Non-Abelian SU(2) Lattice Gauge Theories in Superconducting Circuits

We propose a digital quantum simulator of non-Abelian pure-gauge models with a superconducting circuit setup. Within the framework of quantum link models, we build a minimal instance of a pure SU(2) gauge theory, using triangular plaquettes involving geometric frustration. This realization is the least demanding, in terms of quantum simulation resources, of a non-Abelian gauge dynamics. We present two superconducting architectures that can host the quantum simulation, estimating the requirements needed to run possible experiments. The proposal establishes a path to the experimental simulation of non-Abelian physics with solid-state quantum platforms.

Figure 1

(a) Six tunable-coupling transmon qubits coupled to a single microwave resonator. (b) Six Xmon qubits on a triangular geometry, coupled to a central one. The box 1 in the scheme is implicitly repeated for the sides 2 and 3. Both setups can encode the dynamics of the SU(2) triangular plaquette model schematized in (c), where the left and right gauge degrees of freedom are explicitly depicted.

Figure 2

(a), (b), and (c) Different gauge invariant sectors for a triangular plaquette, together with the sector degeneracy and the action of the Hamiltonian $H_T$ upon them. (d) Extended lattice with periodic boundary conditions. This lattice allows for the $\sum _x{G}^2(x)=0$ sector.

Figure 3

The relative gauge deviation of the digitally simulated gauge value ($⟨{⟩}_D$), versus the ideal value ($⟨{⟩}_I$), $E=(⟨\sum _{\overrightarrow{x}}{\overrightarrow{G}}^2(\overrightarrow{x}){⟩}_I-⟨\sum _{\overrightarrow{x}}{\overrightarrow{G}}^2(\overrightarrow{x}){⟩}_D)/⟨\sum _{\overrightarrow{x}}{\overrightarrow{G}}^2(\overrightarrow{x}){⟩}_I$, as a function of the digital steps $N$ and the simulated phase $\varphi =Jt$. The error decreases with large $N$, and small $\varphi$. The contour plot is interpolated from numerical data at $N=\{1,2,3,4\}$. The initial state is chosen as depicted in Fig. 2.

Figure 4

Digital quantum simulation of pure gauge dynamics for (a) 2 and (b) 3 digital steps. The initial state $|{\mathrm{\Psi }}_0⟩$ is chosen as in Fig. 2. Periodic oscillations between the position and qubit degrees of freedom at the vertices are witnessed by $|⟨{\mathrm{\Psi }}_I|{\mathrm{\Psi }}_0⟩{|}^2$. The theoretical fidelity $|⟨{\mathrm{\Psi }}_I|{\mathrm{\Psi }}_D⟩{|}^2$ increases with the number of digital steps, together with the width of the fidelity bands due to experimental imperfections.