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\justifying Outline of the $Z_2$HM. (a) Shows a sketch of the support of the different terms of Hamiltonian \eqref{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.

\justifying 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.

\justifying 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.

\justifying 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.

\justifying 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 $g > J_{\hexagon}^{\mathrm{eff}}$.

\justifying Trotter circuits for the quantum simulation of the $Z_2$HM. This circuit is designed by repeated composition of the Pauli gadget drawn in Fig.~\ref{fig:fig0} in a specific order (Sect.~\ref{subsec:methods_trotter}) so that the CNOTs commute and they can be combined in blocks $\mathcal{C}$ of depth 3. We divide the matter sites on the lattice into two sets, coloured in blue and green. We place the Pauli gadgets acting on the qubits in each set in parallel, and the sets act sequentially.

\justifying Additional data for the experiments in Fig.~\ref{fig:single_string_dynamics}. (a)-(b) show the transfer of population from the initial endpoints of the string to neighbouring sites in the confined phase. We exclusively observe yo-yo oscillations, also in the matter sites interior to the string, as the bending mode of motion only virtually populates the states with a shorter string. (c)-(d) features the local expectation value of the gauge field at the colored links, interior and exterior to the initial string, respectively. This expectation value approaches a steady value around $\langle \sigma_{(\boldsymbol{n}, \boldsymbol{v})}^z \rangle \simeq 0$, indicating that the gauge field tends to a superposition of every possible configuration in the 2-particle sector. In (e), we plot the dynamics of the occupation in two bivalent nodes of the lattice in the Higgs phase. The grey points do not adjust to the MPS curve because the qubit mapped to the grey site was faulty at the time of the simulation. The propagation of errors introduced by this qubit is well controlled enough that its effect is negligible on the other sites considered. (f) shows the local expectation value of the gauge field in the Higgs phase.

\justifying Results from MPS-based simulations of the single string quench for long times. In (a-b), we plot the occupation at the initial and rotated string endpoints in the confined phase. Green and blue curves overlap with the orange and yellow ones. The longer times accessible to the MPS simulations enable the resolution of more than half oscillation bending mode, corresponding to the high-amplitude oscillation that accompanies the high-frequency yo-yo motion. The occupation in the initial endpoints (a) almost completely vanishes for $t=T_\mathrm{b}/2 \simeq 7.54\lambda^{-1}$, signaling that the string has rotated almost completely. The yo-yo motion is also manifest in the rotated string as the small-amplitude oscillations around $t=T_\mathrm{b}/2$ in (b). (c-d) feature the occupation of initially occupied and empty matter sites, respectively. In the deconfined phase, the system quickly reaches a long-lived steady state with the initially present matter particles spread on the lattice and the gauge field in a superposition of all the different configurations allowed by the gauge symmetry. Revivals are expected at a timescale proportional to the size of the lattice $t_\mathrm{R} \sim \mathcal{O}(N)$. (e-f) show the occupations in the Higgs phase, where long-lived glassy damped oscillations are observed. For small $m$, these oscillations couple the initial state $\ket{\psi_0}$ with $H_I\ket{\psi_0}$, which is only possible if creating matter from the vacuum is energetically cheap. As $m$ increases, spreading is favored over vacuum oscillations, and the glassy oscillations dampen. These MPS-based simulations have been computed using the ``basis update and Galekin'' integrator.

\justifying Single site occupations for the 3-string dynamics. The particle initially placed in the green site, the occupation of which appears in (a), first tunnels out to the neighbouring sites displayed in (b) in a first-order process similar to the yo-yo motion. However, second-order processes prevent it from completely tunneling back to its initial position, and the population in (a) experiences a steady decline that does not translate to an increase in the neighbouring sites of (b). These effects combined translate into the population peak of the broken string configurations plotted in Fig.~\ref{fig:double_string_dynamics} of the main text. In (c), we plot the occupation at the initial string endpoints on the boundary of the lattice. The motion of the particles initially in the purple and gray sites is restricted to neighbouring positions for the times considered in the simulation. The particle initially in the light-yellow site experiences a yoyo motion, and that string endpoint rotates into the dark-yellow site.

\justifying Device characterization for the experiments presented in Fig.\ref{fig:error_mitigation_data}(a) in panel (a), Figs.~\ref{fig:single_string_dynamics}(a)-(b) in panel (b), Figs.~\ref{fig:single_string_dynamics}(e)-(f) in panel (c), Figs.~\ref{fig:single_string_dynamics}(c)-(d) in panel (d) and Figs.~\ref{fig:double_string_dynamics}(a)-(b) in panel (e). For each experiment, we display the qubit-wise CZ gate error on the device connectivity graph and the individual qubit T2 dephasing times, with the specific qubit layout used in each experiment highlighted. Note that the device layouts are isomorphic representations of the simulation layouts illustrated in the corresponding figures. Additionally, we show the T1 relaxation times and readout errors per qubit, as well as a histogram of two-qubit (CZ) gate errors for the qubits included in each experiment.