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arXiv:2403.13461 · Russian Microelectronics

Quantum Control by the Environment: Turing Uncomputability, Optimization over Stiefel Manifolds, Reachable Sets, and Incoherent GRAPE

Alexander Nikolaevich Pechen

Published Dec 1, 2023• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1134/s1063739723600826

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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.
Methods of optimization and control of quantum systems are widely used for various applications in quantum technologies including laser control of chemical reactions, nuclear magnetic resonance, gate generation for quantum computing, describing quantum transport in photosynthesis, etc. Among many topics in quantum control, we briefly outline Riemannian optimization and gradient projection method (GPM) on complex Stiefel manifolds and their relation to quantum control, as well as various related optimization tools such as incoherent GRAPE (InGRAPE) modification of the gradient ascent pulse engineering (GRAPE) algorithm, GPM in the dynamic picture for constrained optimization, and demonstrate operation of GRAPE and GPM for an investigation of the optimization efficiency over unconstrained and constrained quantum control landscapes for an example of a three-level Ξ\Xi Ξ -system having a seventh-order trap.1
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.
Gradient ascent pulse engineering algorithm (GRAPE) is a typical method to solve quantum optimal control problems. However, it suffers from an exponential resource in computing the time evolution of quantum systems with the increasing number of qubits, which is a barrier for its application in large-qubit systems. To mitigate this issue, we propose an iterative GRAPE algorithm (iGRAPE) for preparing a desired quantum state, where the large-scale, resource-consuming optimization problem is decomposed into a set of lower-dimensional optimization subproblems by disentanglement operations. Consequently these subproblems can be solved in parallel with less computing resources. For physical platforms such as nuclear magnetic resonance (NMR) and superconducting quantum systems, we show that iGRAPE can provide up to 13-fold speedup over GRAPE when preparing desired quantum states in systems within 12 qubits. Using a four-qubit NMR system, we also experimentally verify the feasibility of the iGRAPE algorithm.
The GRAPE...jl package (https://github.com/JuliaQuantumControl/GRAPE.jl) implements Gradient Ascent Pulse Engineering, a widely used method of quantum optimal control. Its purpose is to find controls that steer a quantum system in a particular way. This is a prerequisite for next-generation quantum technology, such as quantum computing or quantum sensing. GRAPE...jl exploits the unique strengths of the Julia programming language to achieve both flexibility and numerical performance. It builds on the QuantumControl...jl framework.
Abstract Optimization on matrix manifolds, particularly on complex Stiefel manifolds, plays an important role in various tasks of modern quantum theory. The dynamics of open quantum systems are described by quantum channels, also known as Kraus maps. Properties of the geometric structure of the set of quantum channels are essential for various tasks in diverse areas of quantum theory ranging from quantum information theory to quantum computing and quantum control. This review summarizes the results on representing quantum channels as orbits of a unitary group acting on the complex Stiefel manifold and examines the corresponding quotient space geometry. The implication of this framework for Riemannian optimization is demonstrated with a particular emphasis on the absence of suboptimal extrema for broad classes of quantum control objective functionals.