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arXiv:2403.17388 · Russian Microelectronics

Incoherent GRAPE for Optimization of Quantum Systems with Environmentally Assisted Control

Vadim N. Petruhanov, Alexander Nikolaevich Pechen

Published Dec 1, 2023• 4 citations
arXivHTMLPDF

Abstract

Abstract In this work, we review several results on development and application of incoherent version of GRAPE (Gradient Ascent Pulse Engineering) approach to optimization for open quantum systems driven by both coherent and incoherent controls. In the incoherent control approach, the environment serves as a control together with coherent field, and decoherence rates become generally time-dependent. For a qubit, explicit analytic expressions for evolution of the density matrix were obtained by solving a cubic equation via Cardano method. We discuss applications of incoherent GRAPE method (inGRAPE) to high fidelity gate generation for open one- and two-qubit systems and surprising properties of the underlying control landscapes, forming two groups — smooth single peak landscapes for Hadamard, C-NOT and C-Z gates, and more complicated with two peaks for T (or π/8) gate. For a qutrit, a formulation of the environment-assisted incoherent control with time-dependent decoherence rates is provided.

DOI
10.1134/s1063739723600784

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Abstract In this work, we review several results on development and application of incoherent version of GRAPE (Gradient Ascent Pulse Engineering) approach to optimization for open quantum systems driven by both coherent and incoherent controls. In the incoherent control approach, the environment serves as a control together with coherent field, and decoherence rates become generally time-dependent. For a qubit, explicit analytic expressions for evolution of the density matrix were obtained by solving a cubic equation via Cardano method. We discuss applications of incoherent GRAPE method (inGRAPE) to high fidelity gate generation for open one- and two-qubit systems and surprising properties of the underlying control landscapes, forming two groups — smooth single peak landscapes for Hadamard, C-NOT and C-Z gates, and more complicated with two peaks for T (or π/8) gate. For a qutrit, a formulation of the environment-assisted incoherent control with time-dependent decoherence rates is provided.
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.
In this work, we consider a model of two qubits driven by coherent and incoherent time-dependent controls. The dynamics of the system is governed by a Gorini–Kossakowski–Sudarshan–Lindblad master equation, where coherent control enters into the Hamiltonian and incoherent control enters into both the Hamiltonian (via Lamb shift) and the dissipative superoperator. We consider two physically different classes of interaction with coherent control and study the optimal control problem of state preparation formulated as minimization of the Hilbert–Schmidt distance’s square between the final density matrix and a given target density matrix at some fixed target time. Taking into account that incoherent control by its physical meaning is a non-negative function of time, we derive an analytical expression for the gradient of the objective and develop optimization approaches based on adaptation for this problem of GRadient Ascent Pulse Engineering (GRAPE). We study evolution of the von Neumann entropy, purity, and one-qubit reduced density matrices under optimized controls and observe a significantly different behavior of GRAPE optimization for the two classes of interaction with coherent control in the Hamiltonian.
Abstract The ability to control quantum systems is necessary for many applications of quantum technologies ranging from gate generation in quantum computation to NMR and laser control of chemical reactions. In many practical situations, the controlled quantum systems are open, i.e., interacting with the environment. While often influence of the environment is considered as an obstacle for controlling the systems, in some cases it can be exploited as a useful resource. In this note, we briefly review some results on control of open quantum systems using environment as a resource, including Turing uncomputability of discrete quantum control, parametrization of Kraus maps by points of the Stiefel manifolds and corresponding Riemanninan optimization, control by dissipation and time-dependent decoherence rates, reachable sets, and incoherent GRAPE (Gradient Ascent Pulse Engineering) for gradient-based optimization.
In this work, we consider an environment formed by incoherent photons as a resource for controlling open quantum systems via an incoherent control. We exploit a coherent control in the Hamiltonian and an incoherent control in the dissipator which induces the time-dependent decoherence rates γk(t) (via time-dependent spectral density of incoherent photons) for generation of single-qubit gates for a two-level open quantum system which evolves according to the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation with time-dependent coefficients determined by these coherent and incoherent controls. The control problem is formulated as minimization of the objective functional, which is the sum of Hilbert-Schmidt norms between four fixed basis states evolved under the GKSL master equation with controls and the same four states evolved under the ideal gate transformation. The exact expression for the gradient of the objective functional with respect to piecewise constant controls is obtained. Subsequent optimization is performed using a gradient type algorithm with an adaptive step size that leads to oscillating behaviour of the gradient norm vs. iterations. Optimal trajectories in the Bloch ball for various initial states are computed. A relation of quantum gate generation with optimization on complex Stiefel manifolds is discussed. We develop methodology and apply it here for unitary gates as a testing example. The next step is to apply the method for generation of non-unitary processes and to multi-level quantum systems.
Pulses to steer the time evolution of quantum systems can be designed with optimal control theory. In most cases it is the coherent processes that can be controlled and one optimizes the time evolution toward a target unitary process, sometimes also in the presence of noncontrollable incoherent processes. Here we show how to extend the gradient ascent pulse engineering (GRAPE) algorithm in the case where the incoherent processes are controllable and the target time evolution is a nonunitary quantum channel. We perform a gradient search on a fidelity measure based on Choi matrices. We illustrate our algorithm by optimizing a phase qubit measurement pulse. We show how this technique can lead to a large measurement contrast close to 9999%99. We also show, within the validity of our model, that this algorithm can produce short 1.41.41.4-ns pulses with 98.298.2%98.2 contrast.
We consider an open qutrit system in which the evolution of the density matrix ρ(t)\rho(t)ρ(t) is governed by the Gorini–Kossakowski–Sudarshan–Lindblad master equation with simultaneous coherent (in the Hamiltonian) and incoherent (in the dissipation superoperator) controls. To control the qutrit, we propose to use not only coherent control but also generally time-dependent decoherence rates which are adjusted by the so-called incoherent control. In our approach, the incoherent control makes the decoherence rates time-dependent in a specific controlled manner and within a clear physical mechanism. We consider the problem of maximizing the Hilbert–Schmidt overlap between the final state ρ(T)\rho(T)ρ(T) of the system and a given target state ρtarget\rho_{\text{target}}ρtarget​ , as well as the problem of minimizing the squared Hilbert–Schmidt distance between these states. For both problems, we perform their realifications, derive the corresponding Pontryagin functions, adjoint systems (with two variants of transversality conditions for the two terminal objectives), and gradients of the objectives, and adapt the one-, two-, and three-step gradient projection methods. For the problem of maximizing the overlap, we also adapt the regularized first-order Krotov method. In the numerical experiments, we analyze first the operation of the methods and second the obtained control processes, in respect of considering the environment as a resource via incoherent control.