T-count optimization of approximate quantum Fourier transform

The quantum Fourier transform (QFT) is a ubiquitous quantum operation that is used in numerous quantum computing applications. The major obstacle to constructing a QFT circuit is that numerous elementary gates are required. Among the elementary gates, T gates dominate the cost of fault-tolerant implementation. Currently, the smallest-known T-count required to construct an n-qubit QFT circuit approximated to error O(\varepsilon) is ~8nlog_2(n/\varepsilon). Moreover, the depth of T gates (T-depth) in the approximate QFT circuit is ~2nlog_2(n/\varepsilon). This approximate QFT circuit was constructed using Toffoli gates and quantum adders. In this study, we present a new n-qubit QFT circuit approximated to error O(\varepsilon). Our approximate QFT circuit shows a T-count of ~4nlog_2(n/\varepsilon) and a T-depth of ~nlog_2(n/\varepsilon). Toffoli gates, which account for half of the T-count in the approximate QFT circuit reported in the previous study, are unnecessary in our construction. Quantum adders, which dominate the leading order term of T-depth in our approximate QFT circuit, are arranged in parallel to reduce T-depth.