  We present shadow Hamiltonian simulation, a framework for simulating quantum
dynamics using a compressed quantum state that we call the "shadow state". The
amplitudes of this shadow state are proportional to the expectations of a set
of operators of interest. The shadow state evolves according to its own
Schr\"odinger equation, and under broad conditions can be simulated on a
quantum computer. We analyze a number of applications of this framework to
quantum simulation problems. This includes simulating the dynamics of
exponentially large systems of free fermions, or exponentially large systems of
free bosons, the latter example recovering a recent algorithm for simulating
exponentially many classical harmonic oscillators. Shadow Hamiltonian
simulation can be extended to simulate expectations of more complex operators
such as two-time correlators or Green's functions, and to study the evolution
of operators themselves in the Heisenberg picture.
