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arXiv:2006.00935 · Physical Review A

Hessian-based optimization of constrained quantum control

Mogens Dalgaard, Felix Motzoi, Jesper Hasseriis Mohr Jensen, Jacob Friis Sherson

Published Oct 21, 2020• 32 citations
arXivHTMLPDF

Abstract

Efficient optimization of quantum systems is a necessity for reaching fault-tolerant thresholds. A standard tool for optimizing simulated quantum dynamics is the gradient-based 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 second-order analytical derivatives of the coherent dynamics and find improvements compared to the standard of optimizing with the approximate second-order Broyden-Fletcher-Goldfarb-Shanno algorithm. We demonstrate performance improvements for both the best and the average errors of constrained unitary gate synthesis on a circuit-QED system over a broad range of different gate durations.

DOI
10.1103/physreva.102.042612

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Efficient optimization of quantum systems is a necessity for reaching fault-tolerant thresholds. A standard tool for optimizing simulated quantum dynamics is the gradient-based 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 second-order analytical derivatives of the coherent dynamics and find improvements compared to the standard of optimizing with the approximate second-order Broyden-Fletcher-Goldfarb-Shanno algorithm. We demonstrate performance improvements for both the best and the average errors of constrained unitary gate synthesis on a circuit-QED system over a broad range of different gate durations.
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.
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.
Methods of optimization and control of quantum systems are widely used for various applications in quantum technologies including laser control of chemical reactions, nuclear magnetic resonance, gate generation for quantum computing, describing quantum transport in photosynthesis, etc. Among many topics in quantum control, we briefly outline Riemannian optimization and gradient projection method (GPM) on complex Stiefel manifolds and their relation to quantum control, as well as various related optimization tools such as incoherent GRAPE (InGRAPE) modification of the gradient ascent pulse engineering (GRAPE) algorithm, GPM in the dynamic picture for constrained optimization, and demonstrate operation of GRAPE and GPM for an investigation of the optimization efficiency over unconstrained and constrained quantum control landscapes for an example of a three-level Ξ\Xi Ξ -system having a seventh-order trap.1
Accurate derivatives are important for locally traversing and converging efficiently in quantum optimization landscapes. By deriving analytically exact control derivatives (gradient and Hessian) for unitary control tasks, we show here that the computational feasibility of meeting this accuracy requirement depends on the choice of propagation scheme and problem representation. Even when exact propagation is sufficiently cheap it is, perhaps surprisingly, much more efficient to optimize the (appropriately) approximate propagators: Approximations in the dynamics are traded off for significant complexity reductions in the exact derivative calculations. Importantly, past the initial analytical considerations, only standard numerical techniques are explicitly required with straightforward application to realistic systems. These results are numerically verified for two concrete problems of increasing Hilbert space dimensionality. The best schemes obtain unit fidelity to machine precision, whereas the results for other schemes are separated consistently by orders of magnitude in computation time and in worst-case ten orders of magnitude in achievable fidelity. Since these gaps continually increase with system size and complexity, this methodology allows numerically efficient optimization of very high-dimensional dynamics, e.g., in many-body contexts, operating in the high-fidelity regime.
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
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.