Shadow Hamiltonian Simulation

TL;DR

This paper introduces a novel quantum simulation method using a compressed 'shadow state' that efficiently encodes expectation values, enabling simulation of large systems and complex operators beyond traditional approaches.

Contribution

It proposes the shadow Hamiltonian simulation approach, a new method that efficiently simulates quantum dynamics using a compressed state representation.

Findings

Efficient simulation of large free fermion and boson systems.

Ability to compute complex operator expectations like Green's functions.

Reduces exponential resource requirements for certain quantum simulations.

Abstract

Simulating quantum dynamics is one of the most important applications of quantum computers. Traditional approaches for quantum simulation involve preparing the full evolved state of the system and then measuring some physical quantity. Here, we present a different and novel approach to quantum simulation that uses a compressed quantum state that we call the ``shadow state''. The amplitudes of this shadow state are proportional to the time-dependent expectations of a specific set of operators of interest, and it evolves according to its own Schr\"odinger equation. This evolution can be simulated on a quantum computer efficiently under broad conditions. Applications of this approach to quantum simulation problems include simulating the dynamics of exponentially large systems of free fermions or free bosons, the latter example recovering a recent algorithm for simulating exponentially many classical harmonic oscillators. These simulations are hard for classical methods and also for traditional quantum approaches, as preparing the full states would require exponential resources. Shadow Hamiltonian simulation can also 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.