When can a local Hamiltonian be recovered from a steady state?

TL;DR

This paper investigates the conditions under which local Hamiltonians of quantum spin chains can be reconstructed from a single steady state measurement, introducing methods and theoretical proofs for critical chain length determination.

Contribution

It introduces an alternative energy eigenvalue equation method and proves its equivalence to the homogeneous operator equation method for Hamiltonian recovery.

Findings

Recovery of Hamiltonians is possible at a critical chain length.

The EEE method fully reproduces HOE results.

Analytical expression for the rank of the constraint matrix is derived.

Abstract

With the development of quantum many-body simulator, Hamiltonian tomography has become an increasingly important technique for verification of quantum devices. Here we investigate recovering the Hamiltonians of two spin chains with 2-local interactions and 3-local interactions by measuring local observables. For these two models, we show that when the chain length reaches a certain critical number, we can recover the local Hamiltonian from its one steady state by solving the homogeneous operator equation (HOE) developed in Ref. [1]. To explain the existence of such a critical chain length, we develop an alternative method to recover Hamiltonian by solving the energy eigenvalue equations (EEE). By using the EEE method, we completely recovered the numerical results from the HOE method. Then we theoretically prove the equivalence between the HOE method and the EEE method. In particular, we obtain the analytical expression of the rank of the constraint matrix in the HOE method by using the EEE method, which can be used to determine the correct critical chain length in all the cases.