  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 Positive
Polarity Reed-Muller $PPRM$ 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 literature. The proposed algorithms synthesize
quantum circuits that can applied on IBM quantum computer.
