Characterizing errors in parameter estimation by local measurements

TL;DR

This paper analyzes how local measurement-based methods for estimating quantum Hamiltonian couplings in 1D chains are affected by interactions beyond nearest neighbors, revealing linear error scaling with perturbation strength.

Contribution

It extends existing 1D Hamiltonian estimation schemes to systems with next-nearest neighbor interactions, quantifying the robustness and error scaling under such perturbations.

Findings

Error in coupling estimation scales linearly with perturbation strength

Presence of beyond-nearest neighbor couplings increases average estimation error

Chains of up to 40 sites can be estimated with reasonable accuracy under weak interactions

Abstract

The indirect estimation of couplings in quantum dynamics relies on the measurement of the spectrum and the overlap of eigenvectors with some reference states. This data can be obtained by local measurements on some sites and eliminates the need for full Hamiltonian tomography. For a 1D chain, access to only one edge site is sufficient to compute all the couplings between the adjacent sites, and consequently to reconstruct the full Hamiltonian. However, its robustness in the presence of perturbations remains a critical question, particularly when sites interact with other lattice sites beyond nearest neighbors. Our work studies the applicability of schemes designed for 1D chains to topologies with interactions beyond nearest-neighbour. We treat interactions between the next-nearest sites as perturbation of strength $\varepsilon$ and show that the error in estimation of couplings scales linearly with $\varepsilon$ in the presence of such interactions. Further, we show that on average, the existence of couplings between sites beyond the next-nearest neighbour results in higher error. We also study the length of the chain that can be estimated (up to a fixed precision) as a function of $\varepsilon$, in the presence of next-nearest neighbour interactions. Typically, for weak interactions, chains of 40 sites can be estimated within reasonable error. Thus, we study the robustness of estimation scheme designed for a 1D chain when exposed to such multi-site perturbations, offering valuable insights into its applicability and limitations.