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arXiv:2104.11936 · Physical Review Applied

Optimization of a Controlled- Z Gate with Data-Driven Gradient-Ascent Pulse Engineering in a Superconducting-Qubit System

Zhiwen Zong, Zhenhai Sun, Zhangjingzi Dong, Chongxin Run, Liang Xiang, Ze Zhan, Qianlong Wang, Ying Fei, Yaozu Wu, Wenyan Jin, Cong Xiao, Zhilong Jia, Peng Duan, Jianlan Wu, Yi Yin, Guoping Guo

Published Jun 2, 2021• 16 citations
arXivHTMLPDF

Abstract

The experimental optimization of a two-qubit controlled-$Z$ (cz) gate is realized following two different data-driven gradient-ascent pulse engineering (GRAPE) protocols with the aim of optimizing the gate operator and the output quantum state, respectively. For both GRAPE protocols, the key computation of gradients utilizes mixed information of the input $Z$-control pulse and the experimental measurement. With an imperfect initial pulse in a flattop waveform, our experimental implementation shows that the cz gate is quickly improved and the gate fidelities subject to the two optimized pulses are around 99%. Our experimental study confirms the applicability of the data-driven GRAPE protocols in the gate optimization problem.

DOI
10.1103/physrevapplied.15.064005

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The experimental optimization of a two-qubit controlled-ZZZ (cz) gate is realized following two different data-driven gradient-ascent pulse engineering (GRAPE) protocols with the aim of optimizing the gate operator and the output quantum state, respectively. For both GRAPE protocols, the key computation of gradients utilizes mixed information of the input ZZZ-control pulse and the experimental measurement. With an imperfect initial pulse in a flattop waveform, our experimental implementation shows that the cz gate is quickly improved and the gate fidelities subject to the two optimized pulses are around 99%. Our experimental study confirms the applicability of the data-driven GRAPE protocols in the gate optimization problem.
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 efficient implementation of quantum computing is contingent upon high-fidelity quantum operations. Nevertheless, the fidelity of these operations is constrained by the precision of quantum system evolution control. The optimization of quantum control pulses is essential for improving the manipulation accuracy of superconducting qubits. Traditional optimization methods, including gradient descent and the gradient ascent pulse engineering algorithm, frequently encounter challenges such as slow convergence and susceptibility to local optima in pulse optimization problems. This study introduces an adaptive open-loop optimization algorithm based on the adaptive moment estimation (Adam) optimizer, capitalizing on the benefits of momentum and adaptive learning rate adjustments inherent in Adam. We conduct experimental analysis on the algorithm’s hyperparameters to achieve optimal solutions with a higher fidelity range at a faster convergence speed, effectively improving the fidelity and optimization efficiency of quantum gate operations. Through numerical simulations on the QuTiP platform, we validate the excellent performance of the Adam algorithm in quantum gate optimization. In optimizing the X and SWAP gates, its fidelity improved by 0.03% and 0.0016%, respectively, compared to the GRAPE algorithm. Compared to the CRAB method, the initial convergence speed of the Adam method increased fivefold, enabling it to achieve the target fidelity more rapidly. Future research will investigate closed-loop optimization strategies, utilizing feedback derived from the fidelity results of actual quantum computers to further augment quantum control performance
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 propose a scheme for optimizing two-qubit gate operations in a decoherence-free subspace (DFS) using the gradient ascent pulse engineering (GRAPE) algorithm, where the system consists of nitrogen-vacancy (NV) centers in diamond coupled to a whispering-gallery-mode microresonator. The logical qubits in this scheme are encoded in the DFS, which effectively avoids collective decoherence and provides a long coherence time for high-fidelity quantum operations. The GRAPE algorithm is used to design and optimize the control pulse sequences required for the target operation, thereby compensating for common errors and ensuring that the quantum gate maintains high fidelity under non-ideal conditions. As examples, we successfully implement high-fidelity and robust controlled-phase and SWAP gates. Numerical simulations demonstrate the effectiveness and superiority of the proposed scheme. The strategy of combining encoding protection with optimal control shown in this work can be extended to implement universal quantum gates in complex quantum systems.
In the quest to achieve scalable quantum information processing technologies, gradient-based optimal control algorithms (e.g., grape) are broadly used for implementing high-precision quantum gates, but their performance is often hindered by deterministic or random errors in the system model and the control electronics. In this paper, we show that grape can be taught to be more effective by jointly learning from the design model and the experimental data obtained from process tomography. The resulting data-driven gradient optimization algorithm (d-grape) can in principle correct all deterministic gate errors, with a mild efficiency loss. The d-grape algorithm may become more powerful with broadband controls that involve a large number of control parameters, while other algorithms usually slow down due to the increased size of the search space. These advantages are demonstrated by simulating the implementation of a two-qubit controlled-not gate.
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.