Variational solutions to fermion-to-qubit mappings in two spatial dimensions

Jannes Nys and Giuseppe Carleo

École Polytechnique Fédérale de Lausanne (EPFL), Institute of Physics, CH-1015 Lausanne, Switzerland
Center for Quantum Science and Engineering, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

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Abstract

Through the introduction of auxiliary fermions, or an enlarged spin space, one can map local fermion Hamiltonians onto local spin Hamiltonians, at the expense of introducing a set of additional constraints. We present a variational Monte-Carlo framework to study fermionic systems through higher-dimensional ($\gt$1D) Jordan-Wigner transformations. We provide exact solutions to the parity and Gauss-law constraints that are encountered in bosonization procedures. We study the $t$-$V$ model in 2D and demonstrate how both the ground state and the low-energy excitation spectra can be retrieved in combination with neural network quantum state ansatze.

One can map local fermion Hamiltonians onto local spin Hamiltonians, at the expense of introducing a set of additional (gauge) constraints. We present a variational Monte-Carlo framework to study fermionic systems through higher-dimensional (>1D) Jordan-Wigner transformations.

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Cited by

[1] Alessandro Sinibaldi, Clemens Giuliani, Giuseppe Carleo, and Filippo Vicentini, "Unbiasing time-dependent Variational Monte Carlo by projected quantum evolution", Quantum 7, 1131 (2023).

[2] Yu-An Chen, Alexey V. Gorshkov, and Yijia Xu, "Error-correcting codes for fermionic quantum simulation", SciPost Physics 16 1, 033 (2024).

[3] Anna Dawid, Julian Arnold, Borja Requena, Alexander Gresch, Marcin Płodzień, Kaelan Donatella, Kim A. Nicoli, Paolo Stornati, Rouven Koch, Miriam Büttner, Robert Okuła, Gorka Muñoz-Gil, Rodrigo A. Vargas-Hernández, Alba Cervera-Lierta, Juan Carrasquilla, Vedran Dunjko, Marylou Gabrié, Patrick Huembeli, Evert van Nieuwenburg, Filippo Vicentini, Lei Wang, Sebastian J. Wetzel, Giuseppe Carleo, Eliška Greplová, Roman Krems, Florian Marquardt, Michał Tomza, Maciej Lewenstein, and Alexandre Dauphin, "Modern applications of machine learning in quantum sciences", arXiv:2204.04198, (2022).

[4] Filippo Vicentini, Damian Hofmann, Attila Szabó, Dian Wu, Christopher Roth, Clemens Giuliani, Gabriel Pescia, Jannes Nys, Vladimir Vargas-Calderon, Nikita Astrakhantsev, and Giuseppe Carleo, "NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems", arXiv:2112.10526, (2021).

[5] Adam Wyrzykowski, "Local spin description of fermions on a lattice", arXiv:2206.10393, (2022).

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