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

TL;DR

This work targets efficient, layout-aware decomposition of $C^{n}X$ gates under symmetry constraints. It introduces Stesso, a family of step-decreasing structures (PP-Stesso, MP-Stesso, G-Stesso) that reconfigure standard $C^{2}X$ gates and $X$ gates into symmetry-friendly operators with adjustable support qubits. The approach defines basic shapes (backslash, slash, V, И) and composed sequences that achieve outputs in mixed SOP/ESOP forms, maintaining polynomial complexity and enabling no-ancilla implementations of an $n$-bit binary comparator. The results indicate practical advantages for arithmetic units and quantum search constructs, with applications demonstrated on a 4-bit comparator and potential reductions in quantum resources on common QPU layouts.

Abstract

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.

Figures (8)

• Figure 1: Examples of combined quantum circuits of standard Toffoli gates with the same physical connectivity requirement for qubits $a,b,c,d,e$, utilizing symbol-shape and English-shape structures of Stesso.
• Figure 2: Possible mapping results of qubits $a,b,c,d,e$ in the quantum structures of Rigetti, Google, and IBM QPUs meeting the connectivity requirements in Figure \ref{['fig:Triangle_layout1']}. The red line (bus) indicates that SWAP gates are added based on the distance between two physical qubits. Blue line (bus) means the direct physical connection between two physical qubits. Circles mean the physical qubits of a QPU. The letters ($a,b,c,d,e$) are the logical qubits of standard Toffoli gates.
• Figure 3: Examples of $\backslash$-shape structures with two satisfied rules: (a) a $\backslash$-shape (consisting of three controls, one support, and one target qubit; satisfying the rule of control $\rightarrow$ target qubit) has the same circuit depth and circuit size $D=S= \text{No. C}^2\text{X}=2$, and (b) a $\backslash$-shape (consisting of four controls, two supports, and one target qubit; satisfying the rule of target $\rightarrow$ control qubit) has $D=S= \text{No. C}^2\text{X}=3$.
• Figure 4: Examples of downstairs-shape structures with two satisfied rules: (a) a downstairs-shape (consisting of three controls, one support, and one target qubit; satisfying the rule of control $\rightarrow$ target qubit) has $D= S = \text{No. C}^2\text{X} + \text{No. X}=4$, and (b) a downstairs-shape (consisting of four controls, two supports, and one target qubit; satisfying the rule of target $\rightarrow$ control qubit) has $D=S = \text{No. C}^2\text{X} + \text{No. X}=6$.
• Figure 5: Examples of upstairs-shape structures with two satisfied rules: (a) an upstairs-shape (consisting of four controls, two supports, and one target qubit; satisfying the rule of control $\rightarrow$ target qubit) has $D= S = \text{No. C}^2\text{X} + \text{No. X}=6$, and (b) an upstairs-shape (consisting of three controls, one support, and one target qubit; satisfying the rule of target $\rightarrow$ control qubit) has $D=S = \text{No. C}^2\text{X} + \text{No. X}=4$.

Theorems & Definitions (11)

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