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arXiv:1802.07147 · Frontiers of Physics

Practical pulse engineering: Gradient ascent without matrix exponentiation

Gaurav Bhole, Jonathan A. Jones

Published May 18, 2018• 16 citations
arXivHTMLPDF

Abstract

Since 2005, there has been a huge growth in the use of engineered control pulses to perform desired quantum operations in systems such as nuclear magnetic resonance quantum information processors. These approaches, which build on the original gradient ascent pulse engineering algorithm, remain computationally intensive because of the need to calculate matrix exponentials for each time step in the control pulse. In this study, we discuss how the propagators for each time step can be approximated using the Trotter–Suzuki formula, and a further speedup achieved by avoiding unnecessary operations. The resulting procedure can provide substantial speed gain with negligible costs in the propagator error, providing a more practical approach to pulse engineering.

DOI
10.1007/s11467-018-0791-1

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Since 2005, there has been a huge growth in the use of engineered control pulses to perform desired quantum operations in systems such as nuclear magnetic resonance quantum information processors. These approaches, which build on the original gradient ascent pulse engineering algorithm, remain computationally intensive because of the need to calculate matrix exponentials for each time step in the control pulse. In this study, we discuss how the propagators for each time step can be approximated using the Trotter–Suzuki formula, and a further speedup achieved by avoiding unnecessary operations. The resulting procedure can provide substantial speed gain with negligible costs in the propagator error, providing a more practical approach to pulse engineering.
In designing quantum control, it is generally required to simulate the controlled system evolution with a classical computer. However, computing the time evolution operator can be quite resource consuming since the total Hamiltonian is often hard to diagonalize. In this paper, we mitigate this issue by substituting the time evolution segments with their Trotter decompositions, which reduces the propagator into a combination of single-qubit operations and fixed-time system evolutions. The resulting procedure can provide substantial speed gain with acceptable costs in the propagator error. As a demonstration, we apply the proposed strategy to improve the efficiency of the gradient ascent pulse engineering algorithm for searching optimal control fields. Furthermore, we show that the higher-order Trotter decompositions can provide efficient Ans\"atze for the variational quantum algorithm, leading to improved performance in solving the ground-state problem. The strategy presented here is also applicable for many other quantum optimization and simulation tasks.
More than ten years ago a first step toward quantum error correction (QEC) was implemented [Phys. Rev. Lett. 81, 2152 (1998)]. The work showed there was sufficient control in nuclear magnetic resonance to implement QEC, and demonstrated that the error rate changed from $\ensuremath{\epsilon}$ to $\ensuremath{\sim}{\ensuremath{\epsilon}}^{2}$. In the current work we reproduce a similar experiment using control techniques that have been since developed, such as the pulses generated by gradient ascent pulse engineering algorithm. We show that the fidelity of the QEC gate sequence and the comparative advantage of QEC are appreciably improved. This advantage is maintained despite the errors introduced by the additional operations needed to protect the quantum states.
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.
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.