Fidelity freeze for a random matrix model with off-diagonal perturbation

The concept of fidelity has been introduced to characterize the stability of a quantum-mechanical system against perturbations. The fidelity amplitude is defined as the overlap integral of a wave packet with itself after the development forth and back under the influence of two slightly different Hamiltonians. It was shown by Prosen and Žnidarič in the linear-response approximation that the decay of the fidelity is frozen if the Hamiltonian of the perturbation contains off-diagonal elements only. In the present work the results of Prosen and Žnidarič are extended by a supersymmetry calculation to arbitrary strengths of the perturbation for the case of an unperturbed Hamiltonian taken from the Gaussian orthogonal ensemble and a purely imaginary antisymmetric perturbation. It is found that for the exact calculation the freeze of fidelity is only slightly reduced as compared to the linear-response approximation. This may have important consequences for the design of quantum computers.

Figure 1

Ensemble average of the fidelity amplitude $⟨f_{ϵ}\left(\tau \right)⟩$ with $H_0$ taken from the GOE and a purely imaginary antisymmetric perturbation [solid line, calculated from Eq. (51)] for different perturbation strengths $ϵ$. For comparison, the result from the linear response approximation (dashed line), and for a GOE perturbation (dashed-dotted line) are shown as well.

Figure 2

Ensemble average of the fidelity amplitude $⟨f_{ϵ}\left(\tau \right)⟩$ for an imaginary antisymmetric perturbation as a function of $ϵ$ for three fixed values of $\tau =0.5,1.0,1.5$ (from top to bottom, solid lines). Again the results from the linear response approximation are shown for comparison (dashed lines).