Robustness of energy landscape control to dephasing

Abstract

As shown in previous work, in some cases closed quantum systems exhibit a non-conventional absence of trade-off between performance and robustness in the sense that controllers with the highest fidelity can also provide the best robustness to parameter uncertainty. As the dephasing induced by the interaction of the system with the environment guides the evolution to a more classically mixed state, it is worth investigating what effect the introduction of dephasing has on the relationship between performance and robustness. In this paper we analyze the robustness of the fidelity error, as measured by the logarithmic sensitivity function, to dephasing processes. We show that introduction of dephasing as a perturbation to the nominal unitary dynamics requires a modification of the log-sensitivity formulation used to measure robustness about an uncertain parameter with non-zero nominal value used in previous work. We consider controllers optimized for a number of target objectives ranging from fidelity under coherent evolution to fidelity under dephasing dynamics to determine the extent to which optimizing for a specific regime has desirable effects in terms of robustness. Our analysis is based on two independent computations of the log-sensitivity: a statistical Monte Carlo approach and an analytic calculation. We show that despite the different log-sensitivity calculations employed in this study, both demonstrate that the log-sensitivity of the fidelity error to dephasing results in a conventional trade-off between performance and robustness.

Analyzing and Unifying Robustness Measures for Excitation Transfer Control in Spin Networks

“…We further restrict the dynamics to the single excitation subspace and the case where control is achieved by external bias fields that shift the energy levels of particle n by ∆ n , resulting in an effective single excitation subspace Hamiltonian H ss given by a matrix with diagonal elements ∆ n and off-diagonal elements J mn . The closed system with no interaction with the environment evolves according to ρ(t) = − i ℏ [H ss , ρ(t)], where ρ(t) is density operator describing the state of the system [19], [20].…”

“…For rings, we consider transfers from spin 1 to OUT = 2 through ⌈N/2⌉. All controllers are optimized to maximize fidelity under varying conditions as described in [14], [20]. We evaluate the nominal fidelity error in the LTI formalism as e(T ) = 1 − cr(T ) where c ∈ R 1×N 2 is the transpose of r OUT .…”

“…To model the dephasing processes, we use the set of 1000 dephasing operators specific to spin networks of size N = 5 or 6, as employed in [20], normalized and tested to meet the physical complete positivity constraints [23]. We denote this set of dephasing operators by…”

“…In accordance with [20], we choose the log-sensitivity as one measure of robustness, calculated in two distinct ways: analytically and numerically. In the analytical case we calculate it directly from (7) as in [20]. For a given controller and dephasing process we have…”