Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information

Julien Gacon1,2, Christa Zoufal1,3, Giuseppe Carleo2, and Stefan Woerner1

1IBM Quantum, IBM Research – Zurich, CH-8803 Rüschlikon, Switzerland
2Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
3Institute for Theoretical Physics, ETH Zurich, CH-8092 Zürich, Switzerland

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The Quantum Fisher Information matrix (QFIM) is a central metric in promising algorithms, such as Quantum Natural Gradient Descent and Variational Quantum Imaginary Time Evolution. Computing the full QFIM for a model with $d$ parameters, however, is computationally expensive and generally requires $\mathcal{O}(d^2)$ function evaluations. To remedy these increasing costs in high-dimensional parameter spaces, we propose using simultaneous perturbation stochastic approximation techniques to approximate the QFIM at a constant cost. We present the resulting algorithm and successfully apply it to prepare Hamiltonian ground states and train Variational Quantum Boltzmann Machines.

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Cited by

[1] Tobias Haug, Kishor Bharti, and M. S. Kim, "Capacity and Quantum Geometry of Parametrized Quantum Circuits", PRX Quantum 2 4, 040309 (2021).

[2] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, and Dacheng Tao, "Learnability of Quantum Neural Networks", PRX Quantum 2 4, 040337 (2021).

[3] Johannes Jakob Meyer, "Fisher Information in Noisy Intermediate-Scale Quantum Applications", arXiv:2103.15191.

[4] Tobias Haug and M. S. Kim, "Optimal training of variational quantum algorithms without barren plateaus", arXiv:2104.14543.

[5] Martin Larocca, Nathan Ju, Diego García-Martín, Patrick J. Coles, and M. Cerezo, "Theory of overparametrization in quantum neural networks", arXiv:2109.11676.

[6] Christa Zoufal, David Sutter, and Stefan Woerner, "Error Bounds for Variational Quantum Time Evolution", arXiv:2108.00022.

[7] Tobias Haug and M. S. Kim, "Natural parameterized quantum circuit", arXiv:2107.14063.

[8] Anna Lopatnikova and Minh-Ngoc Tran, "Quantum Natural Gradient for Variational Bayes", arXiv:2106.05807.

[9] Nishant Jain, Brian Coyle, Elham Kashefi, and Niraj Kumar, "Graph neural network initialisation of quantum approximate optimisation", arXiv:2111.03016.

[10] Christa Zoufal, "Generative Quantum Machine Learning", arXiv:2111.12738.

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