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Physical Review A

Accelerating quantum optimal control through iterative gradient-ascent pulse engineering

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

Published Nov 7, 2023• 10 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.108.052603

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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.
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.
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.
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.
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.
An exquisite control over the dynamics of a quantum system is essential for realizing any quantum technology. This thesis presents some practical and efficient strategies for coherent quantum control: a paradigm for controlling quantum dynamics using electromagnetic fields. Coherent control is a common framework for describing the physical implementation of quantum logic gates in several quantum information processing platforms including superconducting qubits and trapped ions. For the purpose of demonstration, however, liquid-state Nuclear Magnetic Resonance (NMR) is used in this thesis as it is an ideal test-bed for developing, testing and benchmarking coherent control tools and techniques. Coherent control was revitalised in 2005 by the development of Gradient Ascent Pulse Engineering (GRAPE), a powerful optimal control technique that is widely used for designing shaped pulses for a variety of tasks. GRAPE, like any other closed-loop optimal control algorithm, requires multiple evaluations of the computationally expensive matrix exponential and the full time-evolution propagator. To this end, I present strategies to sidestep the explicit evaluation of matrix exponentials, thereby offering substantial speed-ups over conventional implementations of the GRAPE algorithm. The results are demonstrated experimentally in NMR, but also in simulations of ultracold molecules and superconducting qubits. Beyond GRAPE, this thesis also considers methods for designing spin-echo sequences for implementing any arbitrary network of controlled-Z gates in a system of qubits with zz-couplings. First, a method based on linear-programming provides near time-optimal sequences in fully-coupled systems having up to a few hundred qubits. Further, an analytic method based on graph colouring finds near time-optimal spin echo sequences in engineered systems with any number of qubits having only nearest and next-nearest neighbour-couplings.
Efficient approaches to quantum control and feedback are essential for quantum technologies, from sensing to quantum computation.Open-loop control tasks have been successfully solved using optimization techniques, including methods such as gradient-ascent pulse engineering (GRAPE) , relying on a differentiable model of the quantum dynamics.For feedback tasks, such methods are not directly applicable, since the aim is to discover strategies conditioned on measurement outcomes.In this work, we introduce feedback GRAPE, which borrows some concepts from model-free reinforcement learning to incorporate the response to strong stochastic (discrete or continuous) measurements, while still performing direct gradient ascent through the quantum dynamics.We illustrate its power considering various scenarios based on cavity-QED setups.Our method yields interpretable feedback strategies for state preparation and stabilization in the presence of noise.Our approach could be employed for discovering strategies in a wide range of feedback tasks, from calibration of multiqubit devices to linear-optics quantum computation strategies, quantum enhanced sensing with adaptive measurements, and quantum error correction.