Nearly tight Trotterization of interacting electrons

Yuan Su; Hsin-Yuan Huang; Earl T. Campbell

doi:10.22331/q-2021-07-05-495

Nearly tight Trotterization of interacting electrons

Yuan Su^1,2, Hsin-Yuan Huang^1,3, and Earl T. Campbell⁴

¹Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125, USA
²Google Research, Venice, CA 90291, USA
³Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA 91125, USA
⁴AWS Center for Quantum Computing, Pasadena, CA 91125, USA

Published:	2021-07-05, volume 5, page 495
Eprint:	arXiv:2012.09194v2
Doi:	https://doi.org/10.22331/q-2021-07-05-495
Citation:	Quantum 5, 495 (2021).

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Abstract

We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting commutativity of the target Hamiltonian, sparsity of interactions, and prior knowledge of the initial state. We achieve this using Trotterization for a class of interacting electrons that encompasses various physical systems, including the plane-wave-basis electronic structure and the Fermi-Hubbard model. We estimate the simulation error by taking the transition amplitude of nested commutators of the Hamiltonian terms within the $\eta$-electron manifold. We develop multiple techniques for bounding the transition amplitude and expectation of general fermionic operators, which may be of independent interest. We show that it suffices to use $\left(\frac{n^{5/3}}{\eta^{2/3}}+n^{4/3}\eta^{2/3}\right)n^{o(1)}$ gates to simulate electronic structure in the plane-wave basis with $n$ spin orbitals and $\eta$ electrons, improving the best previous result in second quantization up to a negligible factor while outperforming the first-quantized simulation when $n=\eta^{2-o(1)}$. We also obtain an improvement for simulating the Fermi-Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons.

► BibTeX data

@article{Su2021nearlytight,
  doi = {10.22331/q-2021-07-05-495},
  url = {https://doi.org/10.22331/q-2021-07-05-495},
  title = {Nearly tight {T}rotterization of interacting electrons},
  author = {Su, Yuan and Huang, Hsin-Yuan and Campbell, Earl T.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {495},
  month = jul,
  year = {2021}
}

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Nearly tight Trotterization of interacting electrons

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