GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls

Abstract

The GRadient Ascent Pulse Engineering (GRAPE) method is widely used for optimization in quantum control. GRAPE is gradient search method based on exact expressions for gradient of the control objective. It has been applied to various coherently controlled closed and open quantum systems. In this work, we adopt the GRAPE method for optimizing objective functionals in open quantum systems driven by both coherent and incoherent controls. In our case, a tailored or engineered environment acts on the controlled system as control via time-dependent decoherence rates γ i ( t ) or, equivalently, via spectral density n ω ( t ) of the environment. To develop the GRAPE approach for this problem, we compute gradient of various objectives for general N-level open quantum systems for the piecewise constant class of control. The case of a single qubit is considered in details and solved analytically. For this case, an explicit analytical expression for evolution and objective gradient is obtained via diagonalization of a 3 × 3 matrix determining the system’s dynamics in the Bloch ball. The diagonalization is obtained by solving a cubic equation via Cardano’s method. The efficiency of the algorithm is demonstrated through numerical simulations for the state-to-state transition problem and its complexity is estimated. Robustness of the optimal controls is also studied.

Optimal remote restoring of quantum states in communication lines via local magnetic field

“…Originally developed for finding NMR pulse sequences for quantum systems [49], it was later extended and applied to various systems [44,[50][51][52][53]. Recently, GRAPE approach was applied to the analysis of quantum control landscapes for single spins with Hamiltonian dynamics and driven by only coherent control in [54] and for single spins interacting with the environment and driven by both coherent control and the environmental drive in [44]. Open GRAPE was developed to realize a CNOT with high fidelities [50].…”

Employing typicality in optimal control theory: Addressing large Hilbert spaces

“…Optimal control theory (OCT) has addressed open-system control problems [6][7][8][9][10][11][12][13][14]. In this case the generator of the dynamics L becomes explicitly time-dependent a function of the control field L(ϵ(t)).…”

Amplification of quantum transfer and quantum ratchet

“…[19][20][21]. In this regard, a constructive significant role of incoherent light (called as incoherent control) for quantum control and quantum transport in general context based on the scheme of environment-assisted quantum control was shown in [28,29] and importance of incoherent light for quantum transport is actively studied now for various quantum systems [30][31][32]. Environment-assisted quantum transport was studied also, e.g., in [33], etc.…”