  Quantum Fourier transform (QFT) is a widely used building block for quantum
algorithms, whose scalable implementation is challenging in experiments. Here,
we propose a protocol of quadratic quantum Fourier transform (QQFT),
considering cold atoms confined in an optical lattice. This QQFT is equivalent
to QFT in the single-particle subspace, and produces a different unitary
operation in the entire Hilbert space. We show this QQFT protocol can be
implemented using programmable laser potential with the
digital-micromirror-device techniques recently developed in the experiments.
The QQFT protocol enables programmable Hamiltonian engineering, and allows
quantum simulations of Hamiltonian models, which are difficult to realize with
conventional approaches. The flexibility of our approach is demonstrated by
performing quantum simulations of one-dimensional Poincar\'{e} crystal physics
and two-dimensional topological flat bands, where the QQFT protocol effectively
generates the required long-range tunnelings despite the locality of the cold
atom system. We find the discrete Poincar\'{e} symmetry and topological
properties in the two examples respectively have robustness against a certain
degree of noise that is potentially existent in the experimental realization.
We expect this approach would open up wide opportunities for optical lattice
based programmable quantum simulations.
