A recent study addresses the problem of "learning" the dense Sachdev-Ye-Kitaev (SYK) Hamiltonian from multiple copies of its Gibbs state. Existing algorithms for Hamiltonian learning typically rely on geometric locality or bounded interaction degree, characteristics that do not apply to the SYK model. This model is distinguished by its quartic interactions, where each overlaps with a number of other interactions of the order of Θ(n³), posing a significant challenge for conventional methods.

The researchers have shown that this difficulty can be overcome by exploiting the inherent random mean-field structure of the SYK model. They have proven that, at any constant temperature, it is possible to learn the entire Hamiltonian to inverse-polynomial accuracy using a polynomial number of samples. This is achieved with high probability over the SYK couplings, representing a significant advance in understanding and manipulating this type of complex system.

Furthermore, the study presents a quasipolynomial-time learning algorithm for the case where the inverse temperature is restricted to be a sufficiently small constant. This qualitatively different algorithm complements the sample-efficient method, offering an alternative pathway for learning SYK Hamiltonians under specific conditions. The findings open new possibilities for research in condensed matter and black hole physics, where the SYK model is relevant.