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Asia and South Pacific Design Automation Conference (ASP-DAC)

Fast synthesis of exact minimal reversible circuits using group theory

Guowu Yang, Xiaoyu Song B, W.N.N. Hung, Marek A. Perkowski

Published Jul 19, 2005• 5 citations
Publisher

Abstract

We present fast algorithms to synthesize exact minimal reversible circuits for various types of gates and costs. By reducing reversible logic synthesis problems to group theory problems, we use the powerful algebraic software GAP to solve such problems. Our algorithms are not only able to minimize for arbitrary cost functions of gates, but also orders of magnitude faster than the existing approaches to reversible logic synthesis. In addition, we show that the Peres gate is a better choice than the standard Toffoli gate in libraries of universal reversible gates.

DOI
10.1109/aspdac.2005.1466509

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We present fast algorithms to synthesize exact minimal reversible circuits for various types of gates and costs. By reducing reversible logic synthesis problems to group theory problems, we use the powerful algebraic software GAP to solve such problems. Our algorithms are not only able to minimize for arbitrary cost functions of gates, but also orders of magnitude faster than the existing approaches to reversible logic synthesis. In addition, we show that the Peres gate is a better choice than the standard Toffoli gate in libraries of universal reversible gates.
We present new algorithms to synthesize exact universal reversible gate library for various types of gates and costs. We use the powerful algebraic software GAP for implementation and examination of our algorithms and the reversible logic synthesis problems have been reduced to group theory problems. It is shown that minimization of arbitrary cost functions of gates and orders of magnitude are faster than its previously counterparts for reversible logic synthesis. Experimental results show that a significant improvement over the previously proposed synthesis algorithm is obtained compared with the existing approaches to reversible logic synthesis.
We present new algorithms to synthesize exact universal reversible gate library for various types of gates and costs. We use the powerful algebraic software GAP for implementation and examination of our algorithms and the reversible logic synthesis problems have been reduced to group theory problems. It is shown that minimization of arbitrary cost functions of gates and orders of magnitude are faster than its previously counterparts for reversible logic synthesis. Experimental results show that a significant improvement over the previously proposed synthesis algorithm is obtained compared with the existing approaches to reversible logic synthesis.
Two questions have been addressed here: how many logic levels are necessary to synthesise a specific instance of a reversible circuit, and how much better to have large gate libraries, and what gates should be required in libraries. Group theory is applied to 3-bit reversible gate synthesis to create a library useful in hierarchical design. It has been shown that arbitrary 3-bit reversible circuit can be synthesised with four-logic levels, using this new gate library. The respective universal library for the four-level synthesis is constructed and optimised it on the level of nuclear magnetic resonance pulses. A very fast algorithm to synthesise arbitrary 3-bit reversible function to gates from our library is also presented. The algorithm demonstrates dramatic speed benefit and results in a maximum of four-level circuit for arbitrary 3-bit reversible function. The gates are optimised on the level of pulses to decrease their cost and allow for objective comparison with standard CNOT, NOT, Toffoli gates (CNT) circuits. This library guarantees a four-level circuit for any 3-qubit reversible function and is also intended to be used in a hierarchical design of larger circuits.
We present an effective algorithm which can construct optimal 3-qubit reversible logic circuits for any given reversible logic gates and costs by constructing a minimal perfect hash function. We also present an algorithm which can automatically construct quantum gate library. In the experiments on 3-qubit synthesis, our algorithm synthesizes all optimal reversible circuits with extremely fast speed--the average speed which synthesizes circuits with minimum cost is 365 times faster than that of best result.
Reversible quantum circuits are a necessary subclass of quantum computation and its realization is required for any quantum computer to be universal. This paper investigates how to synthesis of arbitrary ternary non-reversible logic circuits by adding inputs with constant value and garbage outputs. Group theory has been also used to solve the synthesis of reversible logic circuits. Our algorithm uses the SNT (ternary Swap gate, ternary NOT gate, ternary Toffoli gate) library, by reducing the ternary non-reversible logic circuit synthesis problem to group theory representation. The main result shows the relationship of ternary non-reversible logic circuits and the reversible circuits. The realization approach is constructive and can be further used to develop software for synthesis of arbitrary d-level circuits. This result is significantly different from the binary non-reversible logic circuits.
Synthesis of quantum reversible logic circuits means to automatically construct desired quantum reversible logic circuit with minimal quantum cost.We present a novel and efficient algorithm which can construct almost all optimal 4-qubit reversible logic circuits with various types of gates and minimum length cost based on constructing the shortest coding and the specific topological compression, whose lossless compression ratios of the space of n-qubit circuits is near2×n!.We firstly have created all 3120218828 optimal 4-qubit circuits whose length is less than 9 for the Controlled-Not gate,NOT gate and Toffoli gate library,and our method can achieve 16 steps through cascading created circuits.Our algorithm can not only synthesizes all the 4-qubit benchmark circuits,but also runs extremely fast.