Quantum circuits for elementary arithmetic operations are important not only for implementing Shor's factoring algorithm on a quantum computer but also for understanding the computational power of small quantum circuits, such as linear-size or logarithmic-depth quantum circuits. This paper surveys some recent approaches to constructing efficient quantum circuits for elementary arithmetic operations and their applications to Shor's factoring algorithm. It covers addition, comparison, and the quantum Fourier transform used for addition.

We evaluate the exact number of gates for circuits of Shor's factoring algorithm. We estimate the running time for factoring a large composite such as 576 and 1024 bit numbers by appropriately setting gate operation time. For example, we show that on the condition that each elementary gate is operated within 50 µsec, the running time for factoring 576 bit number is 1 month even if the most effective circuit is adopted. Consequently, we find that if we adopt the long gate operation-time devices or qubit-saving circuits, factorization will not be completed within feasible time on the condition that a new efficient modular exponentiation algorithm will not be proposed. Furthermore, we point out that long gate operation time may become a new problem preventing a realization of quantum computers.