Optimizing Variational Quantum Algorithms with qBang: Efficiently Interweaving Metric and Momentum to Navigate Flat Energy Landscapes

David Fitzek1,2, Robert S. Jonsson1,3, Werner Dobrautz4, and Christian Schäfer1,5

1Department of Microtechnology and Nanoscience, MC2, Chalmers University of Technology, 412 96 Gothenburg, Sweden
2Volvo Group Trucks Technology, 405 08 Gothenburg, Sweden
3Future Technologies, Saab Surveillance, 412 76 Gothenburg, Sweden
4Department of Chemistry and Chemical Engineering, Chalmers University of Technology, 412 96 Gothenburg, Sweden
5Department of Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden

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Variational quantum algorithms (VQAs) represent a promising approach to utilizing current quantum computing infrastructures. VQAs are based on a parameterized quantum circuit optimized in a closed loop via a classical algorithm. This hybrid approach reduces the quantum processing unit load but comes at the cost of a classical optimization that can feature a flat energy landscape. Existing optimization techniques, including either imaginary time-propagation, natural gradient, or momentum-based approaches, are promising candidates but place either a significant burden on the quantum device or suffer frequently from slow convergence. In this work, we propose the quantum Broyden adaptive natural gradient (qBang) approach, a novel optimizer that aims to distill the best aspects of existing approaches. By employing the Broyden approach to approximate updates in the Fisher information matrix and combining it with a momentum-based algorithm, qBang reduces quantum-resource requirements while performing better than more resource-demanding alternatives. Benchmarks for the barren plateau, quantum chemistry, and the max-cut problem demonstrate an overall stable performance with a clear improvement over existing techniques in the case of flat (but not exponentially flat) optimization landscapes. qBang introduces a new development strategy for gradient-based VQAs with a plethora of possible improvements.

Quantum computing is one of the most anticipated technologies of the 21st century, promising to combat the decreasing speed of innovation in classical computing. Considerable challenges for a useful application remain — including a lack of algorithms and fault-tolerant hardware. Variational quantum algorithms mix quantum evaluations with classical optimization to partially circumvent the existing obstacles. However, this composite approach suffers from the inherent quantum feature that the space of possible solutions increases exponentially with the size of the underlying system. Many of those solutions are irrelevant and close in energy, i.e., the gradients of the energy vanish. This poses a considerable challenge for classical optimization, and the most advanced algorithms consider the local metric of the solution space to find an optimal path on this landscape. However, metric-based algorithms remain impractical on quantum devices due to the excessive evaluations needed. In this work, we develop qBang, a hybrid approach that combines state-of-the-art momentum dynamics and instructs every iteration step with curvature information while keeping the number of quantum evaluations comparable to gradient descent. We provide benchmarks for a variety of systems, including combinatoric problems, and quantum chemical systems. Despite its low cost, qBang provides a considerable improvement over its competitors. Furthermore, its flexibility commends the development of a whole new class based on the ideas put forward in this work. The availability of efficient optimization strategies defines the success of variational quantum algorithms, having considerable implications on the near-term use of quantum computing devices.

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[1] Werner Dobrautz, Igor O. Sokolov, Ke Liao, Pablo López Ríos, Martin Rahm, Ali Alavi, and Ivano Tavernelli, "Toward Real Chemical Accuracy on Current Quantum Hardware Through the Transcorrelated Method", Journal of Chemical Theory and Computation acs.jctc.4c00070 (2024).

[2] Davide Castaldo, Marta Rosa, and Stefano Corni, "Fast-forwarding molecular ground state preparation with optimal control on analog quantum simulators", arXiv:2402.11667, (2024).

[3] Erika Magnusson, Aaron Fitzpatrick, Stefan Knecht, Martin Rahm, and Werner Dobrautz, "Towards Efficient Quantum Computing for Quantum Chemistry: Reducing Circuit Complexity with Transcorrelated and Adaptive Ansatz Techniques", arXiv:2402.16659, (2024).

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