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Near-optimal ground state preparation

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Lin Lin1,2 and Yu Tong1

1Department of Mathematics, University of California, Berkeley, CA 94720, USA
2Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

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Abstract

Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.

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Cited by

[1] Kianna Wan, “Exponentially faster implementations of Select(H) for fermionic Hamiltonians”, arXiv:2004.04170, Quantum 5, 380 (2021).

[2] A. Roggero, “Spectral-density estimation with the Gaussian integral transform”, Physical Review A 102 2, 022409 (2020).

[3] Yu Tong, Dong An, Nathan Wiebe, and Lin Lin, “Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functions”, arXiv:2008.13295.

[4] Kianna Wan and Isaac Kim, “Fast digital methods for adiabatic state preparation”, arXiv:2004.04164.

[5] Sirui Lu, Mari Carmen Bañuls, and J. Ignacio Cirac, “Algorithms for quantum simulation at finite energies”, arXiv:2006.03032.

[6] A. E. Russo, K. M. Rudinger, B. C. A. Morrison, and A. D. Baczewski, “Evaluating energy differences on a quantum computer with robust phase estimation”, arXiv:2007.08697.

[7] Yuan Su, Hsin-Yuan Huang, and Earl T. Campbell, “Nearly tight Trotterization of interacting electrons”, arXiv:2012.09194.

[8] Daochen Wang, Xuchen You, Tongyang Li, and Andrew M. Childs, “Quantum exploration algorithms for multi-armed bandits”, arXiv:2007.07049.

[9] Lindsay Bassman, Miroslav Urbanek, Mekena Metcalf, Jonathan Carter, Alexander F. Kemper, and Wibe de Jong, “Simulating Quantum Materials with Digital Quantum Computers”, arXiv:2101.08836.

The above citations are from Crossref’s cited-by service (last updated successfully 2021-02-06 06:08:05) and SAO/NASA ADS (last updated successfully 2021-02-06 06:08:07). The list may be incomplete as not all publishers provide suitable and complete citation data.

Source: https://quantum-journal.org/papers/q-2020-12-14-372/

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