Spectral width and polynomial degree in perfect state transfer

Source abstract

We study the minimum time for perfect state transfer under polynomial Hamiltonians with bounded degree and spectral width. For a strongly cospectral pair and width bound $W$, the optimum, when finite, is an integer multiple of $π/W$, determined by integer interpolation with prescribed parities. For equally spaced supported eigenvalues with alternating signs, we give degree bounds under which every minimizer is affine, and sharp asymptotics for each fixed exact degree. Near-minimizing phase polynomials satisfy a quantitative Chebyshev stability estimate. We determine the optimal transfer time for every degree bound on hypercubes of odd prime dimension. For complementary vertices of $J(2m,m)$, the optimal time at fixed spectral width grows exponentially in $m$ throughout an interval of feasible degrees. We also construct polynomial Hamiltonians showing that every feasible degree $m-t$ with $t=o(m)$ admits subexponential transfer time.