Quantum Modular Multiplication

Quantum modular multiplication circuit is one of the basic quantum computation circuits which are basic functions in quantum algorithms. However, since quantum-quantum modular multipliers require a high cost reversible modular inversion routine for modular multiplication, researchers have been unable to propose a feasible quantum-quantum modular multiplier. In this paper, we proposed efficient quantum-classical modular multipliers and the first quantum-quantum modular multipliers that do not require a reduction stage by transforming the partial product used in multiplication utilizing bit-shift operation. Then, we calculated quantum resource complexity and analyzed it compared to other quantum modular multipliers and utilized ETRI (Electronics and Telecommunications Research Institute) Qcrypton to analyze quantum resource complexity in the practical quantum computing situation. The proposed quantum modular multipliers show an improvement of 50% in terms of gates and circuit depth compared to the most recently proposed high-performance quantum modular multipliers.