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Novel methods for reversible logic synthesis and their application to quantum computing

Marek A. Perkowski, Anas N. Al‐Rabadi

Published Jan 1, 2002• 25 citations
OpenAlex

Abstract

The biggest problems in system design today, and in the future, are the high rate of power consumption, and the emergence of quantum effects for highly dense ICs. The real challenge is to design reliable systems that consume as little power as possible and in which the signals are processed and transmitted at very high speeds with very high signal integrity. Current tools are used to design ICs using only classical design methodologies that apply conventional synthesis constraints such as area, delay, and power. As it was proven, physical processes have to be logically reversible in order to reduce (theoretically eliminate) power consumption. Logical reversibility requires that one can obtain the vector of inputs from the vector of outputs (i.e., backward process) as well as the vector of outputs from the vector of inputs (i.e., forward process). Since only ad-hoc methods were used previously for the reversible synthesis of logic functions, and since systematic and efficient reversible logic synthesis methodologies were significantly missing from previous literature, this dissertation provides several original contributions to reversible logic synthesis by providing a set of tools that can be systematically used to synthesize and evaluate logic functions using reversible logic. This includes, among other new results, Reversible Lattice Structures, Reversible Modified Reconstructability Analysis, Reversible Nets, Reversible Decision Diagrams, and Reversible Cascades. To solve the problem of high signal integrity when processing (computing and transmitting) information using extremely high dense circuits one needs to incorporate the physical quantum mechanical effects that are unavoidable in the nano scales. Since quantum circuits are reversible, and since many of the underlying theorems and formalisms for multiple-valued quantum computing were significantly missing from previous literature, new fundamental foundations for such computations had to be established. These new results include, but not limited to, Quantum Chrestenson Operator, new types of Quantum Decision Trees and Quantum Decision Diagrams as efficient representations for quantum computing, new Composite Basis States, new multiple-valued Einstein-Podolsky-Rosen (EPR) Basis States, and new classes of quantum primitives. Initial evaluations and conclusions for the comparative advantages and disadvantages of the new reversible and quantum computing methodologies are also provided.

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The biggest problems in system design today, and in the future, are the high rate of power consumption, and the emergence of quantum effects for highly dense ICs. The real challenge is to design reliable systems that consume as little power as possible and in which the signals are processed and transmitted at very high speeds with very high signal integrity. Current tools are used to design ICs using only classical design methodologies that apply conventional synthesis constraints such as area, delay, and power.
As it was proven, physical processes have to be logically reversible in order to reduce (theoretically eliminate) power consumption. Logical reversibility requires that one can obtain the vector of inputs from the vector of outputs (i.e., backward process) as well as the vector of outputs from the vector of inputs (i.e., forward process). Since only ad-hoc methods were used previously for the reversible synthesis of logic functions, and since systematic and efficient reversible logic synthesis methodologies were significantly missing from previous literature, this dissertation provides several original contributions to reversible logic synthesis by providing a set of tools that can be systematically used to synthesize and evaluate logic functions using reversible logic. This includes, among other new results, Reversible Lattice Structures, Reversible Modified Reconstructability Analysis, Reversible Nets, Reversible Decision Diagrams, and Reversible Cascades.
To solve the problem of high signal integrity when processing (computing and transmitting) information using extremely high dense circuits one needs to incorporate the physical quantum mechanical effects that are unavoidable in the nano scales. Since quantum circuits are reversible, and since many of the underlying theorems and formalisms for multiple-valued quantum computing were significantly missing from previous literature, new fundamental foundations for such computations had to be established. These new results include, but not limited to, Quantum Chrestenson Operator, new types of Quantum Decision Trees and Quantum Decision Diagrams as efficient representations for quantum computing, new Composite Basis States, new multiple-valued Einstein-Podolsky-Rosen (EPR) Basis States, and new classes of quantum primitives.
Initial evaluations and conclusions for the comparative advantages and disadvantages of the new reversible and quantum computing methodologies are also provided.
Emerging quantum computing technologies are poised to replace standard CMOS logic when the exponential size reduction reaches sub-atomic dimensions. Quantum circuits are reversible and therefore promise the potential of computation without energy loss. This research considers computer aided design (CAD) methods for all major aspects of quantum computing circuit design including logic synthesis, simulation, verification, and testing. The technologies we investigate include quantum cell automata (QCA) and general quantum circuits (QC), with significantly more emphasis on the later. The recently introduced quantum multi-valued decision diagram (QMDD) provides an efficient method to represent and simulate quantum (and other classical reversible) circuits. A major contribution of this dissertation is the development of a sift-like minimization as well as structure metrics based minimization techniques for QMDD. We have used the enhanced QMDD to efficiently simulate quantum circuits as well as quantum vectors. Our early investigation of reversible logic is concerned with a virtual implementation that uses a direct translation of relatively complex binary functions into circuits composed of Fredkin reversible gates. This allowed us to project size and speed complexity of complex reversible circuits, although this direct translation approach fails to minimize garbage inputs and outputs. To achieve proper garbage minimization, one must synthesize reversible logic using gate cascades with appropriate optimization methods embedded that attempt to minimize the total number of lines in the circuit. To that end, we developed a novel QMDD-based tool for cascade logic synthesis that utilizes the QMDD minimized variable order for lexicographical synthesis with garbage minimization included as an optimization criterion. We developed a synthesis tool that investigates the QCA native 3-input majority gates ability to implement complex logic circuits. In particular, we explore the benefit of transforming the logic description into exclusive sum of products (ESOP) forms prior to implementation in the majority gates. We survey recent efforts in establishing the foundation for QC testing and fault tolerant QC. We project the potential use of random tests as well as built in self test (BIST) techniques for future QC. A major contribution of this dissertation is the investigation of partially redundant reversible logic. Detection of partially redundant logic within any design, reversible or irreversible, has ramifications for logic synthesis, for design verification, and for design for test (DFT) issues.
Many-valued quantum circuit synthesis of many-valued reversible expansions and fast transforms is introduced in this paper. Since the reduction of power consumption is a major requirement for the circuit design in future technologies, such as in quantum computing, the main features of several future technologies will include reversibility. Consequently, the new quantum circuits can play an important task in the design of future circuits that consume minimal power. In addition, the new quantum circuit methodology is general and can be used to realize any multiple-valued function in the quantum space. It is also shown that the structural generality advantage that results from the new design methodology is opposed with the increased quantum structural complexity disadvantage which may be considered in any further cost-benefit design analysis.
Reversible logic became a promising alternative to traditional circuits because of its applications in emerging technologies such as quantum computing, low-power design, DNA computing, or nanotechnologies. As a result, synthesis of the respective circuits is an intensely studied topic. However, most synthesis methods are limited, because they rely on a truth table representation of the function to be synthesized. In this paper, the authors present a synthesis approach that is based on Binary Decision Diagrams (BDDs). The authors propose a technique to derive reversible or quantum circuits from BDDs by substituting all nodes of the BDD with a cascade of Toffoli or quantum gates, respectively. Boolean functions containing more than a hundred of variables can efficiently be synthesized. More precisely, a circuit can be obtained from a given BDD using an algorithm with linear worst case behavior regarding run-time and space requirements. Furthermore, using the proposed approach, theoretical results known from BDDs can be transferred to reversible circuits. Experiments show better results (with respect to the circuit cost) and a significantly better scalability in comparison to previous synthesis approaches.
After decades of continuous improvements and shrinking feature sizes, the development of conventional computing technologies faces enormous challenges. In particular, power dissipation in today's computer chips becomes crucial. Reversible computation is a promising alternative to these technologies, where power dissipation can be reduced or even eliminated. Furthermore, reversible logic builds the basis for quantum computation - a completely new way of processing which enables to solve certain problems exponentially faster compared to conventional methods. However, the design of reversible and quantum circuits is significantly different. Thus, new methods e.g. for synthesis, optimization, or verification are needed. This paper provides a brief introduction into reversible circuits and their respective design methods that have been proposed within the last years.
With the continuous shrinkage of transistor sizes in very large scale integrated circuits, power consumption forms a serious concern to be tackled. With their ability to allow for zero energy dissipation, reversible circuits have been considered as promising solution to meetup with low power design requirements. Moreover, advances in their synthesis methodologies can be easily applied to quantum circuits due to the inherited reversibility of the latter. Although numerous algorithms have been proposed in the literature to synthesize reversible circuits and map them into their corresponding quantum circuits, the scalability and computational effort of such algorithms form a serious concern when synthesizing large size input functions. Binary Decision Diagram-based synthesis for reversible circuits has shown great evidence in realizing reversible circuits with low quantum cost through exploiting proper reduction rules for smaller graph size. However, the order of the variables in the decision diagram impacts its overall size, and thus, the cost of its corresponding reversible circuit. While several reordering algorithms have been proposed in this manner, their direct impact on the quantum cost has not been considered. In this article, a Binary Decision Diagram-based algorithm for reversible circuit synthesis is proposed to synthesize reversible circuits for a given Boolean function with low quantum cost through exploiting a linearized relationship between the decision diagram size and the corresponding quantum cost. Thereafter, different decision diagram reordering algorithms have been integrated with the proposed algorithm and compared in terms of their impact on the quantum cost. Experimental results show that Genetic Algorithm-based reordering for decision diagram, supported with, cycle crossover, inverse mutation, and tournament selection, results in the least quantum cost of the output circuit if compared with other algorithms due to its property in preserving the nodes of the decision diagram in their near-optimal locations during the optimiation recipe.
Research on reversible circuits has gained significance due to its application in quantum computations and many further areas such as the design of encoders. At the same time, the use of multiple-valued logic gained importance since this reduces the number of required entities in physical systems (e.g. in a future quantum computer). While most research is still conducted in the Boolean domain, there exist only few approaches which realize reversible circuits for multiple-valued logic. Moreover, most of the previously proposed solutions for synthesis of multiple-valued reversible circuits are not scalable and consider ternary (i.e. 3-valued circuits) only. Instead of overcoming these issues by developing new synthesis approaches for multiple-valued reversible circuits from scratch, we propose to utilize the recent accomplishments in the design of Boolean reversible circuits and to generalize them for multiple-valued logic. To this end, we discuss how to generalize Quantum Multiple-valued Decision Diagram based (QMDD-based) synthesis - a synthesis approach for Boolean reversible circuits which has been proven to be scalable and which has been used in several recently developed design flows. The discussions eventually show how to bridge the development gap between Boolean and multiple-valued logic for reversible circuits.