  Interacting spin networks are fundamental to quantum computing. Data-based
tomography of time-independent spin networks has been achieved, but an open
challenge is to ascertain the structures of time-dependent spin networks using
time series measurements taken locally from a small subset of the spins.
Physically, the dynamical evolution of a spin network under time-dependent
driving or perturbation is described by the Heisenberg equation of motion.
Motivated by this basic fact, we articulate a physics-enhanced machine learning
framework whose core is Heisenberg neural networks. In particular, we develop a
deep learning algorithm according to some physics motivated loss function based
on the Heisenberg equation, which "forces" the neural network to follow the
quantum evolution of the spin variables. We demonstrate that, from local
measurements, not only the local Hamiltonian can be recovered but the
Hamiltonian reflecting the interacting structure of the whole system can also
be faithfully reconstructed. We test our Heisenberg neural machine on spin
networks of a variety of structures. In the extreme case where measurements are
taken from only one spin, the achieved tomography fidelity values can reach
about 90%. The developed machine learning framework is applicable to any
time-dependent systems whose quantum dynamical evolution is governed by the
Heisenberg equation of motion.
