Stesso: A reconfigurable decomposition of $n$-bit Toffoli gates using symmetrical logical structures and adjustable support qubits

An $(n+1)$-bit Toffoli gate is mainly utilized to construct other quantum gates and operators, such as Fredkin gates, arithmetical adders, and logical comparators, where $n \geq 2$. Several researchers introduced different methods to decompose $(n+1)$-bit Toffoli gates in a quantum circuit into a set of standard 3-bit Toffoli gates or a set of elementary quantum gates, such as single-qubit and two-qubit gates. However, these methods are not effectively reconfigurable for linearly connected symmetrical structures (layouts) of contemporary quantum computers, usually utilizing more ancilla qubits. This paper introduces a new structural design method to effectively decompose $(n+1)$-bit Toffoli gates by utilizing configurable ancilla qubits, which we named the ``support qubits". Collectively, we call our decomposition method for symmetrical structures using support qubits the ``step-decreasing structures shaped operators (Stesso)". The main advantage of Stesso is to configurable construct different decomposed operators of various polarities and intermediate sub-circuits, such as Positive Polarity-Stesso, Mixed Polarity-Stesso, and Generalized-Stesso. With Stesso, it has been experimentally proven that $(n+1)$-bit Toffoli gates always have lower quantum costs than using conventional composition methods.