Learning SYK Hamiltonians

Abstract: We study the problem of learning the dense Sachdev--Ye--Kitaev (SYK) Hamiltonian from copies of its Gibbs state. Existing algorithms for Hamiltonian learning typically rely on geometric locality or bounded interaction degree and therefore do not apply to SYK, where each quartic interaction overlaps with $Θ(n<sup>3)$ others. We show that this obstruction can be overcome by exploiting the random mean-field structure of the model. At any constant temperature, we prove that with high probability over the SYK couplings, the entire Hamiltonian can be learned to inverse-polynomial accuracy using polynomially many samples. Furthermore, when the inverse temperature is restricted to be a sufficiently small constant, we construct a quasipolynomial-time learning algorithm which is qualitatively different from the sample-efficient algorithm.