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.

翻译：量子模加法器是最基础且多用途的量子计算操作之一。它们有助于实现更高复杂度的函数，如减法和乘法，这些函数应用于量子密码分析、量子图像处理和通信安全等领域。据我们所知，目前尚无现成的量子模 $(2n+1)$ 加法器设计。在本工作中，我们提出了四种专门针对模 $(2n+1)$ 加法设计的量子加法器。这些加法器可以同时提供常规和与模 $(2n+1)$ 的和，增强了其在基于余数系统的算术运算中的应用。我们的第一个设计 QMA1 是一种新型的量子模 $(2n+1)$ 加法器。第二个提出的加法器 QMA2 优化了 QMA1 中量子门的利用率，使得 CNOT 门数量减少了 37.5%，CNOT 深度减少了 46.15%，并且 Toffoli 门及其深度均减少了 26.5%。我们提出了第三个加法器 QMA3，它采用了基于动态电路的零重置特性来重用量子比特，从而节省了 25% 的量子比特数量。我们的第四个设计 QMA4 展示了引入额外零重置以获得更纯净零态的优势，从而减少了量子态制备误差。值得注意的是，我们在基于 Eagle R1 架构的 127 量子比特量子计算机 IBM Washington 上，使用所提出的模 $(2n+1)$ 加法器的 5 量子比特配置进行了实验，结果表明 QMA1 的误差降低了 28.8%，其中：(i) 18.63% 的误差降低源于 QMA2 中门数量和深度的减少，(ii) 2.53% 的误差降低源于 QMA3 中量子比特数量的减少，以及 (iii) 7.64% 的误差降低源于 QMA4 中应用了额外的零重置。