog errors can be made low enough and coherence times long enough to solve problems of practical interest.