D. Michael Miller

Multiple-Valued Logic:Concepts and Representations

Abstract Multiple Valued Logic: Concepts and Representations begins with a survey of the use ofmultiple-valued logic in several modern application areas including electronic design automation algorithms and circuit design. The mathematical basis and concepts of various algebras and systems of multiple valued logic are provided including comparisons among various systems and examples of their application. The book also provides an examination of alternative representations of multiple-valued logic suitable for implementation as data structures in automated computer applications. Decision diagram structures for multiple valued applications are described in detail with particular emphasis on the recently developed quantum multiple valued decision diagram.Table of Contents: Multiple Valued Logic Applications / MVL Concepts and Algebra / Functional Representations / Reversible andQuantum Circuits / Quantum Multiple-Valued Decision Diagrams / Summary / Bibliography

Spectral Techniques in VLSI CAD

Preface. 1. Introduction. 2. The Boolean Domain. 3. The Spectral Domain. 4. Decisions Diagrams. 5. Computation of Spectral Coefficients. 6. BDD Minimization. 7. Logic Synthesis. 8. Logic Verification. 9. Concluding Remarks. References. Index.

Reducing Reversible Circuit Cost by Adding Lines

Additional lines are required to implement an irreversible function as a reversible circuit. The emphasis, particularly in automated synthesis methods, has been on using the minimal number of additional lines. In this paper, we show that circuit cost reductions can be achieved by adding additional lines. We present an algorithm for line addition that can be targeted to reducing the quantum cost of a circuit or the transistor count for a CMOS implementation. Experimental results show that the cost reduction can be significant even if (1) only a small number of lines (even one) is added and (2) other circuit optimizations have already been applied.

Elementary Quantum Gate Realizations for Multiple-Control Toffoli Gates

A new method for determining elementary quantum gate realizations for multiple-control Toffoli (MCT) gates is presented. The realization for each MCT gate is formed as a composition of realizations of smaller MCT gates. A marking algorithm which is more effective than the traditional moving rule is used to optimize the final circuit. The main improvement is that the resulting circuits make significantly better use of ancillary lines than has been achieved in earlier approaches. Initial results are also presented for circuits with nearest-neighbour communication. These results show that the overall approach is not as effective for that problem indicating that research on direct synthesis of nearest-neighbour quantum circuits should be considered. While, the results presented are for the NCV quantum gate library (i.e. for quantum circuits composed of NOT gates, controlled-NOT gates, and controlled-V=V+ gates), the approach can be applied to other libraries of elementary quantum gates.

Decision Diagram Techniques for Micro- and Nanoelectronic Design Handbook

FUNDAMENTALS OF DECISION DIAGRAM TECHNIQUES Introduction Data Structures Graphical Data Structures AND-EXOR Expressions, Trees, and Diagrams Arithmetic Representations Word-Level Representations Spectral Techniques Information-Theoretical Measures Event-Driven Analysis DECISION DIAGRAM TECHNIQUES FOR SWITCHING FUNCTIONS Introduction Classification of Decision Diagrams Variable Ordering in Decision Diagrams Spectral Decision Diagrams Linearly Transformed Decision Diagrams Decision Diagrams for Arithmetic Circuits Edge-Valued Decision Diagrams Word-Level Decision Diagrams Minimization via Decision Diagrams Decision Diagrams for Incompletely Specified Functions Probabilistic Decision Diagram Techniques Power Consumption Analysis using Decision Diagrams Formal Verification of Circuits Ternary Decision Diagrams Information-Theoretical Measures in Decision Diagrams Decomposition Using Decision Diagrams Complexity of Decision Diagrams Programming of Decision Diagrams DECISION DIAGRAM TECHNIQUES FOR MULTIVALUED FUNCTIONS Introduction Multivalued Functions Spectral Transforms of Multivalued Functions Classification of Multivalued Decision Diagrams Event-Driven Analysis in Multivalued Systems SELECTED TOPICS OF DECISION DIAGRAM TECHNIQUES Introduction Three-Dimensional Techniques Decision Diagrams in Reversible Logic Decision Diagrams on Quaternion Groups Linear Word-Level Decision Diagrams Fibonacci Decision Diagrams Techniques of Computing via Taylor-Like Expansions Developing New Decision Diagrams Historical Perspectives and Open Problems APPENDICES Appendix A: Algebraic Structures for the Fourier Transform on Finite Groups Appendix B: Fourier Analysis on Groups Appendix C: Discrete Walsh Functions Appendix D: The Basic Operations for Ternary and Quaternary Logic INDEX

Realizing reversible circuits using a new class of quantum gates

Quantum computing offers a promising alternative to conventional computation due to the theoretical capacity to solve many important problems with exponentially less complexity. Since every quantum operation is inherently reversible, the desired function is often realized in reversible logic and then mapped to quantum gates. We consider the realization of reversible circuits using a new class of quantum gates. Our method uses a mapping that grows at a very low linear rate with respect to the number of controls. Results show that, particularly for medium to large circuits, our method yields substantially smaller quantum gate counts than do prior approaches.

Transforming MCT circuits to NCVW circuits

Mapping a circuit of reversible gates to a circuit of elementary quantum gates is a key step in synthesizing quantum realizations of Boolean functions. The library containing NOT, controlled-NOT and controlled square-root-of-NOT gates has been considered extensively. In this paper, we extend the library to include fourth-root-of-NOT gates. Experimental results using REVLIB benchmark circuits show that using this extended library results in smaller quantum circuits.

Embedding of Large Boolean Functions for Reversible Logic

Reversible logic represents the basis for many emerging technologies and has recently been intensively studied. However, most of the Boolean functions of practical interest are irreversible and must be embedded into a reversible function before they can be synthesized. Thus far, an optimal embedding is guaranteed only for small functions, whereas a significant overhead results when large functions are considered. We study this issue in this article. We prove that determining an optimal embedding is coNP-hard already for restricted cases. Then, we propose heuristic and exact methods for determining both the number of additional lines and a corresponding embedding. For the approaches, we considered sum of products and binary decision diagrams as function representations. Experimental evaluations show the applicability of the approaches for large functions. Consequently, the reversible embedding of large functions is enabled as a precursor to subsequent synthesis.

Synthesizing Reversible Circuits for Irreversible Functions

Many reversible circuit synthesis procedures have been proposed. A common feature of most methods is that the initial specification must be a completely-specified reversible function. However, often the desired functionality is a, possibly incompletely-specified, irreversible function. In this paper, we consider how to fully automate the process of synthesizing a reversible function given an irreversible specification with particular emphasis on how to embed an irreversible function into a reversible specification. Systematic procedures are presented and results for benchmark problems show the methods produce very good results compared to earlier methods.

Lowering the Quantum Gate Cost of Reversible Circuits

One approach to determining a quantum circuit is to first synthesize a circuit composed of binary reversible gates and to then map that circuit to an equivalent quantum gate realization. This paper considers the mapping phase with the goal of reducing the number of quantum gates required. Our method is based on novel line labeling and gate moving procedures. Results are presented for the quantum library: NOT, controlled-NOT, and the square-root-of-NOT gates (V and V+). The approach is applicable to other quantum gate libraries.