Nearly tight Trotterization of interacting electrons

Yuan Su1,2, Hsin-Yuan Huang1,3, 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

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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.

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Cited by

[1] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu, "A Theory of Trotter Error", arXiv:1912.08854.

[2] Anthony Ciavarella, Natalie Klco, and Martin J. Savage, "Trailhead for quantum simulation of SU(3) Yang-Mills lattice gauge theory in the local multiplet basis", Physical Review D 103 9, 094501 (2021).

[3] Paul K. Faehrmann, Mark Steudtner, Richard Kueng, Maria Kieferova, and Jens Eisert, "Randomizing multi-product formulas for improved Hamiltonian simulation", arXiv:2101.07808.

[4] Lin Lin and Yu Tong, "Heisenberg-limited ground state energy estimation for early fault-tolerant quantum computers", arXiv:2102.11340.

[5] Dong An, Di Fang, and Lin Lin, "Time-dependent unbounded Hamiltonian simulation with vector norm scaling", arXiv:2012.13105.

[6] Shouzhen Gu, Rolando D. Somma, and Burak Şahinoğlu, "Fast-forwarding quantum evolution", arXiv:2105.07304.

[7] Torin F. Stetina, Anthony Ciavarella, Xiaosong Li, and Nathan Wiebe, "Simulating Effective QED on Quantum Computers", arXiv:2101.00111.

[8] Patrick Rall, "Faster Coherent Quantum Algorithms for Phase, Energy, and Amplitude Estimation", arXiv:2103.09717.

[9] Natalie Klco, Alessandro Roggero, and Martin J. Savage, "Standard Model Physics and the Digital Quantum Revolution: Thoughts about the Interface", arXiv:2107.04769.

[10] Qi Zhao and Xiao Yuan, "Exploiting anticommutation in Hamiltonian simulation", arXiv:2103.07988.

The above citations are from SAO/NASA ADS (last updated successfully 2021-07-27 04:28:38). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2021-07-27 04:28:37).