Variational quantum amplitude estimation

Kirill Plekhanov, Matthias Rosenkranz, Mattia Fiorentini, and Michael Lubasch

Cambridge Quantum Computing Limited, SW1P 1BX London, United Kingdom

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We propose to perform amplitude estimation with the help of constant-depth quantum circuits that variationally approximate states during amplitude amplification. In the context of Monte Carlo (MC) integration, we numerically show that shallow circuits can accurately approximate many amplitude amplification steps. We combine the variational approach with maximum likelihood amplitude estimation [Y. Suzuki et al., Quantum Inf. Process. 19, 75 (2020)] in variational quantum amplitude estimation (VQAE). VQAE typically has larger computational requirements than classical MC sampling. To reduce the variational cost, we propose adaptive VQAE and numerically show in 6 to 12 qubit simulations that it can outperform classical MC sampling.

Amplitude estimation is an important quantum algorithm that has a wide range of applications. However, running it on current quantum computers is challenging. In our work, we explore the possibility of realizing amplitude estimation with constant-depth quantum circuits by making use of variational quantum algorithms. To benchmark our algorithms, we consider mean calculations of shifted and univariate Gaussian, Cauchy-Lorentz and log-normal probability distributions. We find that an adaptive implementation of the algorithms significantly reduces the variational cost and can be more efficient than classical Monte Carlo sampling. These results pave the way for the efficient realization of amplitude estimation on current gate-based quantum devices.

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