Quantum Optimal Control via Semi-Automatic Differentiation

Michael H. Goerz, Sebastián C. Carrasco, and Vladimir S. Malinovsky

DEVCOM Army Research Laboratory, 2800 Powder Mill Road, Adelphi, MD 20783, USA

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Abstract

We develop a framework of "semi-automatic differentiation" that combines existing gradient-based methods of quantum optimal control with automatic differentiation. The approach allows to optimize practically any computable functional and is implemented in two open source Julia packages, $\tt{GRAPE.jl}$ and $\tt{Krotov.jl}$, part of the $\tt{QuantumControl.jl}$ framework. Our method is based on formally rewriting the optimization functional in terms of propagated states, overlaps with target states, or quantum gates. An analytical application of the chain rule then allows to separate the time propagation and the evaluation of the functional when calculating the gradient. The former can be evaluated with great efficiency via a modified GRAPE scheme. The latter is evaluated with automatic differentiation, but with a profoundly reduced complexity compared to the time propagation. Thus, our approach eliminates the prohibitive memory and runtime overhead normally associated with automatic differentiation and facilitates further advancement in quantum control by enabling the direct optimization of non-analytic functionals for quantum information and quantum metrology, especially in open quantum systems. We illustrate and benchmark the use of semi-automatic differentiation for the optimization of perfectly entangling quantum gates on superconducting qubits coupled via a shared transmission line. This includes the first direct optimization of the non-analytic gate concurrence.

In quantum optimal control theory, we seek to find control fields, e.g., the amplitude of a microwave pulse in a superconducting circuit, to steer a quantum system in some way, e.g., to implement an entangling gate for a quantum computer. A standard method is to iteratively improve the control field based on the gradient of the error with respect to the amplitudes of the control field at each point in time, based on a numerical simulation of the quantum dynamics. In recent years, automatic differentiation (AD) has been adapted from machine learning to quantum control to easily obtain the gradients for arbitrary, even non-analytical optimization functionals. However, AD has an exorbitant numerical overhead, which prevents it from scaling to problems of larger size. Here, we develop the concept of "semi-automatic differentiation" (semi-AD) that eliminates this overhead and can optimize for arbitrary functionals with the same numerical cost as the traditional GRAPE method. We have implemented semi-AD in the Julia programming language in the GRAPE.jl and Krotov.jl packages, part of the QuantumControl.jl framework, and use this implementation to benchmark against both a full-AD optimization and traditional GRAPE. We also demonstrate the power of the approach by directly optimizing the non-analytical entangling power of a quantum gate for the first time.

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Cited by

[1] Georg Raithel, Alisher Duspayev, Bineet Dash, Sebastián C. Carrasco, Michael H. Goerz, Vladan Vuletić, and Vladimir S. Malinovsky, "Principles of tractor atom interferometry", Quantum Science and Technology 8 1, 014001 (2023).

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