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arXiv:quant-ph/0305134

Representation of Boolean Quantum Circuits as Reed-Muller Expansions

Ahmed Younes, Julian Francis Miller

Published May 22, 2003• 0 citations
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Abstract

In this paper we show that there is a direct correspondence between quantum Boolean operations and certain forms of classical (non-quantum) logic known as Reed-Muller expansions. This allows us to readily convert Boolean circuits into their quantum equivalents. A direct result of this is that the problem of synthesis and optimization of quantum Boolean logic can be tackled within the field of Reed-Muller logic.

DOI
10.48550/arxiv.quant-ph/0305134

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In this paper we show that there is a direct correspondence between quantum Boolean operations and certain forms of classical (non-quantum) logic known as Reed-Muller expansions. This allows us to readily convert Boolean circuits into their quantum equivalents. A direct result of this is that the problem of synthesis and optimization of quantum Boolean logic can be tackled within the field of Reed-Muller logic.
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This paper presents how to synthesize efficient Boolean quantum circuits (BQC) from Reed-Muller (RM) expansions of Boolean functions. Firstly, the method of Younes et al [A. Younes and J. F. Miller, International Journal of Electronics 91, 431 (2004)] is generalized to synthesize BQC for multiple outputs function. The transformation is straightforward. A technique rearranging the modified Toffoli gates is also presented to eliminate additive NOT gates as much as possible. Secondly, the approach synthesizing BQC without work qubits is proposed which is simple and high-efficient. The necessary conditions for the RM expansions which can be synthesized by the proposed method are derived too.
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.