# Determining system Hamiltonian from eigenstate measurements without correlation functions

## TL;DR

This paper introduces a novel algorithm that reconstructs local Hamiltonians solely from local eigenstate measurements, bypassing the need for nonlocal correlation data, and demonstrates its effectiveness through numerical tests.

## Contribution

It presents a new method to determine local Hamiltonians using only local measurements, reformulating the problem as an optimization task and employing machine learning techniques.

## Key findings

- Successful numerical reconstruction of random local Hamiltonians.
- Local measurements suffice to recover the full Hamiltonian in generic cases.
- The method avoids reliance on nonlocal correlation functions.

## Abstract

Local Hamiltonians arise naturally in physical systems. Despite its seemingly `simple' local structure, exotic features such as nonlocal correlations and topological orders exhibit in eigenstates of these systems. Previous studies for recovering local Hamiltonians from measurements on an eigenstate $|\psi\rangle$ require information of nonlocal correlation functions. In this work, we develop an algorithm to determine local Hamiltonians from only local measurements on $|\psi\rangle$, by reformulating the task as an unconstrained optimization problem of certain target function of Hamiltonian parameters, with only polynomial number of parameters in terms of system size. We also develop a machine learning-based-method to solve the first-order gradient used in the algorithm. Our method is tested numerically for randomly generated local Hamiltonians and returns promising reconstruction in the desired accuracy. Our result shed light on the fundamental question on how a single eigenstate can encode the full system Hamiltonian, indicating a somewhat surprising answer that only local measurements are enough without additional assumptions, for generic cases.