  A generic architecture of n-bit quantum operators is proposed for
cost-effective transpilation, based on the layouts and the number of n neighbor
physical qubits for IBM quantum computers, where n >= 3. This proposed
architecture is termed "GALA-n quantum operator". The GALA-n quantum operator
is designed using the visual approach of the Bloch sphere, from the visual
representations of the rotational quantum operations for IBM native gates
(square root of X, X, RZ, and CNOT). In this paper, we also proposed a new
formula for the quantum cost, which calculates the total numbers of native
gates, SWAP gates, and the depth of the final transpiled quantum circuits. This
formula is termed the "transpilation quantum cost". After transpilation, our
proposed GALA-n quantum operator always has a lower transpilation quantum cost
than that of conventional n-bit quantum operators, which are mainly constructed
from costly n-bit Toffoli gates.
