Minimization of AND-XOR Expressions with Decoders for Quantum Circuits

TL;DR

This paper proposes a novel decoder-based approach for minimizing quantum cost in reversible quantum circuits by introducing Multi-Valued Input Fixed Polarity Reed-Muller forms, reducing input qubits in Toffoli gates through new algorithms.

Contribution

It introduces MVI-RM forms and two algorithms for three-level circuit synthesis, improving quantum circuit minimization with decoders.

Findings

Decoder-based circuits reduce input qubits in Toffoli gates.

New algorithms effectively find MVI-FPRM forms.

Significant reduction in quantum cost achieved.

Abstract

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.