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arXiv · 2311.04379

A square-root speedup for finding the smallest eigenvalue

Alex Kerzner·Vlad Gheorghiu·Michele Mosca·Thomas Guilbaud·Federico Carminati·Fabio Fracas·Luca Dellantonio
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Abstract

We describe a quantum algorithm for finding the smallest eigenvalue of a Hermitian matrix. This algorithm combines Quantum Phase Estimation and Quantum Amplitude Estimation to achieve a quadratic speedup with respect to the best classical algorithm in terms of matrix dimensionality, i.e., $\widetilde{\mathcal{O}}(\sqrt{N}/\epsilon)$ black-box queries to an oracle encoding the matrix, where $N$ is the matrix dimension and $\epsilon$ is the desired precision. In contrast, the best classical algorithm for the same task requires $\Omega(N)\text{polylog}(1/\epsilon)$ queries. In addition, this algorithm allows the user to select any constant success probability. We also provide a similar algorithm with the same runtime that allows us to prepare a quantum state lying mostly in the matrix's low-energy subspace. We implement simulations of both algorithms and demonstrate their application to problems in quantum chemistry and materials science.

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Alex Kerzner, Vlad Gheorghiu, Michele Mosca, Thomas Guilbaud, Federico Carminati, Fabio Fracas, Luca Dellantonio. 2023-11-07. A square-root speedup for finding the smallest eigenvalue. https://arxiv.org/abs/2311.04379

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