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arXiv:1712.01780 · Physical Review A

Data-driven gradient algorithm for high-precision quantum control

Re-Bing Wu, Bing Chu, D.H. Owens, HERSCHEL A. RABITZ

Published Apr 24, 2018• 63 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.97.042122

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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.
In the hardware implementation of quantum computation, the control pulses for quantum states and gate operations must be calibrated to correct errors induced by unknown drifts of system parameters. The calibration can be iteratively done by checking the control precision, but the required randomized benchmarking or quantum tomography is experimentally expensive. In this paper, we propose that the control calibration and the quantum tomography can be taken as two separate but collaborative learning processes. Combining the gradient-descent pulse engineering and the adaptive tomography, the resulting c-GRAPE algorithm can substantially reduce the high cost of measurements without sacrificing the control accuracy. The effectiveness is demonstrated by numerical simulation examples.
High-precision control of quantum states and gate operations is essential to the hardware implementation of quantum computation. Recently, online calibration has become an important tool for correcting errors induced by parameter shifts or environmental noises in the underlying quantum control systems. However, the experimental cost for acquiring information through quantum tomography (for state or gate reconstruction) is very high, especially when many iterations are to be done. In this paper, we propose a novel scheme that integrates the gradient-descent optimization of quantum control pulses with the adaptive learning of quantum tomography as two interactive processes, which updates the control iteratively with the progressively refined state tomography. This scheme, which we call c-GRAPE, can greatly improve the calibration efficiency by substantial reduction the experimental cost for tomography without sacrificing the control precision.
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.
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.
Efficient optimization of quantum systems with large state or control spaces is a necessity for reaching fault tolerant thresholds. A standard tool for optimizing simulated quantum dynamics is the gradient-based \textsc{grape} algorithm, which has been successfully applied in a wide range of different branches of quantum physics. In this work, we derive and implement exact 2nd2^{\mathrm{nd}}2nd order analytical Hessian of the coherent dynamics and find improvements compared to the approximate 2nd2^{\mathrm{nd}}2nd order \textsc{bfgs} standard optimization. We demonstrate performance improvements for both the best and average errors of constrained unitary gate synthesis on a circuit-\textsc{qed} system over a broad range of different gate durations.
Quantum control optimization algorithms are routinely used to synthesize optimal quantum gates or to realize efficient quantum state transfers. The computational resource required for the optimization is an essential consideration in order to scale toward quantum control of larger registers. Here, we propose and demonstrate the use of a machine learning method, specifically the recommender system (RS), to deal with the challenge of enhancing computational efficiency. Given a sparse database of a set of products and their customer ratings, RS is used to efficiently predict unknown ratings. In the quantum control problem, each iteration of a numerical optimization algorithm typically involves evaluating a large number of parameters, such as gradients or fidelities, which can be tabulated as a rating matrix. We establish that RS can rapidly and accurately predict elements of such a sparse rating matrix. Using this approach, we expedite a gradient ascent based quantum control optimization, namely GRAPE, and demonstrate the faster construction of two-qubit CNOT gate in registers with up to 8 qubits. We also describe and implement the enhancement of the computational speed of a hybrid algorithm involving simulated annealing as well as gradient ascent. Moreover, the faster construction of three-qubit Toffoli gates further confirmed the applicability of RS in larger registers.