Scalable cold-atom quantum simulator of a $3+1$D U$(1)$ lattice gauge theory with dynamical matter

TL;DR

The paper tackles the challenge of simulating ab-initio real-time dynamics of a $3+1$D lattice gauge theory by proposing a scalable cold-atom quantum simulator for a $3+1$D $U(1)$ lattice gauge theory expressed as a quantum-link model. It maps the LGT to a 3D Bose-Hubbard system with restricted local Hilbert spaces and stabilizes gauge invariance through linear gauge protection, then benchmarks the approach with tree tensor networks against the ideal gauge theory. An analog quantum error-mitigation scheme is introduced to suppress unwanted first-order processes, significantly improving agreement with the target dynamics. This work constitutes a concrete, experimentally feasible step toward realistic quantum simulators of higher-dimensional lattice gauge theories, enabling access to real-time dynamics and regimes beyond classical computability.

Abstract

The stated overarching goal of the highly active field of quantum simulation of high-energy physics (HEP) is to achieve the capability to study \textit{ab-initio} real-time microscopic dynamics of $3+1$D quantum chromodynamics (QCD). However, existing experimental realizations and theoretical proposals for future ones have remained restricted to one or two spatial dimensions. Here, we take a big step towards this goal by proposing a concrete experimentally feasible scalable cold-atom quantum simulator of a U$(1)$ quantum link model of quantum electrodynamics (QED) in three spatial dimensions, employing \textit{linear gauge protection} to stabilize gauge invariance. Using tree tensor network simulations, we benchmark the performance of this quantum simulator through near- and far-from-equilibrium observables, showing excellent agreement with the ideal gauge theory. Additionally, we introduce a method for \textit{analog quantum error mitigation} that accounts for unwanted first-order tunneling processes, vastly improving agreement between quantum-simulator and ideal-gauge-theory results. Our findings pave the way towards realistic quantum simulators of $3+1$D lattice gauge theories that can probe regimes well beyond classical simulability.

Figures (6)

• Figure 1: Schematic for our proposed scalable cold-atom analog quantum simulator. (a) Example of a transition between a zero-particle (bare) vacuum state to a charge-proliferated one in the QLM and BHM representations. We study this transition by varying the mass with a ramping protocol as reported in Fig. \ref{['fig:Ramp dynamics']}. (b) We show how to simulate the desired dynamics in cold atoms trapped in an optical superlattice. The $\delta$ term in Hamiltonian \ref{['eq:BHM_hamiltonian']} restricts the Hilbert space in the desired subspace required to faithfully capture the QLM dynamics. The $\gamma$ term creates a tilt in potential and is used to prevent dynamics that would violate Gauss law. (c) Mapping between the BHM and the QLM. Using the staggered-fermion representation, a single atom on an odd (even) site represents a "$-$" ("$+$") particle (antiparticle), which can be viewed as an electron (positron). The red (blue) links represent eigenstates of $\hat{S}^z$ with eigenvalue $\pm\frac{1}{2}$, which correspond to double (zero) occupied link sites in the BHM, respectively (see text).
• Figure 2: Illustration of a $3$d optical superlattice structure with gauge-invariant hopping along each axis. Atoms are prepared in each layer.
• Figure 3: Time evolution of the average matter occupation during the ramp protocol. Starting from the bare vacuum state, the system evolves into a charge-proliferated regime. The BHM hopping term $J(t)$ follows a quadratic profile, maximizing at $t=6$ where the renormalized mass vanishes ($m_{\text{ren}} = 0$), and smoothly decreasing to zero at both $t=0$ and the final time $t=t_f$. This choice of ramp profile ensures that the system begins and ends in an easily readable state, simplifying experimental protocols for measurements. The dynamics exhibits strong sensitivity to all parameters, making careful tuning essential for achieving the desired charge proliferation. The insert shows how the ratio $m_\text{eff}/\kappa_\text{eff}$ varies throughout the dynamics.
• Figure 4: Time evolution of the matter occupation in the wake of global quenches, obtained using TTN simulations on an $8\times8\times4$ lattice for the BHM model, with $J = 0.03$ kHz. a Simulation starting from a bare vacuum state with $m_{\text{eff}} = 0$ throughout the dynamics. b Simulation starting from a charge proliferated state with $m_{\text{eff}} = 0$ throughout the dynamics. c Simulation starting from a bare vacuum state with $U = 2\delta$ which implies $m_{\text{eff}}/\kappa_{\text{eff}} \simeq -2.79$. d Simulation starting from a charge proliferated state with $m_{\text{eff}} = 0$ throughout the dynamics. c Simulation starting from charge proliferated state with $U = 2\delta$ which implies $m_{\text{eff}}/\kappa_{\text{eff}} \simeq -2.79$.
• Figure 5: Comparison of the average matter occupation for raw simulation data, error-mitigated data, and the exact QLM dynamics. The system undergoes a quench from the vacuum state with an initial mass-to-hopping ratio of $m_{\text{eff}}/\kappa_{\text{eff}} = -2.79$. The BHM hopping parameter is held constant at $J = 0.03$ kHz during the time evolution. The error mitigation scheme substantially reduces deviations from the target QLM behavior, demonstrating the effectiveness of the proposed mitigation method.