Quantum Resources in Non-Abelian Lattice Gauge Theories: Nonstabilizerness, Multipartite Entanglement, and Fermionic Non-Gaussianity

Abstract

Lattice gauge theories (LGTs) represent one of the most ambitious goals of quantum simulation. From a practical implementation perspective, non-Abelian theories present significantly tougher challenges than Abelian LGTs. However, it is unknown whether this is also reflected in increased values of quantum resources relating to the complexity of simulating quantum many-body models. Here, we compare three paradigmatic measures of quantum resources -- stabilizer Rényi entropy, generalized geometric measure of entanglement, and fermionic antiflatness -- for pure-gauge theories on a ladder with Abelian $\mathbb{Z}_N$ as well as non-Abelian $D_3$ and SU(2) gauge symmetries. We find that non-Abelian symmetries are not necessarily inherently harder to simulate than Abelian ones, but rather the required quantum resources depend nontrivially on the interplay between the group structure, superselection sector, and encoding of the gauge constraints. Our findings help indicate where quantum advantage could emerge in simulations of LGTs, both in NISQ and fault-tolerant eras.

Figures (5)

• Figure 1: (a) Sketch of the (2+1)D LGT flux ladder and the three different mappings to one-dimensional chains, depending on the underlying gauge symmetry: (b) $\mathbb{Z}_N$, (c) $\mathrm{D}_3$, (d) SU(2). (e) Sketch of the maximum values of the resources across the phase diagrams of the different models we consider: Multipartite entanglement $G_2$, stabilizer Rényi entropy $\mathcal{M}_2$, and fermionic antiflatness $\mathcal{F}_2$. All values except the maximal GGM for $\mathrm{D}_3$ remain below those of Haar random states (fermionic antiflatness is not computed for $\mathrm{D}_3$).
• Figure 2: (a) Expectation value of the plaquette operator, (b) GGM, (c) SRE density, and (d) FAF density, computed for the ground states of SU(2), $\mathbb{Z}_2$, and $\mathrm{D}_3$ pure LGTs. For all considered theories, results converge fast with increasing system size. The derivative of $G_2$, see the inset of panel (b), peaks at the same position as $\mathcal{M}_2$. SU(2) displays sizable values of all quantum resources at small coupling, where quantum fluctuations proliferate. In contrast, the studied LGTs with discrete groups display mixed regimes, which are easy in terms of one resource and hard for another. FAF is only computed for SU(2) and $\mathbb{Z}_2$, where the Jordan--Wigner transformation enables a clear mapping between Majorana and qubit operators.
• Figure 3: Impact of superselection sector $k$ and group order $N$ on quantum resources, illustrated for a $\mathbb{Z}_N$ LGT, for $N = 2,3,4,5,6$ and $k=0,1,2$, with $L=4$ fixed. (a) In all cases considered, SRE peaks when electric and magnetic terms compete. (b) GGM has a significant peak in the crossover region only in the absence of background charges $k=0$. Both quantities may reach a plateau at strong coupling, whose emergence and height critically depend on the combination of $k$ and $N$.
• Figure 4: Examples of gauge-invariant configurations in the representation (electric) basis for the different models considered in this work. We highlight how these states appear after the mapping onto effective (1+1)D chains. The local basis for $\mathbb{Z}_2$ and SU(2) is $\{\ket{\uparrow}, \ket{\downarrow}\}$, which correspond to the eigenstates of the Pauli-$Z$ operator. Their gauge-invariant states also share the same representation, while their Hamiltonians are markedly different. For $\mathbb{Z}_N$, instead, we represent the eigenstates of the generalized phase ($Z$) operator with the corresponding angles, the $N$th-roots of unity, on the plane. For $\mathrm{D}_3$, the circles identify the possible combinations of the three irreps---identity $e$, parity $p$, and fundamental $\tau$---into group-singlets. The internal structure of two-dimensional representations is not depicted.
• Figure 5: Energy gaps vs coupling $g^2$ for three different models considered in this work: $\mathbb{Z}_2$, SU(2) (a), and $\mathrm{D}_3$ (b). Neither displays significant finite-size scaling, indicating the lack of a phase transition but rather a smooth crossover between the electric and magnetic regimes. The panel (b) highlights the level crossing between different symmetry sectors of the global $\mathrm{D}_3$ gauge transformation. Solid lines indicate the gap with the first excited states belonging to the same symmetry sector of the ground state, while dashed lines point to the manifold that transforms nontrivially under the global $\mathrm{D}_3$ symmetry and becomes degenerate with the ground state at $g^2=0$.