Renormalized dual basis for scalable simulations of 2+1D compact quantum electrodynamics

TL;DR

The paper addresses the persistent challenge of infinite-dimensional gauge sectors in Hamiltonian lattice gauge theories by introducing Renormalized Dual Basis (RDB), a gauge-invariant, locality-friendly truncation scheme built from the single-plaquette problem. By dualizing to plaquettes and strings, RDB separates local and nonlocal contributions, enabling a one-time classical precomputation of a variational, coupling-aware local basis that renormalizes across the full lattice. The authors demonstrate improved plaquette accuracy and resource efficiency for 2+1D $\text{U}(1)$ LGTs on small lattices, and establish scalability to larger lattices through tensor-network methods, with strong performance both in pure-gauge and cQED settings (including dynamical matter). The results suggest RDB is a robust, scalable pathway toward accurate quantum- or quantum-inspired simulations of gauge theories across coupling regimes, with potential extensions to non-Abelian groups and higher dimensions.

Abstract

The classical and quantum simulation of lattice gauge theories (LGTs) with Lie groups is hindered by the infinite-dimensional Hilbert space of gauge degrees of freedom. In a recent work [Phys. Rev. X 15, 031065 (2025)], we introduced a new truncation scheme -- here renamed as Renormalized Dual Basis (RDB) -- based on the resolution of the single-plaquette problem, and demonstrated its performance for SU(2) LGTs. In this paper, we apply the RDB to compact quantum electrodynamics (cQED) in three spacetime dimensions (2+1D). We variationally determine the ground state of the theory for small lattices with periodic (for pure gauge) and open (in presence of fermionic matter) boundary conditions, achieving improved precision for the plaquette operator compared to previous approaches. By leveraging tensor networks, we extend the study to larger lattices and demonstrate the scalability of the method. Overall, we show that the RDB provides an efficient description across all coupling regimes.

Figures (8)

• Figure 1: Representation of the Renormalized Dual Basis as applied to $\text{U}(1)$ lattice gauge theories. (a) We get a finite Hamiltonian by truncating the dual Hamiltonian (see Fig. \ref{['2d_lattice_vars']}), built of plaquettes, in an optimal local basis. The local basis for the $\bf n$ plaquette is obtained from the resolution of a single-plaquette problem for arbitrary basis parameter $g_{\bf n}$ and is variationally optimized by minimizing the energy over $g_{\bf n}$, see Sec. \ref{['LocalBasisU1']}. (b) Ground (in dark orange), first- (in light blue), and second-excited (in dark blue) states of the single-plaquette problem by varying coupling strength $g_{\bf n}$, from weak $(g_{\bf n}\ll1)$ to strong $(g_{\bf n}\gg1)$ coupling. The single-plaquette eigenbasis can be determined with arbitrary precision for any $g_{\bf n}$. (c) The expectation values of the plaquette operator $\langle\square\rangle$, defined in Eq. \ref{['HB_expval_definition']}, are computed by retaining $L_{\text{max}} +1$ states per plaquette and quickly converge to the exact result for any value of the coupling constant $\beta=(2g^2)^{-1}$.
• Figure 2: Representation of the fundamental gauge variables in the Kogut--Susskind formulation (left lattice), the links $U_\mu({\bf n})$ and the electric fields $E_\mu({\bf n})$, and the dual variables (right lattice), the plaquettes $P_{{\bf n}}$ and the rotators $R_{{\bf n}}$. In the case of periodic boundary conditions, we have to consider the large loops wrapping the lattice, i.e., $\mathcal{L}_{\hat{x},\hat{y}}$, and the absence of the plaquette in the reference site ${\bf n}_R\equiv(0,N-1)$ (dark grey cross). Dynamical fermions (staggered formulation) are replaced in the dual formulation by electric strings connecting the origin to the other sites Haase2021resourceefficient.
• Figure 3: Relative precision on the plaquette expectation value for the 2+1D $\text{U}(1)$ LGT on the minimal torus as a function of the total size of the Hilbert space. We compare different formulations and basis choices: “Quantum” (blue circles) Haase2021resourceefficient, “Dual" (green circles), “B&G” (orange circles) BauerPRD2023, “RDB” (red circles) Fontana2024, and “Improved RDB” (purple circles), see Sec. \ref{['hilbert space']}. The dashed vertical light coral and grey lines highlight the physically relevant cases of $\text{dim}(\mathcal{H}_{\text{phys}})=27,\;125$, corresponding to three and five retained states per plaquette, respectively.
• Figure 4: Application of the RDB to cQED. (a) Logarithm of the relative precision on the plaquette expectation value for the single plaquette cQED, obtained with the RDB truncated to $L_{\text{max}}=2$ (3 gauge states), as a function of the fermion mass $m$ and kinetic energy $\kappa$. The exact result has been taken to be the highest truncation computed, $L_{\text{max}}=10$ (11 gauge states). Three different coupling strengths are used: strong coupling ($\beta=0.1$), intermediate coupling ($\beta=1$) and weak coupling ($\beta=10$). (b) Plaquette expectation value for the 2+1D cQED on a single plaquette as a function of $\beta=\frac{1}{2g^2}$. We compare the formulation of Haase2021resourceefficient used in a recent trapped-ion computation Meth2023Oct (from where we extract the data), with the RDB for the truncations $L_{\text{max}}\in\{2,10\}$. The results with $L_{\text{max}}=10$ is taken as exact and the inset shows the biggest error when the gauge field is truncated to a qutrit. Results obtained by truncating to the electric basis have been added to assist in the comparison. The gauge sector Hilbert space dimension is indicated in parentheses.
• Figure 5: Relative precision on the plaquette expectation value for the 2+1D $\text{U}(1)$ LGT on a $4\times4$ lattice (left) and a $2\times30$ lattice (right), as a function of $\beta=\frac{1}{2g^2}$. In both cases the exact result are taken to be the one minimizing the energy at each $\beta$. For the $4\times4$ lattice these were either $L_{\text{max}}=5$($8$) with (without) variational optimization. For the $2\times30$ lattice these are either $L_{\text{max}}=7$($9$) with (without) variational optimization.