Quantum computing makes use of wave-function superposition to provide a computational system with remarkable possibilities. A computing bit is a 0 or 1, but a quantum bit, a qubit, is a superposition of two quantum states, of, say, an atom. A number n of qubits will then be in a superposition of 2n distinct states. An operation on such a state then does 2n simultaneous operations, demonstrating quantum parallelism. The possibilities of quantum computing are demonstrated via discussion of Deutch’s problem, Shor’s period-finding algorithm, and Grover’s search algorithm. Shor’s algorithm might allow the factoring of large numbers, which might enable the decoding of many encryption schemes. Details are given for the solution of Deutch’s problem. The analysis introduces certain operators, the NOT and Hadamard, that are used in computer logic.
Computers today become faster by becoming smaller. By reducing the size of transistors, engineers are able to fit more of them on a given size microprocessor, thus increasing the processor's computational power. This process cannot continue forever. It has been estimated that sometime within the next two decades, at the current rate, engineers will be faced with the problem of building something that is smaller than an atom. This is as far as our current "classical" computing paradigm will take us. Quantum computing is a potential solution to this problem. This paradigm of computing seeks to directly exploit quantum mechanical phenomena to perform calculations or in some way boost computational efficiency. Some problems can theoretically be solved on a quantum computer exponentially faster than on a classical computer. An overview of the subject is provided here with emphasis on quantum information processing and physical realizations of a quantum computer. Included is a brief history of the subject, a discussion of quantum computing notions such as parallelism and entanglement and their use in quantum algorithms, and conjecture on the prospects and pitfalls of this theory.
The Deutsch-Jozsa algorithm is one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm. It was proposed by Deutsch and Jozsa in 1992 with improvements by Cleve, Ekert, Macchiavello, and Mosca in 1998. The Deutsch-Jozsa problem is a promise problem and we can equivalently describe it as a partial function ${\text{DJ}}_{n}^{0}:{{0,1}}^{n}\ensuremath{\rightarrow}{0,1}$ defined as
DJn0(x)=1 for
∣x∣=n/2,DJn0(x)=0 for
∣x∣=0,n, and it is undefined for the rest of the cases, where
n is even, and
∣x∣ is the Hamming weight of
x. The optimal quantum algorithm needs only one query to compute
DJn0 but the classical deterministic algorithm requires ${2}^{n\ensuremath{-}1}+1$ queries to compute it in the worse case. In this article, we generalize the Deutsch-Jozsa problem as
DJnk(x)=1 for
∣x∣=n/2,DJnk(x)=0 for
∣x∣ in the set ${0,1,...,k,n\ensuremath{-}k,n\ensuremath{-}k+1,...,n}$, and it is undefined for the rest of the cases, where $0\ensuremath{\le}k<n/2$. In particular, we give and prove an optimal exact quantum query algorithm with complexity
k+1 for computing the generalized Deutsch-Jozsa problem
DJnk. It is clear that the case of
k=0 is in accordance with the Deutsch-Jozsa problem. Also, we give a method for finding the approximate and exact degrees of symmetric partial Boolean functions.
Quantum computing takes fully advantage of the superposition principle to increase greatly (even exponentially) the speed of calculations, relative to the classical approach. The Deutsch-Jozsa algorithm is the simplest quantum algorithm illustrating this power. Unfortunately, the standard derivation involves several ingenious steps which usually leave students feeling that they could never have figured out the algorithm by themselves. We present here a different formulation of the problem which allows students to derive the algorithm using only basic knowledge of quantum mechanics and linear algebra.
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)
→ Electron-Address-Transaction
→Electron-Address-Quantaction (EAQ)
→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.
Quantum Computation is an interdisciplinary field formed by the merge of quantum effect, information theory and algorithms. The future of computing lies with the quantum computers which may be a hypothetical paradigm but surely will be a reality very soon.This book describes about some quantum algorithms and programs which will prove as the foundation of quantum computers in the years to come.In quantum computers we exploit quantum effects to compute in ways that are faster or moreefficient than, or even impossible, on conventional computers. Quantum computers use aspecific physical implementation to gain a computational advantage over conventionalcomputers. Properties called superposition and entanglement may, in some cases, allow anexponential amount of parallelism. Also, special purpose machines like quantumcryptographic devices use entanglement and other peculiarities like quantum uncertainty.Quantum computing combines quantum mechanics, information theory, and aspects ofcomputer science. The field is a relatively new one that promises secure data transfer,dramatic computing speed increases, and may take component miniaturization to itsfundamental limit.