A fast quantum circuit for addition with few qubits

We show how to construct a fast quantum circuit for computing the sum of two $n$-bit binary numbers with few qubits. The constructed circuit uses $O(n/\log n)$ ancillary qubits and its depth and size are $O(\log n)$ and $O(n)$, respectively. The number of ancillary qubits is asymptotically less than that in Draper et al.'s quantum carry-lookahead adder, and the depth and size are asymptotically the same as those of Draper et al.'s. Moreover, we show that the circuit is useful for constructing an efficient quantum circuit for Shor's factoring algorithm.