Holographically Emergent Gauge Theory in Symmetric Quantum Circuits

We develop a novel holographic framework to study dynamical phases in random quantum circuits with a global symmetry $G$. Viewing the circuit as a tensor network, we decompose it into two parts: a symmetric layer, which defines an emergent gauge wavefunction in one higher dimension, and a non-symmetric layer, composed of random multiplicity tensors. For $G\,{=}\,\mathbb{Z}_N$ symmetric circuits consisting of local unitary gates interspersed with local symmetric noise channels, averaging over the non-symmetric layer yields a dynamically generated noisy $\mathbb{Z}_{N}$ surface code. This allows us to interpret $\mathbb{Z}_{N}$ symmetric circuits in the volume-law phase as quantum error-correcting codes with a distinguished set of logical spin states that inherit the topological protection of the bulk code. By establishing equality of bulk and boundary coherent information, we show that quantum information encoded in these logical states is maximally protected against symmetric noise up to a finite threshold. We further study weakly monitored $\mathbb{Z}_{N}$ symmetric circuits which exhibit a charge-sharpening transition. We show that the point at which the observer gains classical information about the global charge coincides with the point at which measurements destroy the underlying quantum information encoded in the bulk surface code. This also allows for a natural interpretation of the sharpening transition as a confinement transition in the gauge theory. For $N\,{\leq}\,4$, weak measurements drive a single transition from a charge-fuzzy phase with exponential sharpening time $t_{\#}\sim e^{L}$ to a charge-sharp phase with $t_{\#}\sim \mathcal{O}(1)$. On the other hand, for $N>4$, the circuit can enter an intermediate phase with a linear sharpening time $t_{\#}\sim \mathcal{O}(L)$. In this regime, the bulk gauge theory realizes a Coulomb phase with emergent gapless photons.