  Quantum Fourier transform (QFT) is a key ingredient of many quantum
algorithms where a considerable amount of ancilla qubits and gates are often
needed to form a Hilbert space large enough for high-precision results. Qubit
recycling reduces the number of ancilla qubits to one but imposes the
requirement of repeated measurements and feedforward within the coherence time
of the qubits. Moreover, recycling only applies to certain cases where QFT can
be carried out in a semi-classical way. Here, we report a novel approach based
on two harmonic resonators which form a high-dimensional Hilbert space for the
realization of QFT. By employing the all-resonant and perfect state-transfer
methods, we develop a protocol that transfers an unknown multi-qubit state to
one resonator. QFT is performed by the free evolution of the two resonators
with a cross-Kerr interaction. Then, the fully-quantum result can be localized
in the second resonator by a projective measurement. Qualitative analysis shows
that a 2^10-dimensional QFT can be realized in current superconducting quantum
circuits which paves the way for implementing various quantum algorithms in the
noisy intermediate-scale quantum (NISQ) era.
