Numerical estimation of reachable and controllability sets for a two-level open quantum system driven by coherent and incoherent controls

Abstract

The article considers a two-level open quantum system, whose evolution is governed by the Gorini-Kossakowski-Lindblad-Sudarshan master equation with Hamiltonian and dissipation superoperator depending, correspondingly, on piecewise constant coherent and incoherent controls with constrained magnitudes. Additional constraints on controls' variations are also considered. The system is analyzed using Bloch parametrization of the system's density matrix. We adapt the section method for obtaining outer parallelepipedal and pointwise estimations of reachable and controllability sets in the Bloch ball via solving a number of problems for optimizing coherent and incoherent controls with respect to some objective criteria. The differential evolution and dual annealing optimization methods are used. The numerical results show how the reachable sets' estimations depend on distances between the system's initial states and the Bloch ball's center point, final times, constraints on controls' magnitudes and variations. * arXiv version of the article [Oleg V. Morzhin and Alexander N. Pechen, "Numerical estimation of reachable and controllability sets for a two-level open quantum system driven by coherent and incoherent controls",

Математика Квантовых Технологий

“…Эта система рассматривалась в последующих работах [29][30][31]33]. Здесь ρ(t) является (2 × 2)матрицей плотности, т.е.…”

“…Если {h k } N j=1 = {(±1, 0, 0), (0, ±1, 0), (0, 0, ±1)} (здесь N = 6), то вместо Φ 1 (u) рассматриваем следующий целевой функционал: В работах [34,49] получение внешней параллелепипедной оценки для МД рассматривалось только как первый шаг в методе сечений для оценивания МД. В нашей работе [33] метод сечений был адаптирован для оценки МД и МУ системы (3.1) с кусочно постоянными управлениями v, n при наличии или отсутствии дополнительных ограничений на вариации этих управлений наряду с ограничениями на величины самих v и n.…”

Optimal control for state preparation in two-qubit open quantum systems driven by coherent and incoherent controls via GRAPE approach

“…Uncomputability of discrete quantum control was shown via establishing a relation with Diophantine equations and tenth Hilbert problem [39]. Numerical optimization schemes for a qubit driven by coherent and incoherent controls were studied in [40,41,42,43,44,45] for various objective criteria, for studying reachable and controllability sets, and for exploiting machine learning.…”

“…Such class of controls was used in [42] for considering these parameters together with T as outputs in the regression problem for obtaining suboptimal solutions of the time-minimal problem; here certain machine learning techniques were used. In [44], the problem of minimizing the Hilbert-Schmidt distance with a fixed final time was used for numerical estimation of reachable and controllability sets of a one-qubit system in the Bloch ball. In [41], the Uhlmann-Jozsa fidelity of the final density matrix, ρ(T ), for the one-qubit system driven by piecewise continuous coherent and incoherent controls was studied.…”

Reachable sets for two-level open quantum systems driven by coherent and incoherent controls

“…Controllability for coherently controlled open systems with GKSL master equations was investigated in [36,37]. Time-optimal control of dissipative two-level open quantum systems was studied using geometric control theory and other methods in [38,39,40,41,42,22,43,44]. Local properties such as the absence of traps for controlling a qubit have also been proved [45,46,47].…”