Derivatives of the hamiltonian eigenvalues with respect to parameters

L. M. Garrido, P. Pascual

Liubov A. Markovich, S. Malikis, Stefano Polla +1 more

The phase shift rules enable the estimation of the derivative of a quantum state with respect to phase parameters, providing valuable insights into the behavior and dynamics of quantum systems. This capability is essential in quantum simulation tasks where understanding the behavior of complex quantum systems is of interest, such as simulating chemical reactions or condensed matter systems. However, parameter shift rules are typically designed for Hamiltonian systems with equidistant eigenvalues. For systems with closely spaced eigenvalues, effective rules have not been established. We provide insights about the optimal design of a parameter shift rule, tailored to various sorts of spectral information that may be available. The proposed method lets derivatives be calculated for any system, regardless of how close the eigenvalues are to each other. It also optimizes the number of phase shifts, which reduces the amount of gate resources needed.

H. H. Aly, Roger C. Hill

Gennadi Aleksandrovich Sardanashvily

We provide geometric quantization of a completely integrable Hamiltonian system in the action-angle variables around an invariant torus with respect to the angle polarization. The carrier space of this quantization is the pre-Hilbert space of smooth complex functions on the torus. A Hamiltonian of a completely integrable system in this carrier space has a countable spectrum. If it is degenerate, its eigenvalues are countably degenerate. We study nonadiabatic perturbations of this Hamiltonian by a term depending on classical time-dependent parameters. It is originated by a connection on the parameter space, and is linear in the temporal derivatives of parameters. One can choose it commuting with a degenerate Hamiltonian of a completely integrable system. Then the corresponding evolution operator acts in the eigenspaces of this Hamiltonian, and is an operator of parallel displacement along a curve in the parameter space.

Francisco Marcelo Fernández

If the Hamiltonian operator depends on a parameter λ, which may be a particle charge or mass, a field strength, or simply a dummy parameter artificially introduced to apply perturbation theory, then the eigenvalues and eigenfunctions of Hˆ will also depend on that parameter. If we differentiate < |(Hˆ − E)| >= 0 with respect to λ 〈 ∂ ∂λ ∣∣∣(Hˆ − E)∣∣∣ 〉 + 〈 ∣∣∣∣∣ ∂(Hˆ − E) ∂λ ∣∣∣∣∣ 〉 + 〈 ∣∣∣(Hˆ − E)∣∣∣ ∂ ∂λ 〉 = 0 (3.5) and take into account equation (3.2), then we obtain the Hellmann-Feynman theorem [25] ∂E ∂λ < | >= 〈 ∣∣∣∣∣ ∂Hˆ ∂λ ∣∣∣∣∣ 〉 , (3.6) or ∂E ∂λ = 〈 ∂Hˆ ∂λ 〉 (3.7) if is normalized to unity. In what follows we show that the general results just derived facilitate the application of pertur- bation theory.

Leon Bess