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International Journal of Electronics

Representation of Boolean quantum circuits as reed–Muller expansions

Ahmed Younes, Julian F. Miller

Published Jul 1, 2004• 37 citations
Publisher

Abstract

In this paper we show that there is a direct correspondence between Boolean quantum 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 Boolean quantum circuits can be tackled within the field of Reed–Muller logic.

DOI
10.1080/00207210412331272643

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In this paper we show that there is a direct correspondence between Boolean quantum 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 Boolean quantum circuits can be tackled within the field of Reed–Muller logic.
Abstract There is a direct equivalence between Boolean functions represented in Reed-Muller logic and Boolean Quantum Circuits. Different polarity Reed-Muller expansions will give different Boolean quantum circuits with different cost for the same Boolean function. For a given Boolean function with n variables there are 2n possible expansions. Searching for the expansion that gives a Boolean quantum circuit with minimum quantum cost within the search space is a hard problem for large n. This paper will use genetic algorithms to find the fixed/mixed polarity Reed-Muller expansion that gives a Boolean quantum circuit with minimum quantum cost to optimize the circuit realization of a given Boolean function.
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—Quantum Boolean circuit synthesis issues are becoming a key area of research in the domain of quantum computing. For gate-level synthesis, minterm based and Reed-Muller canonical decomposition techniques are adopted as common approaches. Physical implementation of quantum circuits have inherent constraints and hence nearest neighbour template of input lines is gaining importance. In this work, we present a brief analysis of the various Fixed Polarity Reed Muller (FPRM) expressions for a given quantum Boolean circuit and also introduce the rules for the nearest neighbour template-based synthesis of these forms. The corresponding circuit costs are evaluated.