Learning effective Hamiltonians for adaptive time-evolution quantum algorithms

TL;DR

This paper demonstrates that adaptive Trotter protocols, combined with quantum Hamiltonian learning, can reliably approximate the target Hamiltonian in digital quantum simulations, ensuring controlled errors in local and global dynamics.

Contribution

It introduces a method to use quantum Hamiltonian learning to improve adaptive Trotter algorithms, ensuring bounded errors in the generator of the dynamics.

Findings

Deviations from the target Hamiltonian remain bounded over time.

Adaptive Trotter algorithms reliably simulate local dynamics.

Global quantum states are controllably approximated.

Abstract

Digital quantum simulation of many-body dynamics relies on Trotterization to decompose the target time evolution into elementary quantum gates operating at a fixed equidistant time discretization. Recent advances have outlined protocols enabling more efficient adaptive Trotter protocols, which have been shown to exhibit a controlled error in the dynamics of local observables and correlation functions. However, it has remained open to which extent the errors on the actual generator of the dynamics, i.e., the target many-body Hamiltonian, remain controlled. Here, we propose to use quantum Hamiltonian learning to numerically obtain the effective Hamiltonian and apply it on the recently introduced ADA-Trotter algorithm as a concrete demonstration. Our key observation is that deviations from the target generator remain bounded on all simulation times. This result suggests that the ADA-Trotter not only generates reliable digital quantum simulation of local dynamics, but also controllably approximates the global quantum state of the target system. Our proposal is sufficiently general and readily applicable to other adaptive time-evolution algorithms.