Real-Time Dynamics in a (2+1)-D Gauge Theory: The Stringy Nature on a Superconducting Quantum Simulator

TL;DR

This work demonstrates real-time quantum simulation of string dynamics in a (2+1)-D Z2-Higgs lattice gauge theory on superconducting qubits, revealing confinement-driven string oscillations, endpoint bending, and multi-string fragmentation. By combining a tailored Hamiltonian embedding on heavy-hex IBM hardware with advanced error suppression, mitigation, and a Gauss-sector-based correction, the authors resolve dynamical string phenomena that have been inaccessible to classical simulations. Tensor-network methods (DMRG/BUG integrator) validate the observed dynamics and guide interpretation, bridging effective string descriptions with observable non-perturbative gauge phenomena. The results establish a practical pathway for studying non-equilibrium gauge dynamics on quantum devices and motivate scaling to larger systems and more complex gauge groups with fault-tolerant prospects.

Abstract

Understanding the confinement mechanism in gauge theories and the universality of effective string-like descriptions of gauge flux tubes remains a fundamental challenge in modern physics. We probe string modes of motion with dynamical matter in a digital quantum simulation of a (2+1) dimensional gauge theory using a superconducting quantum processor with up to 144 qubits, stretching the hardware capabilities with quantum-circuit depths comprising up to 192 two-qubit layers. We realize the $Z_2$-Higgs model ($Z_2$HM) through an optimized embedding into a heavy-hex superconducting qubit architecture, directly mapping matter and gauge fields to vertex and link superconducting qubits, respectively. Using the structure of local gauge symmetries, we implement a comprehensive suite of error suppression, mitigation, and correction strategies to enable real-time observation and manipulation of electric strings connecting dynamical charges. Our results resolve a dynamical hierarchy of longitudinal oscillations and transverse bending at the end points of the string, which are precursors to hadronization and rotational spectra of mesons. We further explore multi-string processes, observing the fragmentation and recombination of strings. The experimental design supports 300,000 measurement shots per circuit, totaling 600,000 shots per time step, enabling high-fidelity statistics. We employ extensive tensor network simulations using the basis update and Galerkin method to predict large-scale real-time dynamics and validate our error-aware protocols. This work establishes a milestone for probing non-perturbative gauge dynamics via superconducting quantum simulation and elucidates the real-time behavior of confining strings.

Figures (10)

• Figure 1: Outline of the $Z_2$HM. (a) Shows a sketch of the support of the different terms of Hamiltonian \ref{['eq:hamiltonian']} and the gauge transformation operators $G_n$. In (b), we sketch the phase diagram of the model and present data for the energy gap from large-scale density matrix renormalization group (DMRG) computations. We use stars to highlight the value of the microscopic parameters $(m, g, \lambda)$ used for the real-time quantum and MPS simulations sketched in (c). We consider three distinct sets of values for these $\{(5, 2, 1), (5, 0.01, 1), (0.3, 0.5, 1)\}$, corresponding to each of the static phases of the model, and find three dynamical regimes. The structure of the Trotter circuits implementing the real-time evolution is displayed in (d). These quantum circuits are built by repeated, ordered composition of the Pauli gadget depicted in (e). $\mathcal{C}$ is a dense block of CNOT gates with depth 3, which are grouped after commutation. The two-qubit depth of these circuits is $D=6N_eL$ for $N_e$ edges on the simulated lattice and $L$ Trotter depth.
• Figure 2: Single string dynamics at different points of the phase diagram. (a) Features the occupation at the initial endpoints of the string in the confined phase in a $2\times2$ lattice (35 qubits) and Trotter $dt=0.15$ (2280 two-qubit gates). The yo-yo and bending modes are distinguished as short-period oscillations and a steady decrease in the mean occupation, respectively. The occupation in the endpoints of the rotated strings in this regime is shown in (b). Here, only bending and vacuum fluctuations are present. The colors of the curves indicate the site where the local occupation operators are measured. (c-d) show the dynamics of the occupation in the deconfined phase in a $3 \times 3$ lattice (68 qubits) with $dt=0.125$ (4286 two-qubit gates), where matter spreads all over the lattice and the system reaches a quasi-stationary state. (e-f) show the dynamics in the Higgs phase for the $3 \times 3$ lattice and $dt=0.15$. In this regime, the local occupations present a long-lived, damped oscillating behavior. Shaded regions indicate $70\%$ bootstrapping confidence.
• Figure 3: Dynamics of string-like correlators $\langle S_k \rangle$ for the initial 3-string and broken string configurations in a $2\times2$ lattice in the confined phase $m = 5$, $g=2$. For each configuration $k$, $S_k$ is defined as the product of occupation operators in the matter sites indicated in the inset of (a). These operators quantify the population of each configuration. In (b), we highlight the population of the broken string configurations. We set $dt=0.125$ for the Trotter circuits.
• Figure 4: Addressing the different sources of error in the simulation. (a) displays data for the dynamics of the local occupation in a trivalent matter site measured in the completely glassy regime $m=0$, $g=0$ in a $7\times 3$ lattice (144 qubits) for different error mitigation settings. Even though these dynamics can be reproduced with one Trotter layer, we choose $dt=0.25$ (7872 two-qubit gates) to evaluate the performance of the device with increasing depth. We progressively increase the number of error cancellation techniques introduced in the simulations and observe good convergence to the analytical expression. In (b), we quantify the number of flips measured by the GSC decoder for the different settings in (a). Dots represent the mean of the flip count distribution for the 300000 shots, and the shaded region indicates one standard deviation. (c) shows the occupation in the upper initial string endpoint from quantum simulations with varying $dt$ in the confined regime $m = 5$, $g = 2$, in the $2\times 2$ lattice (35 qubits). Every simulation reproduces the MPS results up to $dt=0.2$. In (d) we display the results of a quantum simulation of the $Z_2$HM in a (1+1)-dimensional chain of length $L=21$ (41 qubits) for $m=5$, $g=0.8$. The smaller dimensionality of the system constrains the propagation of errors, which allows performing a fair comparison of the mirror and Clifford ODR calibration circuit performance displayed in (e). We empirically observe that both calibration circuits perform similarly for short depth, but the mirror circuits are superior for long times, when more errors have accumulated, as Cliffordized circuits result in an effective change of the Hamiltonian coupling constants in the calibration.
• Figure 5: DMRG results for the ground state. Panel (a) shows the simulated symmetric flower-like flakes. Numbers indicate system sizes for $R \in [0,1,2]$. The largest simulated system contains 19 hexagons and $N_n = 54$ nodes. In panels (b-c), the average magnetizations of the matter and gauge qubits are shown, respectively, for system size $R=2$ and bond dimension $M=256$. The bottom left corner with low magnetization corresponds to the Higgs region, the large red region with magnetization close to one indicates the confined regime. Close to the horizontal axis, the matter spins are fully polarized, but the gauge spins are not: this indicates a narrow deconfined region. In panel (d), the ground state entanglement (von Neumann) entropy is shown between the two system halves if the flake of size $R=2$ is cut symmetrically by a vertical line at the middle. The bond dimension is again $M=256$. Vertical red lines at the bottom right indicate the specific cuts of panels (e-g) for which higher accuracy (up to $M=1024$) DMRG calculations have been performed. In these panels, vertical dashed lines indicate the roughly proposed phase boundary $\tilde{g} \approx J_{\hexagon}^{\mathrm{eff}}$ between the deconfined and confined phases. We observe elevated entropy for $g< J_{\hexagon}^{\mathrm{eff}}$, and also the DMRG did not fully converge even for $M=1024$. In contrast, we observe low entropy and reasonable convergence for $g > J_{\hexagon}^{\mathrm{eff}}$.