Shadow Hamiltonian Simulation
Abstract
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 Schrodinger 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
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 Schrodinger 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
