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arXiv:2212.02806

Iterative Gradient Ascent Pulse Engineering algorithm for quantum optimal control

Yuquan Chen, Yajie Hao, Ze Wu, Bi-Ying Wang, Ran Liu, Yanjun Hou, Jiangyu Cui, Man‐Hong Yung, Xinhua Peng

Published Dec 6, 2022• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2212.02806

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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.
Quantum optimal control, a powerful toolbox for engineering an optimal control field modulation that most precisely implements a desired quantum operation in the best way possible, has evolved into one of the cornerstones for enabling quantum technologies. The gradient ascent pulse engineering (GRAPE) algorithm is a widely used method in quantum optimal control, which has achieved great success in different physical platforms. However, its computational complexity increases exponentially with the number of qubits, making it challenging to be implemented for large-scale quantum systems. To mitigate this issue, we present the iterative GRAPE algorithm (iGRAPE), which reduces the optimization problem into a series of lower-dimensional subproblems by incorporating disentanglement operations. Our numerical simulations on physical platforms such as nuclear magnetic resonance and superconducting quantum systems demonstrate that iGRAPE significantly enhances state preparation speed. Specifically, compared to GRAPE, iGRAPE achieves up to a five-fold acceleration in preparing Greenberger--Horne--Zeilinger states using a 12-qubit implementation, and up to a 13-fold acceleration for arbitrary state preparation with eight qubits. To further validate our findings, we conduct experimental validation of iGRAPE on a four-qubit nuclear magnetic resonance system. Overall, iGRAPE offers an efficient solution for implementing optimal control in large-scale quantum systems, holding great potential for advancing quantum technologies during the noisy intermediate-scale quantum era.
We present an iterative optimal control method of quantum systems, aimed at an implementation of a desired operation with optimal fidelity. The update step of the method is based on the linear response of the fidelity to the control operators, and its projection onto the mode functions of the corresponding operator. Our method extends methods such as gradient-ascent pulse engineering (GRAPE) and variational quantum algorithms, by determining the fidelity gradient in a hyperparameter-free manner, and using it for a multiparameter update, capitalizing on the multimode overlap of the perturbation and the mode functions. This directly reduces the number of dynamical trajectories that need to be evaluated in order to update a set of parameters. We demonstrate this approach, and compare it to the standard GRAPE algorithm, for the example of a quantum gate on two qubits, demonstrating a clear improvement in convergence and optimal fidelity of the generated protocol.
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.
Abstract As a crucial component of quantum technologies, quantum optimal control enables the high-fidelity engineering of desired quantum states and operations. Conventional approaches, such as the widely used gradient ascent pulse engineering (GRAPE) algorithm, typically rely on dense pulse sequences with a large number of control parameters, leading to inefficient optimization and increased sensitivity to experimental imperfections. In this work, we propose the alternating control–evolution (ACE) protocol, a flexible framework that constructs quantum operations by interleaving elementary control operations with tunable free evolutions, thereby enabling the design of sparse pulse sequences that exploit intrinsic system dynamics. This design substantially reduces the number of control parameters while retaining high expressibility and control fidelity. Numerical simulations on nuclear magnetic resonance systems show that the ACE protocol achieves more than a 20-fold reduction in the number of control parameters compared to GRAPE, with comparable fidelity in state preparation tasks. We further experimentally validate the ACE protocol on a four-qubit NMR platform by preparing a Greenberger–Horne–Zeilinger (GHZ) state with a fidelity of 99.52%. These results demonstrate that the ACE protocol provides an efficient and experimentally robust strategy for quantum optimal control, particularly suitable for the noisy intermediate-scale quantum (NISQ) era.
Zero-field nuclear magnetic resonance (NMR) has been recently developed as a complementary tool for nondestructive structural investigations of matter. However, precise and efficient control in such systems is still at the beginning. Based on the controllability of spin systems, we theoretically study the issue of quantum control in zero-field NMR by using a numerical optimization method based on the gradient ascent pulse engineering (GRAPE) algorithm. A set of quantum gates, including single-qubit and two-qubit gates, are achieved as examples. In particular, we show that homonuclear spins can be individually manipulated in zero-field NMR by the GRAPE method, which extends the ability to manipulate spin systems at zero magnetic field. Quantum control realized by the numerical GRAPE method features the properties of high fidelity, robustness, and hardware-friendly characteristics. High-fidelity quantum control in zero-field NMR systems might provide promising applications in material science, chemical analysis, quantum information processing, and fundamental physics.