ACCELERATING SPECTRAL CLUSTERING ON QUANTUM AND ANALOG PLATFORMS

The paper shows that evolving a block-Hamiltonian built from the normalized incidence matrix yields a wave equation in u1 with Lsym = BB^T, and that time-delay DMD applied to a single-node scalar time series recovers exact Laplacian eigenvalues and (scaled) eigenvector components; it then argues that sign-based assignments reproduce spectral clustering and that, under the stated analog/quantum primitives, the overall complexity is O(N). These points are established via the Hamiltonian reduction (their Eqs. (25)–(28)), the DMD lemmas (single-output Hankel factorization) and the end-to-end complexity table (analog O(1) MVM; quantum O(N) dynamics; analog O(M log M) eigendecomposition; O(log N) linear solves) . The candidate solution reproduces the same core argument with an explicit Vandermonde/Hankel factorization that yields A = V_K Λ V_K^+, recovering μj = e^{iζj} and amplitudes proportional to v^{(j)}_ℓ, and then identifies the Fiedler component by the smallest nonzero frequency, with a consistent O(N) complexity under the same primitives. Minor deviation: the paper presents discrete-time coefficients pj, qj = (1 ± i tan(ζj/2))/2 for the per-mode exponential split, whereas the candidate notes pj = qj = 1/2 under continuous-time zero-velocity initial conditions; this is a parameterization difference that does not affect the recovered eigenvalues/eigenvectors or clustering rule, and the paper itself uses the continuous Schrödinger reduction as well as the discrete scheme . Overall, both are correct and substantially aligned.