Hamiltonian Determination with Restricted Access in Transverse Field Ising Chain

Abstract

We propose a method to evaluate parameters in the Hamiltonian of the Ising chain under site-dependent transverse fields, with a proviso that we can control and measure one of the edge spins only. We evaluate the eigenvalues of the Hamiltonian and the time-evoultion operator exactly for a 3-spin chain, from which we obtain the expectation values of σ x of the first spin. The parameters are found from the peak positions of the Fourier transform of the expectation value. There are four assumptions in our method, which are mild enough to be satisfied in many physical systems.

Estimation of coupling constants of a three-spin chain: a case study of Hamiltonian tomography with nuclear magnetic resonance

“…This paves the way for the identification of spins and couplings that are off-resonant or would usually be drowned by background noise. Whereas the methods of [5,8] rely on Fourier analysis, which can only be applied in systems clean enough for sufficiently long time data acquisition, our method can be applied in more noisy cases. We have shown that there is competition in the observed evolution between the amount of data acquired and the amount of noise coming in.…”

“…In contrast, in the present work we estimate the coupling constants by fitting the time domain data without making use of the spectra. The materials presented in this paper therefore provide the first step toward the full verification of the scheme discussed in [5,8]. While nonlinear least-square data fitting is not computationally efficient for large systems, we find that it is very suitable for the three-spin chain and much more robust against relaxation than the method discussed in [5,8].…”

“…The materials presented in this paper therefore provide the first step toward the full verification of the scheme discussed in [5,8]. While nonlinear least-square data fitting is not computationally efficient for large systems, we find that it is very suitable for the three-spin chain and much more robust against relaxation than the method discussed in [5,8]. Our finding thus paves the way for indirect estimation of a Hamiltonian of a small-scale noisy system, where direct methods are not applicable.…”

“…We emphasize that the method used here is different from that presented in [5,8], where the coupling constants are estimated from the energy eigenvalues found from the spectral peaks, which are evaluated from long time evolution of the spin at one end of the chain. In contrast, in the present work we estimate the coupling constants by fitting the time domain data without making use of the spectra.…”