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Synthesis of Linear Reversible Circuits and EXOR-AND-based Circuits for Incompletely Specified Multi-Output Functions

Ben Schaeffer

Published Jul 21, 2017• 0 citations
Publisher

Abstract

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.

DOI
10.15760/etd.3444

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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.
This paper introduces new algorithms to synthesize reversible functions using EXOR-sum of Products-of-EXOR-sums (EPOE) structures. The motivation for using these structures is to reduce the number of as well as the sizes of multiple controlled Toffoli gates, and thus the quantum cost. To achieve these reductions the paper generalizes from existing 2-level AND-EXOR structures (ESOP) commonly used in reversible logic to a mixture of 3-level EXOR-AND-EXOR structures and ESOPs. Our approach can be applied to reversible and permutative quantum circuits to synthesize single output functions on to an output line, with no additional ancilla bits. A comparison of the ESOP minimizer EXORCISM-4 and two variants of the EPOE minimizer, called EPOEM-1s and EPOEM-2, is presented. The results show that EPOE circuits do in fact achieve the above-stated cost reductions, in particular when expressed in terms of Maslov's quantum cost, the metric commonly used in quantum circuit synthesis.
Previously, the synthesis of reversible and quantum permutative circuits based on products of Exclusive-OR (EXOR) sums of inputs (POE) was explored as an approach to reduce quantum cost. Herein, the mathematical representation used in POE expressions is formally detailed and contrasted with the pseudoproduct-based theory of Luccio and Pagli. The theory and practice of product transformation, or computation of equivalent POE expressions, is presented. To demonstrate the generality of the mathematical representations outside of the reversible domain, a new method to synthesize incompletely specified functions as EXOR-AND-OR circuits is introduced. The method is heuristic. It first uses spectral information from the Walsh-Hadamard transform to compute a set of POE implicants for a function, and then uses product transformation to minimize both the total number of literals and unique EXOR sums (ignoring polarity) used in the POE implicants. The product transformation minimization extends to multioutput functions. The new method has no performance guarantees, i.e., it does not have a fallback mode of minimum or near-minimum sum of products; therefore, it is suited to a tournament synthesis approach. When synthesizing benchmark functions of up to 26 inputs, the new method is shown to typically produce lower cost circuits than heuristic pseudoproduct-based methods.
This paper introduces a new synthesis approach called Products of EXOR-sums (POE) and demonstrates its utility when employed in an Exclusive-Or-Sum-of-Products (ESOP) based algorithm for reversible and permutative quantum circuits restricted to single output functions. Unlike previous ESOP approaches such as EXORCISM-4 which realize functions as an EXOR-ed sum of product literals, this new algorithm realizes functions as an EXOR-ed sum of POEs (EPOE). A comparison of EXORCISM-4 and EPOE circuits shows that the latter approach often produces circuits with significantly lower quantum costs.
This paper introduces a new logic structure for reversible quantum circuit synthesis. Our synthesis method aims to minimize the quantum cost of reversible quantum circuits with decoders. In this method, multi-valued input, binary output (MVI) functions are utilized as a mathematical concept only, but the circuits are binary. We introduce the new concept of ``Multi-Valued Input Fixed Polarity Reed-Muller (MVI-RM)" forms. Our decoder-based circuit uses three logical levels in contrast to commonly-used methods based on Exclusive-or Sum of Products (ESOP) with two levels (AND-XOR expressions), realized by Toffoli gates. In general, the high number of input qubits in the resulting Toffoli gates is a problem that greatly impacts the quantum cost. Using decoders decreases the number of input qubits in these Toffoli gates. We present two practical algorithms for three-level circuit synthesis by finding the MVI-FPRM: products-matching and the newly developed butterfly diagrams. The best MVI-FPRM forms are factorized and reduced to approximate Multi-Valued Input Generalized Reed-Muller (MVI-GRM) forms.
Reversible logic circuits have been historically motivated by theoretical research in low-power, and recently attracted interest as components of the quantum algorithm, optical computing and nanotechnology. However due to the intrinsic property of reversible logic, traditional irreversible logic design and synthesis methods cannot be carried out. Thus a new set of algorithms are developed correctly to synthesize reversible logic circuit. This paper presents a comprehensive literature review with comparative study on heuristic based reversible logic synthesis. It reviews a range of heuristic based reversible logic synthesis techniques reported by researchers (BDD-based, cycle-based, search-based, non-search-based, rule-based, transformation-based, and ESOP-based). All techniques are described in detail and summarized in a table based on their features, limitation, library used and their consideration metric. Benchmark comparison of gate count and quantum cost are analysed for each synthesis technique. Comparing the synthesis algorithm outputs over the years, it can be observed that different approach has been used for the synthesis of reversible circuit. However, the improvements are not significant. Quantum cost and gate count has improved over the years, but arguments and debates are still on certain issues such as the issue of garbage outputs that remain the same. This paper provides the information of all heuristic based synthesis of reversible logic method proposed over the years. All techniques are explained in detail and thus informative for new reversible logic researchers and bridging the knowledge gap in this area.