The discrete Fourier transform (DFT) underpins many classical algorithms and is a fundamental unitary operator for quantum information processing. Implementing the N-dimensional DFT in photonic integrated circuits (PICs) is limited by the cascades o
The discrete Fourier transform (DFT) underpins many classical algorithms and is a fundamental unitary operator for quantum information processing. Implementing the N-dimensional DFT in photonic integrated circuits (PICs) is limited by the cascades of Mach-Zehnder interferometers that current architectures require. Here we propose bosonic Hamiltonians that realize the N-dimensional DFT through a single stage of multimode evolution, complemented only by phase shifters before and after the interaction region, in a geometry suited to 3D waveguides. Modeling the system as a graph, where edges correspond to couplings and the vertices are the waveguides, we obtain analytical solutions for complete graphs up to N=6 and numerical solutions up to N=31. For non-complete graphs, different propagation constants are required in the Hamiltonian. We report all solutions for N\leq 8, partial exploration for N=9, and selected cases for N=10, together with three conjectures that guide the numerical search for N \geq 11. These configurations circumvent the vanishing evanescent coupling strength imposed by the waveguide separation, and a closed-form sensitivity criterion selects those that are admissible as a waveguide layout and least sensitive to fabrication error. We also uncover the missing non-affine parameters of the 6-dimensional DFT, and show that the scaling law for implementing the N-dimensional DFT with our building blocks is O(N\log\log{N}). This allows assembling the 2520-dimensional DFT with only 2625 interferometers, in contrast to the \approx 3\times 10^6 of Reck and Clements architectures.