Evolution-Free Hamiltonian Parameter Estimation through Zeeman Markers

We provide a protocol for Hamiltonian parameter estimation which relies only on the Zeeman effect. No time-dependent quantities need to be measured; it fully suffices to observe spectral shifts induced by fields applied to local “markers.” We demonstrate the idea with a simple tight-binding Hamiltonian and numerically show stability with respect to Gaussian noise on the spectral measurements. Then we generalize the result to show applicability to a wide range of systems, including quantum spin chains, networks of qubits, and coupled harmonic oscillators, and suggest potential experimental implementations.

Figure 1

We estimate the parameters $c_n$ of a Hamiltonian $H$ given by Eq. (1) by actively modifying it to $H^{\prime }$, applying a time-independent local field on the “Zeeman marker” (orange). Performing spectroscopy on $H$ and $H^{\prime }$ yields two spectra, $\{e_k\}$ and $\{e_n^{\prime }\}$, from which we compute the $c_n$ through Eqs. (2) and (3).

Figure 2

Average error in the estimation of the $c_n$ in Eq. (1). Shown is $\sum _{n=1}^{N-1}(c_n-{\widehat{c}}_n{)}^2/(N-1)$ averaged over 10000 samples for chains of length $N=2,\dots ,20$. The true couplings were taken to be $c_n=1$ and the estimated couplings ${\widehat{c}}_n$ were numerically computed using the tomography scheme described in the text. The field parameter was chosen to be large ($f=10$). The four lines correspond to Gaussian noise in the spectral measurements of standard deviations 0.01–0.08.