Shadow Hamiltonian simulation

Shadow Hamiltonian simulation is a recently proposed framework for simulating quantum dynamics using a compressed quantum state that we refer to as the "shadow state". The amplitudes of this shadow state are proportional to the expectations of a set of operators of interest, and the shadow state evolves according to its own Schrödinger equation. This evolution can be simulated on a quantum computer, and the simulation is efficient under broad conditions. Shadow Hamiltonian simulation enables the efficient solution to numerous problems in quantum simulation that would otherwise require exponential resources using traditional methods. Examples include simulating dynamics of exponentially large quantum systems of free fermions or bosons, and simulating dynamics of exponentially large systems of classical oscillators. The framework can be extended to other problems, including the simulation of the evolution of operators in the Heisenberg picture and learning unitary oracles.