Classical Hardness of Learning Functions of Hamiltonians

תקציר מקורי באנגליתarXiv:2610.01141v1 Announce Type: cross Abstract: Morohoshi, Nakayama, Manabe, and Mitarai proposed a physically motivated quantum machine learning problem in which the goal is to predict quantities of the form $\operatorname{Tr}[f(H)\rho]$ from classical descriptions of a Hamiltonian $H$ and a quantum state $\rho$, where $f$ is an unknown function. We call this problem Hamiltonian function learning in this paper. They constructed an efficient quantum learning algorithm under suitable conditions, while leaving a rigorous proof of average-case classical hardness open. In this paper, we rigorously prove the average-case classical hardness for two distribution-specific Hamiltonian function learning problems for $f_{\cos,\pi}(\lambda)=\cos(\pi\lambda)$ and $f_{\exp,\beta}(\lambda)=e^{-\beta\la