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Journal of Circuits Systems and Computers

REDUCING QUANTUM COST OF REVERSIBLE CIRCUITS FOR HOMOGENEOUS BOOLEAN FUNCTIONS

Ahmed Younes

Published Oct 26, 2010• 6 citations
Publisher

Abstract

Homogeneous Boolean functions have many applications in computing systems, e.g., cryptography. This paper presents a factorization algorithm for reducing the quantum cost of the reversible circuits for that class of Boolean functions. The algorithm reduces the multi-calculation of any common parts of the circuit. This allows Homogeneous Boolean related applications to be implemented efficiently on novel computing paradigms such as quantum computers and low power devices.

DOI
10.1142/s0218126610006736

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Homogeneous Boolean functions have many applications in computing systems, e.g., cryptography. This paper presents a factorization algorithm for reducing the quantum cost of the reversible circuits for that class of Boolean functions. The algorithm reduces the multi-calculation of any common parts of the circuit. This allows Homogeneous Boolean related applications to be implemented efficiently on novel computing paradigms such as quantum computers and low power devices.
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It is a well-known fact in logic design that synthesis of some special classes of Boolean functions is often easier than the synthesis of a general unrestricted specification. In reversible logic, well-scaled synthesis methods with a reasonably small cost of the associated implementation have been found for only a few classes of functions. This includes synthesis of multiple-output symmetric and reversible linear functions. The author presents an efficient reversible/quantum synthesis method for the class of multiple-output symmetric functions. The method is purely theoretical, therefore its scaling on functions with a large number of inputs/outputs requires minimal resources. The author calculates garbage, i.e. the number of outputs that are not required by the function specification, the number of reversible gates, and the quantum cost of the presented implementations. The proposed approach is then applied to the synthesis of benchmark functions. Comparison of the designs to the previously reported implementations is favourable.
Quantum computers, more specifically quantum circuits, take on the eyes with their computational promises such as reversibility. In this paper, we perform synthesis and optimization of quantum circuits. In the first part, we propose a fast synthesis algorithm that implements any given reversible Boolean function with quantum gates. Instead of an exhaustive search on every given function, our algorithm creates a library of essential functions and performs sorting. As an example, to implement 4 bit circuits we only use 120 essential functions out of all 20922789888000 functions. In the second part, we optimize our circuits by using new templates. The proposed templates mainly consist of Toffoli gates with negative and positive controlling lines. These templates also show us that optimum area solutions proposed in the literature are not actually optimum; they can be improved.
This paper studies synthesis and optimization of reversible circuits composed of Toffoli gates with negative and positive control lines. The proposed synthesis algorithm performs sorting among optimal implementations of certain functions - called as essential functions - to implement any reversible Boolean function. Essential functions comprise very small amount of all functions. For example, to implement 3 bit circuits, 28 essential functions out of all 40320 functions are needed. The proposed optimization algorithm considers both reversible and quantum circuit costs. First, reversible cost is reduced by considering adjacent gate pairs. Then, inner quantum structures of the gates are investigated and quantum optimization is performed. The proposed algorithms are evaluated on benchmark circuits in comparison with the results in the literature.
Abstract The construction of quantum computers is based on the synthesis of low-cost quantum circuits. The quantum circuit of any Boolean function expressed in a Positive Polarity Reed-Muller (PPRM) expansion can be synthesized using Multiple-Control Toffoli (MCT) gates. This paper proposes two algorithms to construct a quantum circuit for any Boolean function expressed in a PPRM expansion. The Boolean function can be expressed with various algebraic forms, so there are different quantum circuits can be synthesized for the Boolean function based on its algebraic form. The proposed algorithms aim to map the MCT gates into the NCV gates for any quantum circuit by generating a simple algebraic form for the Boolean function. The first algorithm generates a special algebraic form for any Boolean function by rearrangement of terms of the Boolean function according to a predefined degree of term d term , then synthesizes the corresponding quantum circuit. The second algorithm applies the decomposition methods to decompose MCT circuit into its elementary gates followed by applying a set of simplification rules to simplify and optimize the synthesized quantum circuit. The proposed algorithms achieve a reduction in the quantum cost of synthesized quantum circuits when compared with relevant work in the literature. The proposed algorithms synthesize quantum circuits that can applied on IBM quantum computer.