SISSA : Overcoming exponential volume scaling in quantum simulations of lattice gauge theories

Kane, Christopher F.; Grabowska, Dorota M.; Bauer, Christian W.; Nachman, Benjamin

doi:10.22323/1.430.0016

Article
Report number	arXiv:2212.04619
Title	Overcoming exponential volume scaling in quantum simulations of lattice gauge theories
Author(s)	Kane, Christopher F. (Arizona U.) ; Grabowska, Dorota M. (Washington U., Seattle ; CERN) ; Nachman, Benjamin (LBL, Berkeley) ; Bauer, Christian W. (LBL, Berkeley)
Publication	2023
Imprint	2022-12-08
Number of pages	10
Note	11 pages, 2 figures, Proceedings of the 39th Annual International Symposium on Lattice Field Theory (Lattice 2022), August 8-13 2022, Bonn, Germany
In:	PoS LATTICE2022 (2023) 016
In:	39th International Symposium on Lattice Field Theory (Lattice 2022), Bonn, Germany, 8 - 13 Aug 2022, pp.016
DOI	10.22323/1.430.0016
Subject category	quant-ph ; General Theoretical Physics ; hep-lat ; Particle Physics - Lattice
Abstract	Real-time evolution of quantum field theories using classical computers requires resources that scale exponentially with the number of lattice sites. Because of a fundamentally different computational strategy, quantum computers can in principle be used to perform detailed studies of these dynamics from first principles. Before performing such calculations, it is important to ensure that the quantum algorithms used do not have a cost that scales exponentially with the volume. In these proceedings, we present an interesting test case: a formulation of a compact U(1) gauge theory in 2+1 dimensions free of gauge redundancies. A naive implementation onto a quantum circuit has a gate count that scales exponentially with the volume. We discuss how to break this exponential scaling by performing an operator redefinition that reduces the non-locality of the Hamiltonian. While we study only one theory as a test case, it is possible that the exponential gate scaling will persist for formulations of other gauge theories, including non-Abelian theories in higher dimensions.
Copyright/License	CC-BY-NC-ND-4.0 preprint: (License: CC BY 4.0) publication: © 2022-2025 The author(s)

$Visual representation of the cosine terms in $H_B$ for $N_p=16$. Each operator $\mathcal{O}_p$ represents a magnetic field operator $B_p$, where the left(right) figure shows the original(weaved) basis. Operators inside the same box also appear in a single cosine of $H_B$. Boxes with different colors or line-styles correspond to different cosine terms. Left: The blue dashed rectangles with a single operator correspond to the local $\cos(B_p)$ terms in the original basis, and the red solid rectangle that contains every operator corresponds to the $\cos(\sum_{p=1}^{N_p} B_p)$ term in the original basis. Right: The blue dashed and dotted rectangles show, in the weaved basis, the increased connectivity of the previously local $\cos(B_p)$ terms. The red solid rectangle shows how the previously maximal connectivity of the $\cos(\sum_{p=1}^{N_p} B_p)$ term is reduced in the weaved basis.$ $Visual representation of the cosine terms in $H_B$ for $N_p=16$. Each operator $\mathcal{O}_p$ represents a magnetic field operator $B_p$, where the left(right) figure shows the original(weaved) basis. Operators inside the same box also appear in a single cosine of $H_B$. Boxes with different colors or line-styles correspond to different cosine terms. Left: The blue dashed rectangles with a single operator correspond to the local $\cos(B_p)$ terms in the original basis, and the red solid rectangle that contains every operator corresponds to the $\cos(\sum_{p=1}^{N_p} B_p)$ term in the original basis. Right: The blue dashed and dotted rectangles show, in the weaved basis, the increased connectivity of the previously local $\cos(B_p)$ terms. The red solid rectangle shows how the previously maximal connectivity of the $\cos(\sum_{p=1}^{N_p} B_p)$ term is reduced in the weaved basis.$ Show more plots

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