bibkey: barriga2026dftbosonic authors: Edgar Barriga et al. year: 2026 title: “N-dimensional discrete Fourier transform via bosonic Hamiltonian” doi: null url: https://arxiv.org/abs/2609.05644 claim: “A waveguide array with propagation constants beta_l and couplings C_jk evolves by U = exp(-i C z); it realizes the N-dimensional discrete Fourier transform F_N (entries omega^(xy)/sqrt N, omega = exp(-2 pi i/N)) when input and output phase shifters give Phi_out U Phi_in = F_N. The graph has the waveguides as vertices and the nonzero couplings as edges. Section III.D conjectures edge ranges for the solutions; for N = 10 the third conjecture, with l = 5, gives at least N l / 2 = 25 edges. Section II.B asks whether the polygonal model (zero-diagonal real circulant couplings with C_l = C_{N-l}) can always yield F_N.” strata_touched:

• D5/S3/Quantum/Dynamics/BarrigaTenModeFourierEdgeRefutation
• D5/S3/Quantum/Dynamics/PolygonalFourierCouplings license: citation-only triage: anchor

N-dimensional discrete Fourier transform via bosonic Hamiltonian

arXiv:2609.05644v1 (primary quant-ph; v1 only). Section II models a waveguide array by the single-particle coupling matrix $\mathcal{C}$ with the propagation constants on the diagonal and the evanescent couplings off it, $U=e^{-i\mathcal Cz}$,with$z$ absorbed into the coefficients. The target is

$F_N=\frac{1}{\sqrt{N}}\left({\omega }^{jk}\right)$, where $\omega =e^{-2i\pi /N}$,

reached after input and output phase shifters, ${\Phi }^{\mathrm{out}}U{\Phi }^{\mathrm{in}}=F_N$.

Section II.B (the polygonal model) takes the circulant coupling matrix with rows $(0,C_1,\dots ,C_{N-1})$, each row rotated one place to the right of the previous one, with “the evanescent coupling strength felt by the waveguides has symmetry $C_l=C_{N-l}$”, reports analytic solutions for $N=4,5,6$ and numerical ones up to $N=31$, and ends:

Whether the polygonal model can always yield solutions to obtain $F_N$ from Eq.~\eqref{eq:Coupling_polygon} or not is beyond our theoretical efforts.

The formal answer is D5/S3/Quantum/Dynamics/PolygonalFourierCouplings.result: for every $N$ there are strictly positive polygonal couplings and diagonal unimodular phases with ${\Phi }^{\mathrm{out}}{e}^{-i\mathcal{C}}{\Phi }^{\mathrm{in}}=F_N$. An explicit choice, argued and checked numerically in #13954 but not formalized, is ${\mathcal{C}}_{jk}=\pi /(2N{\sin }^2(\pi (j-k)/N))$ for $j\ne k$ and ${\mathcal{C}}_{jj}=0$; at $N=4,5,6$ it gives the paper’s analytic solutions (the hexagon of Appendix B has $C_2=\pi /9$; the main text’s $\pi /6$ is a misprint). Section III.D (“Observations and conjectures to continue”) states three conjectures; the third reads:

\textbullet\ Third, all the other possible solutions for non-prime $N$ are in the following range of edges $|E(R_{\ell })|\le |E|<|E(K_{N-1})|-\ell ,$ where $R_{\ell }$ denotes $\ell$-regular graphs (same number of edges per vertex) and $|E(R_{\ell })|=N\ell /2$.

For $N=10$ the same section fixes $\ell$: “This strongly suggests that there should be a $\ell$-regular graph for $N=10$ as well, and it should be $\ell =5$.” The authors report four 5-regular graphs (25 edges) realizing $F_{10}$.

Verified locator

• URL: https://arxiv.org/abs/2609.05644 (v1; source file DFT.tex, md5 f6ff1ec31c30e8afc4674b5ef3ebf3b3).