  Quantum modular adders are one of the most fundamental yet versatile quantum
computation operations. They help implement functions of higher complexity,
such as subtraction and multiplication, which are used in applications such as
quantum cryptanalysis, quantum image processing, and securing communication. To
the best of our knowledge, there is no existing design of quantum modulo
$(2n+1)$ adder. In this work, we propose four quantum adders targeted
specifically for modulo $(2n+1)$ addition. These adders can provide both
regular and modulo $(2n+1)$ sum concurrently, enhancing their application in
residue number system based arithmetic. Our first design, QMA1, is a novel
quantum modulo $(2n+1)$ adder. The second proposed adder, QMA2, optimizes the
utilization of quantum gates within the QMA1, resulting in 37.5% reduced CNOT
gate count, 46.15% reduced CNOT depth, and 26.5% decrease in both Toffoli gates
and depth. We propose a third adder QMA3 that uses zero resets, a dynamic
circuits based feature that reuses qubits, leading to 25% savings in qubit
count. Our fourth design, QMA4, demonstrates the benefit of incorporating
additional zero resets to achieve a purer zero state, reducing quantum state
preparation errors. Notably, we conducted experiments using 5-qubit
configurations of the proposed modulo $(2n+1)$ adders on the IBM Washington, a
127-qubit quantum computer based on the Eagle R1 architecture, to demonstrate a
28.8% reduction in QMA1's error of which: (i) 18.63% error reduction happens
due to gate and depth reduction in QMA2, and (ii) 2.53% drop in error due to
qubit reduction in QMA3, and (iii) 7.64% error decreased due to application of
additional zero resets in QMA4.
