  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.
