A Cost-Effective Layout-Aware Quantum Circuit Synthesis For Triangular, Square, and Heavy-Hex Layouts

The quantum layout and the mapping of logical to physical qubits are crucial in quantum circuit synthesis for a real quantum computer. Circuits that include large $n$-bit Toffoli gates ($n \geq 3$), such as those designed from cost-expensive gates and hard-to-decompose Exclusive-or Sum of Products (ESOP) expressions, have complications of effective mappings into contemporary quantum layouts, such as the square grid and heavy-hex layouts. These complications are primarily caused by the limited connectivity among the physical qubits in such layouts, leading to the insertion of many additional SWAP gates. This paper introduces a new quantum circuit synthesis methodology by exploring the advantage of a Positive Davio lattice (PDL) as an intermediate representation to create our proposed triangular layout and layout-aware circuits. From these circuits, we introduce and form the SWAT gate, composed of a SWAP gate followed by a 3-bit Toffoli gate. To illustrate the usefulness of our method for existing industrial quantum layouts, we also introduce cost-effective mappings of the resulting circuits onto square grid and heavy-hex layouts without additional SWAP gates. This is done with the help of the SWAT gate. Our research highlights PDLs as an efficient tool for layout-aware quantum circuit synthesis.