  Quantum addition based on the quantum Fourier transform can be an integral
part of a quantum circuit and proved to be more efficient than the existing
classical ripple carry adder. Our study includes identifying the quantum
resource required in a quantum adder in any arbitrary dimension and its
relationship with the performance indicator in the presence of local noise
acting on the circuit and when a limited number of controlled rotation
operations is permitted, a procedure known as banding. We analytically prove an
upper bound on the number of the controlled rotation gates required to
accomplish the quantum addition up to an arbitrary defect in the fidelity
between the desired and imperfect output. When the environment interacts with
individual qudits, we establish a connection between quantum coherence and
fidelity of the output. Interestingly, we demonstrate that when banding is
employed in the presence of noise, approximate circuits of constant depth
outperform circuits with a higher number of controlled rotations, establishing
a complementary relationship between the approximate quantum adder and the
strength of the noise. We exhibit that utilizing magnetic fields to prepare an
initial state that evolves according to a one-dimensional spin chain for a
specific amount of time can be a potential technique to implement quantum
addition circuits in many-body systems.
