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.

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.

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.

"Over the last few years, research on reversible logic emerged as an important topic in many directions starting from synthesis towards test, debugging and verification as well as arithmetic designs. The motivation behind reversible computation comes from low power dissipation and close relation to quantum circuits, which, in the near future, could become a competitor to current classical circuits. As reversible circuits are still relatively new, the biggest research impact is on synthesis of such circuits. In the first part of this thesis, we present a synthesis approach to realize large reversible circuits based on classical technology mapping. The irreversible nature of most of the original algorithms makes the synthesis of reversible circuits from irreversible specifications a challenging task. A large part of the existing algorithms, although optimized in garbage bits and gate counts, are restricted to small functions, while some approaches address large functions but are costly in terms of gate count, additional lines and quantum cost. A synthesis solution for large circuits with less quantum cost and garbage bits is presented in this thesis by avoiding permutation based reversible embedding.In addition, we present an indirect way of realizing arithmetic circuits avoiding the direct translation of classical truth table with better performance with respect to various reversible parameters. We develop an improved reversible controlled adder/subtractor with overflow detection to enhance reliability. We use this adder/subtractor module with slight modification to implement some complex designs such as reversible square-root circuit, comparator for signed numbers and finally a new integrated module of reversible arithmetic logic unit, which encapsulates most of the operations in classical realization with less number of control lines. This module intends to perform the basic mathematical operations of addition, subtraction with overflow detection, comparison, as well as logic operations AND, OR, XOR and some negated logical functions such as NAND, NOR and XNOR including implication. Thus our design is very efficient and versatile with less number of lines and quantum cost.Apart from synthesis and designs, testing must also be brought onboard to accommodate the reliable implementation of reversible logic. Our final part of the thesis addresses this issue. To date, most reversible circuit fault models include stuck-at-value, missing gate fault and control point faults of Toffoli network. Now-a-days, the synthesis process is not restricted to standard reversible gates, rather some designs especially arithmetic circuits include other gates. In such realization, failures can happen due to erroneous replacements or incorrect cascading of gates, which cannot be defined with existing fault model alone. Thus in this thesis, we present two fault models namely gate replacement fault and wire replacement fault which target circuits implemented using any reversible gate library. To test such faults, three testing schemes are proposed by adopting the conventional testing methods for irreversible circuits based on Boolean Satisfiability (SAT) formulation. In particular, a new Reversible Test Miter is constructed, which, along with backtracking, speed up detection gate and wire replacement faults with less memory. In addition, on a different study, the testing feature of modular reversible design is investigated and presented in this thesis showing that the same test set of basic block is applicable for cascaded design. We hope our effort on synthesis, design and test of reversible circuits will enrich their viable technological realization." --

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.

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.

At this time the synthesis of reversible circuits for quantum computing is an active area of research. In the most restrictive quantum computing models there are no ancilla lines and the quantum cost, or latency, of performing a reversible form of the AND gate, or Toffoli gate, increases exponentially with the number of input variables. In contrast, the quantum cost of performing any combination of reversible EXOR gates, or CNOT gates, on n input variables requires at most O(n2/log2n) gates. It was under these conditions that EXOR-AND-EXOR, or EPOE, synthesis was developed. In this work, the GF(2) logic theory used in EPOE is expanded and the concept of an EXOR-AND product transform is introduced. Because of the generality of this logic theory, it is adapted to EXOR-AND-OR, or SPOE, synthesis. Three heuristic spectral logic synthesis algorithms are introduced, implemented in a program called XAX, and compared with previous work in classical logic circuits of up to 26 inputs. Three linear reversible circuit methods are also introduced and compared with previous work in linear reversible logic circuits of up to 100 inputs.