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

TL;DR

This work tackles the challenge of efficient quantum modulo arithmetic by introducing four modulo $(2n+1)$ adders. The authors design two static, fully reversible adders (QMA1 and QMA2) and two dynamic, non-reversible adders leveraging zero resets (QMA3 and QMA4) to reduce qubit count and state-preparation errors. Experimental validation on IBM's 127-qubit IBM Washington platform demonstrates a cumulative 28.8% reduction in error from QMA1 to QMA4, driven by gate-depth reductions and improved state initialization. The results highlight practical pathways for integrating modulo $(2n+1)$ arithmetic into quantum subroutines for subtraction, multiplication, and exponentiation within Residue Number Systems and related quantum applications.

Abstract

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.

Figures (5)

• Figure 1: Quantum gates: (a) CNOT Gate. (b) Toffoli Gate.
• Figure 2: Proposed quantum modulo (2n + 1) adder QMA1 for n=4 in a 5-qubit configuration. The QMA1 takes the inputs (a0:a4) and (b0:b4) to generate Modulo Sum (m0:m4). The Sum (s0:s6) and input (b0:b4) are retained. QMA1 is the basic interpretation of Algorithm \ref{['alg:1']}, using quantum full adder to calculate (a + b) modulo 2n as intermediate Sum, followed by NOR operation to compute $\overline{(S\textsubscript{n+1} \vee S\textsubscript{n})}$. Another quantum full adder adds these with Sn+1 2n.
• Figure 3: Optimized quantum modulo (2n + 1) adder QMA2 for n=4 in a 5-qubit configuration. The inputs (a0:a4) and (b0:b4) are used to generate Modulo Sum (m0:m4) and Sum (s0:s5). Unlike QMA1, QMA2 uses a quantum half adder instead of a full adder to reduce gate count.
• Figure 4: Proposed quantum modulo (2n + 1) adders QMA3 and QMA4 for n=4 in a 5-qubit configuration. Zero reset helps in reusing a qubit by resetting it to $|0$⟩. The zero resets in the box (a) are in both QMA3 and QMA4, while zero resets in the box (b) are only in QMA4. QMA3 uses the zero resets in the box (a) to help reuse input (b0:b4) to generate Modulo Sum (m0:m4). Sum (s0:s5) is generated in parallel as before. QMA4 utilizes zero resets in box (b), in addition to zero resets present in box (a) for QMA3, to reduce quantum state preparation errors.
• Figure 5: The above graph compares the proposed quantum modulo (2n + 1) adders, QMA1 to QMA4, for n=4 in a 5-qubit configuration, with NMED representing error and Figure Of Merit (FOM) representing resource usage. FOM = (Circuit Width x Toffoli Depth). For both NMED and FOM, lower is better. From QMA1 to QMA3, the NMED falls along with the reduction in FOM, showing the impact of lower quantum resource usage. However, the 7.64% reduction in NMED from QMA3 to QMA4, without any drop in FOM, reflects reduction in quantum state preparation errors due to additional resets.