  Quantum Arithmetic faces limitations such as noise and resource constraints
in the current Noisy Intermediate Scale Quantum (NISQ) era quantum computers.
We propose using Distributed Quantum Computing (DQC) to overcome these
limitations by substituting a higher depth quantum addition circuit with
Residue Number System (RNS) based quantum modulo adders. The RNS-based
distributed quantum addition circuits possess lower depth and are distributed
across multiple quantum computers/jobs, resulting in higher noise resilience.
We propose the Quantum Superior Modulo Addition based on RNS Tool (QSMART),
which can generate RNS sets of quantum adders based on multiple factors such as
depth, range, and efficiency. We also propose a novel design of Quantum
Diminished-1 Modulo (2n + 1) Adder (QDMA), which forms a crucial part of
RNS-based distributed quantum addition and the QSMART tool. We demonstrate the
higher noise resilience of the Residue Number System (RNS) based distributed
quantum addition by conducting simulations modeling Quantinuum's H1 ion
trap-based quantum computer. Our simulations demonstrate that RNS-based
distributed quantum addition has 11.36% to 133.15% higher output probability
over 6-bit to 10-bit non-distributed quantum full adders, indicating higher
noise fidelity. Furthermore, we present a scalable way of achieving distributed
quantum addition higher than limited otherwise by the 20-qubit range of
Quantinuum H1.
