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Ultimate Algorithm for Quantum Computers

Koji Nagata, Tadao Nakamura

Published Mar 1, 2019• 0 citations
OpenAlex

Abstract

We first propose herein a novel parallel computation, even though today's algorithm methodology for quantum computing, for all of the combinations of values in variables of a logical function. Our concern so far has been to obtain an attribute of some function. In fact such a task is only for one task problem solving. However, we could treat positively the plural evaluations of some logic function in parallel instead of testing the function for finding out its attribute. In fact, these evaluations of the function are naturally included and evaluated, in parallel, in normal quantum computing discussing a function in a Boolean algebra stemmed from atoms in it. As is naturally understandable with mathematics, quantum computing with qubit systems naturally is included and exemplified by a Boolean algebra, which treats only both 0 and 1. Namely, the theory in this paper is quite natural in logical sense even though physics domain. Therefore, quantum computing has an ability to solve some mathematical problems described in a Boolean algebra. The reason why we positively introduce a Boolean algebra here is because we have multiple evaluations of a function in quantum computing general.

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Kuantum hesaplamayla mantıksal fonksiyonların ağırlıklarının ayırt edilmesi.

Uyanık, Kıvanç

In this thesis, we investigate solvability of the weight decision problem of two Boolean functions by quantum computation. In particular, we study this problem first from a general quantum operator discrimination perspective and second from a direct algorithmic viewpoint. As quantum operator discrimination approach is concerned, we give two different formulations for two different cases. In one, the unitary transformations that correspond to the function evaluation are applied in a parallel fashion and in the other, they are applied only sequentially. Since the parallel case can always be simulated with a serial architecture, we put more emphasis on the serial approach and present a superior result in the serial setting. Specifically we show that any protocol with a serial application of p function evaluations interspersed with p − 1 generic unitary operators in between can be uniquely mapped to a density matrix acting on some other Hilbert space. In the direct approach, we generalize Grover’s iteration in such a way that it can be run deterministically for the discrimination of Boolean functions. We show that sure-success weight distinguishability problem of two Boolean functions using a certain number of evaluations can be reduced to the problem of determining whether a point lies inside the convex hull of a curve. This convex analysis problem is further translated into a system of algebraic equations of a single variable. These equations are solved analytically for the case of single and two evaluations. For more evaluations numerical methods are utilized.

•Jan 1, 2014•0 citations
We first propose herein a novel parallel computation, even though today's algorithm methodology for quantum computing, for all of the combinations of values in variables of a logical function. Our concern so far has been to obtain an attribute of some function. In fact such a task is only for one task problem solving. However, we could treat positively the plural evaluations of some logic function in parallel instead of testing the function for finding out its attribute. In fact, these evaluations of the function are naturally included and evaluated, in parallel, in normal quantum computing discussing a function in a Boolean algebra stemmed from atoms in it. As is naturally understandable with mathematics, quantum computing with qubit systems naturally is included and exemplified by a Boolean algebra, which treats only both 0 and 1. Namely, the theory in this paper is quite natural in logical sense even though physics domain. Therefore, quantum computing has an ability to solve some mathematical problems described in a Boolean algebra. The reason why we positively introduce a Boolean algebra here is because we have multiple evaluations of a function in quantum computing general.
Nature is the relations between processes and phenomena. Nothing exists in a Universe without relations. Computing is the transaction of relations between data using control and execution mechanisms. Quantum relations are a superposition of particles and their states. Quantum computing is unconditional transactions of relations between qubit data. Entanglement is a nonlocal superposition of deterministic states. A quantum computer is an analog device for solving combinatorial problems in parallel. Practically focused definitions of the concepts of quantum computing represent a path to the development of scalable quantum parallel algorithms for solving combinatorial problems. Any algorithm can be reduced to a sequence of operations without conditions, because any truth table is a collection of a complete system of conditions-states. Any sequence of actions can always be reduced to one parallel operation. Conditions and sequences arise only when the developer uses previously created primitives-constructs to build an always non-optimal calculator. The deterministic paradigm of creating quantum computing by using photon transactions on the electrons of an atom can exclude the use of quantum logic. The evolutionary path of quantum computing from the classical one: Memory-Address-Transaction (MAT) →\rightarrow→ Electron-Address-Transaction →\rightarrow→Electron-Address-Quantaction (EAQ)→\rightarrow→State-Superposition- Logic. What is calculated on a quantum computer can be calculated in parallel on a classical computer due to the redundancy of memory. MAT-computing implements any algorithms through transactions (read-write) in memory. Qubit-vector models for describing functionalities differ from the known truth tables in terms of compactness of description and technological implementation of parallel algorithms for the synthesis and analysis of digital devices and SoC-components. An example is given – a memory-driven algorithm for simulating digital devices based on qubit-vector forms of describing functionalities for a significant increase in the productivity of computing processes by parallel execution of logical operations. Comparative evaluations of the use of qubit models and methods show an increase in the efficiency of algorithms for simulating digital devices in comparison with tabular ones. The superposition of classical, quantum and analog computing is integrally presented, which makes it possible to search the best solutions for recognition and decision making.
Abstract The fundamental relationship between quantum physics and discrete mathematics is examined. A method for representing Boolean functions in the form of unitary transformations is described. The question of the connection of Zhegalkin polynomials defining the algebraic normal form of a Boolean function with quantum circuits is considered. It is shown that the quantum information language provides a simple algorithm for constructing the Zhegalkin polynomial based on the truth table. The developed methods and algorithms are generalized to the case of an arbitrary Boolean function with a multibit domain of definition and a multibit set of values, as well as to the case of multivalued (kkk-value) logic when k=pk = pk=p is a prime number. The developed approach is important for the implementation of quantum computer technologies and is the foundation for the transition from classical computer logic to quantum hardware.
Here, we propose a new type of quantum algorithm for determining the 2N2^N2N values of a function. By measuring the single output state, we determine all the values of f(x)f(x)f(x) for all xxx simultaneously. This is very interesting indeed: the quantum circuit gives us the ability to determine a perfect property of f(x)f(x)f(x), namely, f(x)f(x)f(x). This is faster than a classical apparatus by a factor of 2N2^N2N.
Quantum computing aims to outperform classical computers for some specific calculations. It relies on the additional resources given by the qubit superposition states, which can be used to design more efficiently. Reversibility of quantum computation may help in solving NP problems, which are easy in one direction but hard in the opposite sense. Quantum computers can enhance data processing and analysis tasks. The potential impact of quantum algorithms on optimization problems, simulation of quantum systems, and machine learning tasks such as clustering and classification. It also includes efforts to standardize quantum hardware and software interfaces, enabling interoperability and accelerating the adoption of quantum computing. The barriers we face while adopting Quantum computing are the accuracy of the information, lack of technical experts, and organizational interest in adopting the new process. We find one more critical barrier is the lack of standards of secure communication techniques while adopting quantum computing. While numerous challenges lie ahead, the promise of quantum computing in solving complex problems and driving innovation is too significant to ignore. We will examine the existing quantum algorithms and applications that have demonstrated quantum advantage or potential utility in various domains. The quantum computing field is expected to transition from physical qubits to error-corrected logical qubits. AI is likely to be unleashed by quantum computing for formidable progress advances in speed, cost, and algorithm learning function. Quantum computing solves one or two problems, with little practical use, and costs a fortune to run. Additionally, we will assess the challenges associated with scaling up quantum systems, addressing the issues of quantum coherence, error correction, and noise reduction.
Quantum computation is one of the very promising new fields of the new millennium for basic research in theoretical computer science and computational mathematics. Quantum computers have stimulated great interests due to their promise of being able to solve problems considered infeasible on current computers. From a computer scientist's and mathematician's point of view, the growth in the capacities of the underlying hardware is vital since an inefficient algorithm can solve the problem with the faster new hardware. For decades, this attitude was irrelevant but the deadline to change it is near. That is why quantum computation has led to rapid developments in physics, mathematics and computer science.This tutorial paper summarizes our year long series of multidisciplinary seminars on the topics of using computational algebra in quantum computation for finding new theoretical models for the upcoming generation of computers together with finding the models' computational abilities.
In this thesis, we investigate solvability of the weight decision problem of two Boolean functions by quantum computation. In particular, we study this problem first from a general quantum operator discrimination perspective and second from a direct algorithmic viewpoint. As quantum operator discrimination approach is concerned, we give two different formulations for two different cases. In one, the unitary transformations that correspond to the function evaluation are applied in a parallel fashion and in the other, they are applied only sequentially. Since the parallel case can always be simulated with a serial architecture, we put more emphasis on the serial approach and present a superior result in the serial setting. Specifically we show that any protocol with a serial application of p function evaluations interspersed with p − 1 generic unitary operators in between can be uniquely mapped to a density matrix acting on some other Hilbert space. In the direct approach, we generalize Grover’s iteration in such a way that it can be run deterministically for the discrimination of Boolean functions. We show that sure-success weight distinguishability problem of two Boolean functions using a certain number of evaluations can be reduced to the problem of determining whether a point lies inside the convex hull of a curve. This convex analysis problem is further translated into a system of algebraic equations of a single variable. These equations are solved analytically for the case of single and two evaluations. For more evaluations numerical methods are utilized.