  Given the recent developments in quantum techniques, modeling the physical
Hamiltonian of a target quantum many-body system is becoming an increasingly
practical and vital research direction. Here, we propose an efficient strategy
combining maximum likelihood estimation, gradient descent, and quantum
many-body algorithms. Given the measurement outcomes, we optimize the target
model Hamiltonian and density operator via a series of descents along the
quantum likelihood gradient, which we prove is negative semi-definite with
respect to the negative-log-likelihood function. In addition to such
optimization efficiency, our maximum-likelihood-estimate Hamiltonian learning
respects the locality of a given quantum system, therefore, extends readily to
larger systems with available quantum many-body algorithms. Compared with
previous approaches, it also exhibits better accuracy and overall stability
toward noises, fluctuations, and temperature ranges, which we demonstrate with
various examples.
