CALA-$n$: A Quantum Library for Realizing Cost-Effective 2-, 3-, 4-, and 5-bit Gates on IBM Quantum Computers using Bloch Sphere Approach, Clifford+T Gates, and Layouts

We introduce a new quantum layout-aware approach to realize cost-effective $n$-bit gates using the Bloch sphere, for $2 \le n \le 5$ qubits. These $n$-bit gates are entirely constructed from the Clifford+T gates, in the approach of selecting sequences of rotations visualized on the Bloch sphere. This Bloch sphere approach ensures to match the quantum layout for synthesizing (transpiling) these $n$-bit gates into an IBM quantum computer. Various standard $n$-bit gates (Toffoli, Fredkin, etc.) and their operational equivalent of our proposed $n$-bit gates are examined and evaluated, in the context of the final quantum costs, as the final counts of generated IBM native gates. In this paper, we demonstrate that all our $n$-bit gates always have lower quantum costs than those of standard $n$-bit gates after transpilation. Hence, our Bloch sphere approach can be used to build a quantum library of various cost-effective $n$-bit gates for different layouts of IBM quantum computers.