This book opens the door to a new interesting and ambitious world of reversible and quantum computing research. It presents the state of the art required to travel around that world safely. Top world universities, companies and government institutions are in a race of developing new methodologies, algorithms and circuits on reversible logic, quantum logic, reversible and quantum computing and nano-technologies. In this book, twelve reversible logic synthesis methodologies are presented for the first time in a single literature with some new proposals. Also, the sequential reversible logic circuitries are discussed for the first time in a book. Reversible logic plays an important role in quantum computing. Any progress in the domain of reversible logic can be directly applied to quantum logic. One of the goals of this book is to show the application of reversible logic in quantum computing. A new implementation of wavelet and multiwavelet transforms using quantum computing is performed for this purpose. Researchers in academia or industry and graduate students, who work in logic synthesis, quantum computing, nano-technology, and low power VLSI circuit design, will be interested in this book.

For the first time in book form, this comprehensive and systematic monograph presents methods for the reversible synthesis of logic functions and circuits. It is illustrated with a wealth of examples and figures that describe in detail the systematic methodologies of synthesis using reversible logic.

The LNCS journal Transactions on Computational Science reflects recent developments in the field of Computational Science, conceiving the field not as a mere ancillary science but rather as an innovative approach supporting many other scientific disciplines. The journal focuses on original high-quality research in the realm of computational science in parallel and distributed environments, encompassing the facilitating theoretical foundations and the applications of large-scale computations and massive data processing. It addresses researchers and practitioners in areas ranging from aerospace to biochemistry, from electronics to geosciences, from mathematics to software architecture, presenting verifiable computational methods, findings, and solutions and enabling industrial users to apply techniques of leading-edge, large-scale, high performance computational methods. This, the 24th issue of the Transactions on Computational Science journal, guest edited by Himanshu Thapliyal and Nagarajan Ranganathan, is devoted to the topic of reversible computing. It is comprised of eight selected papers on reversible energy recovery designs, design of reversible logic gates and arithmetic circuits in optical computing, reversible basic linear algebra subprograms, quantum circuit description language, and reversible circuit and logic synthesis.

At first sight, quantum computing is completely different from classical computing. Nevertheless, a link is provided by reversible computation. Whereas an arbitrary quantum circuit, acting on ?? qubits, is described by an ?? × ?? unitary matrix with ??=2??, a reversible classical circuit, acting on ?? bits, is described by a 2?? × 2?? permutation matrix. The permutation matrices are studied in group theory of finite groups (in particular the symmetric group ????); the unitary matrices are discussed in group theory of continuous groups (a.k.a. Lie groups, in particular the unitary group U(??)). Both the synthesis of a reversible logic circuit and the synthesis of a quantum logic circuit take advantage of the decomposition of a matrix: the former of a permutation matrix, the latter of a unitary matrix. In both cases the decomposition is into three matrices. In both cases the decomposition is not unique.

The development of computing machines found great success in the last decades. But the ongoing miniaturization of integrated circuits will reach its limits in the near future. Shrinking transistor sizes and power dissipation are the major barriers in the development of smaller and more powerful circuits. Reversible logic p- vides an alternative that may overcome many of these problems in the future. For low-power design, reversible logic offers signi?cant advantages since zero power dissipation will only be possible if computation is reversible. Furthermore, quantum computation pro?ts from enhancements in this area, because every quantum circuit is inherently reversible and thus requires reversible descriptions. However, since reversible logic is subject to certain restrictions (e.g. fanout and feedback are not directly allowed), the design of reversible circuits signi?cantly differs from the design of traditional circuits. Nearly all steps in the design ?ow (like synthesis, veri?cation, or debugging) must be redeveloped so that they become applicable to reversible circuits as well. But research in reversible logic is still at the beginning. No continuous design ?ow exists so far. Inthisbook,contributionstoadesign?owforreversiblelogicarepresented.This includes advanced methods for synthesis, optimization, veri?cation, and debugging.

This dissertation is devoted to efficient automated logic synthesis of reversible circuits using various gate types and initial specifications. These Reversible circuits are of interest to several modern technologies, including Nanotechnology, Quantum computing, Quantum Dot Cellular Automata, Optical computing and low power adiabatic CMOS, but so far the most important practical application of reversible circuits is in quantum computing. Logic synthesis methodologies for reversible circuits are very different than those for classical CMOS or other technologies. The focus of this dissertation is on synthesis of reversible (permutative) binary circuits. It is not related to general unitary circuits that are used in quantum computing and which exhibit quantum mechanical phenomena such as superposition and entanglement. The interest in this dissertation is only in logic synthesis aspects and not in physical (technological) design aspects of reversible circuits. Permutative quantum circuits are important because they include the class of oracles and blocks that are parts of oracles, such as comparators or arithmetic blocks, counters of ones, etc. Every practical quantum algorithm, such as the Grover Algorithm, has many permutative circuits. These circuits are also used in Shor Algorithm (integer factorization), simulation of quantum systems, communication and many other quantum algorithms. Designing permutative circuits is therefore the major engineering task that must be solved to practically realize a quantum algorithm. The dissertation presents the theory that leads to MP (Multi-Path) algorithm, which is currently the top minimizer of reversible circuits with no ancilla bits. Comparison of MP with other 2 leading software tools is done. This software allows to minimize functions of more variables and with smaller quantum cost that other CAD tools. Other software developed in this dissertation allows to synthesize reversible circuits for functions with "don't cares" in their initial specifications. Theory to realize functions from relational representations is also given. Our yet other software tool allows to synthesize reversible circuits for new types of reversible logic, for which no algorithm was ever created, using the so-called "pseudo-reversible" gates called Y-switches.