Novel Optimized Designs of Modulo 2n+1 Adder for Quantum Computing

Quantum modular adders are one of the most fundamental yet versatile quantum computation operations. They help implement the 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 ( $2^{n}+1$ ) adder (QMA). In this work, we propose four quantum adders targeted specifically for modulo ( $2^{n}+1$ ) addition. These adders can provide both regular and modulo ( $2^{n}+1$ ) sum concurrently, enhancing their application in residue number system-based arithmetic. Our first design, QMA1, is a novel quantum modulo ( $2^{n}+1$ ) 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 $|0$ $\rangle $ state, reducing quantum state preparation errors. Notably, we conducted experiments using 5-qubit configurations of the proposed modulo ( $2^{n}+1$ ) 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 do the following: 1) 18.63% error reduction happens due to gate/depth reduction in QMA2; 2) 2.53% drop in error due to qubit reduction in QMA3; and 3) 7.64% error decreased due to application of additional zero resets in QMA4.