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Control modular addition is a core arithmetic function, and we must consider the computational cost for actual quantum computers to realize efficient implementation. To achieve a low computational cost in a control modular adder, we focus on minimizingKQ (where K is the number of logical qubits required by the algorithm, and Q is the elementary gate step), defined by the product of the number of qubits and the depth of the circuit. In this article, we construct an efficient control modular adder with small KQ by using relative-phase Toffoli gates in two major types of quantum computers: fault-tolerant quantum computers (FTQ) on the logical layer and noisy intermediate-scale quantum computers (NISQ). We give a more efficient construction compared with Van Meter and Itoh&#x2019;s, based on a carry-lookahead adder. In FTQ, <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula> gates incur heavy cost due to distillation, which fabricates ancilla for running <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula> gates with high accuracy but consumes a lot of especially prepared ancilla qubits and a lot of time. Thus, we must reduce the number of <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula> gates. We propose a new control modular adder that uses only 20&#x0025; of the number of <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula> gates of the original. Moreover, when we take distillation into consideration, we find that we minimize <inline-formula><tex-math notation="LaTeX">$\text{KQ}_{T}$</tex-math></inline-formula> (the product of the number of qubits and <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula>-depth) by running <inline-formula><tex-math notation="LaTeX">$\Theta (n / \sqrt{\log n})$</tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula> gates simultaneously. In NISQ, <sc>cnot</sc> gates are the major error source. We propose a new control modular adder that uses only 35&#x0025; of the number of <sc>cnot</sc> gates of the original. Moreover, we show that the <inline-formula><tex-math notation="LaTeX">$\text{KQ}_{\text{CX}}$</tex-math></inline-formula> (the product of the number of qubits and <sc>cnot</sc>-depth) of our circuit is 38&#x0025; of the original. Thus, we realize an efficient control modular adder, improving prospects for the efficient execution of arithmetic in quantum computers.

Efficient Construction of a Control Modular Adder on a Carry-Lookahead Adder Using Relative-Phase Toffoli Gates