Quantum Optimal Control via Semi-Automatic Differentiation
Michael H. Goerz, Sebastián C. Carrasco, Vladimir S. Malinovsky
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,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="monospace">G</mml:mi><mml:mi mathvariant="monospace">R</mml:mi><mml:mi mathvariant="monospace">A</mml:mi><mml:mi mathvariant="monospace">P</mml:mi><mml:mi mathvariant="monospace">E</mml:mi><mml:mo mathvariant="monospace">.</mml:mo><mml:mi mathvariant="monospace">j</mml:mi><mml:mi mathvariant="monospace">l</mml:mi></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="monospace">K</mml:mi><mml:mi mathvariant="monospace">r</mml:mi><mml:mi mathvariant="monospace">o</mml:mi><mml:mi mathvariant="monospace">t</mml:mi><mml:mi mathvariant="monospace">o</mml:mi><mml:mi mathvariant="monospace">v</mml:mi><mml:mo mathvariant="monospace">.</mml:mo><mml:mi mathvariant="monospace">j</mml:mi><mml:mi mathvariant="monospace">l</mml:mi></mml:mrow></mml:math>, part of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="monospace">Q</mml:mi><mml:mi mathvariant="monospace">u</mml:mi><mml:mi mathvariant="monospace">a</mml:mi><mml:mi mathvariant="monospace">n</mml:mi><mml:mi mathvariant="monospace">t</mml:mi><mml:mi mathvariant="monospace">u</mml:mi><mml:mi mathvariant="monospace">m</mml:mi><mml:mi mathvariant="monospace">C</mml:mi><mml:mi mathvariant="monospace">o</mml:mi><mml:mi mathvariant="monospace">n</mml:mi><mml:mi mathvariant="monospace">t</mml:mi><mml:mi mathvariant="monospace">r</mml:mi><mml:mi mathvariant="monospace">o</mml:mi><mml:mi mathvariant="monospace">l</mml:mi><mml:mo mathvariant="monospace">.</mml:mo><mml:mi mathvariant="monospace">j</mml:mi><mml:mi mathvariant="monospace">l</mml:mi></mml:mrow></mml:math>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.