Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule

Leonardo Banchi1,2 and Gavin E. Crooks3,4

1Department of Physics and Astronomy, University of Florence, via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy
2INFN Sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy
3X, the moonshot factory (x.company), Mountain View, CA, USA
4Berkeley Institute for Theoretical Science, Berkeley, CA, USA

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Abstract

Hybrid quantum-classical optimization algorithms represent one of the most promising application for near-term quantum computers. In these algorithms the goal is to optimize an observable quantity with respect to some classical parameters, using feedback from measurements performed on the quantum device. Here we study the problem of estimating the gradient of the function to be optimized directly from quantum measurements, generalizing and simplifying some approaches present in the literature, such as the so-called parameter-shift rule. We derive a mathematically exact formula that provides a stochastic algorithm for estimating the gradient of any multi-qubit parametric quantum evolution, without the introduction of ancillary qubits or the use of Hamiltonian simulation techniques. The gradient measurement is possible when the underlying device can realize all Pauli rotations in the expansion of the Hamiltonian whose coefficients depend on the parameter. Our algorithm continues to work, although with some approximations, even when all the available quantum gates are noisy, for instance due to the coupling between the quantum device and an unknown environment.

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Cited by

[1] Johannes Jakob Meyer, "Gradients just got more flexible", Quantum Views 5, 50 (2021).

[2] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum (NISQ) algorithms", arXiv:2101.08448.

[3] Johannes Jakob Meyer, Johannes Borregaard, and Jens Eisert, "A variational toolbox for quantum multi-parameter estimation", arXiv:2006.06303.

[4] Andrea Mari, Thomas R. Bromley, and Nathan Killoran, "Estimating the gradient and higher-order derivatives on quantum hardware", Physical Review A 103 1, 012405 (2021).

[5] Stefano Barison, Filippo Vicentini, and Giuseppe Carleo, "An efficient quantum algorithm for the time evolution of parameterized circuits", arXiv:2101.04579.

[6] Jakob S. Kottmann, Abhinav Anand, and Alán Aspuru-Guzik, "A Feasible Approach for Automatically Differentiable Unitary Coupled-Cluster on Quantum Computers", arXiv:2011.05938.

[7] Ieva Čepaitė, Brian Coyle, and Elham Kashefi, "A Continuous Variable Born Machine", arXiv:2011.00904.

[8] Paolo Braccia, Filippo Caruso, and Leonardo Banchi, "How to enhance quantum generative adversarial learning of noisy information", arXiv:2012.05996.

The above citations are from Crossref's cited-by service (last updated successfully 2021-05-06 22:00:52) and SAO/NASA ADS (last updated successfully 2021-05-06 22:00:53). The list may be incomplete as not all publishers provide suitable and complete citation data.

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