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We propose a simplified mathematical construction of the quantum Fouriertransform which is suited for systems described by Ising-type Hamiltonians. Bycontrast to the standard Cooley-Tuckey scheme, which prescribes sequences ofCPHASE gates, our implementation is based on one-qubit gates and a freeevolution process. We also show how to obtain a quadratic speed-up by applyingthe conditional interactions simultaneously. Thus rather than O($N^2$) ourimplementation time scales as O($N$). Finally, we show a realization of ourmethod with homogeneous microwave driven ion traps in a magnetic field withgradient.

Simplified implementation of the quantum Fourier transform with Ising-type Hamiltonians: Example with ion traps