Holographically Emergent Gauge Theory in Symmetric Quantum Circuits

TL;DR

This work develops a holographic, two-layer tensor-network framework that maps symmetric random circuits onto a bulk G-gauge theory in one higher dimension, enabling a bulk–boundary correspondence between mixed-state circuit dynamics and gauge-theoretic observables. For unitary dynamics with abelian G, the bulk gauge state is deconfined and, under averaging, becomes a Z_N string-net superposition, yielding emergent topological protection and a bulk surface-code correspondence that preserves coherent information on the boundary. When non-unitary noise or weak measurements are included, the framework predicts a charge-sharpening transition that coincides with a decodability threshold in the bulk surface code, with distinct behavior depending on N and the presence of a Coulomb (QLRO) phase for N>4. The results provide a concrete, holographic view of mixed-state orders in symmetric circuits and suggest robust logical memory protected by bulk topological codes under decoherence, as well as a precise diagnostic via non-contractible ’t Hooft loops. The approach is general and paves the way for exploring non-abelian symmetries and extensions to higher-Lindblad dynamics in open quantum systems.

Abstract

We develop a novel holographic framework for mixed-state phases in random quantum circuits, both unitary and non-unitary, 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 random non-symmetric layer, which consists of random multiplicity tensors. For unitarity circuits, the bulk gauge state is deconfined, but under a generic non-unitary circuit (e.g. channels), the bulk gauge theory can undergo a decoherence-induced phase transition: for $G\,{=}\,\mathbb{Z}_N$ with local symmetric noise, the circuit can act as a quantum error-correcting code with a distinguished logical subspace inheriting the $\mathbb{Z}_N$-surface code's topological protection. We then identify that the charge sharpening transition from the measurement side is complementary to a decodability transition in the bulk: noise of the bulk can be interpreted as measurement from the environment. For $N\,{\leq}\,4$, weak measurements drive a single transition from a charge-fuzzy phase with sharpening time $t_{\#}\sim e^{L}$ to a charge-sharp phase with $t_{\#}\sim \mathcal{O}(1)$, corresponding to confinement that destroys logical information. For $N>4$, measurements generically generate an intermediate quasi-long-range ordered Coulomb phase with gapless photons and purification time $t_{\#}\sim \mathcal{O}(L)$.

Figures (11)

• Figure 1: Summary.(a) Monitored Symmetric Circuit. The circuit consists of two parts: local symmetric gates and local charge measurements. One can map a symmetric monitored random circuit into the tensor network. (b) Two-layer tensor network. The top layer of the model is constructed using the Clebsch-Gordon coefficients of the group and realizes a gauge theory. The bottom layer is comprised of random multiplicity tensors. (c) Sharpening transition. Charge sharpening occurs when local charge measurements extract the global charge of the spin chain. At the same time, information encoded in the gauge wavefunction is destroyed. Therefore, fuzzy (sharp) phase corresponds to information protected (lost) in the gauge theory. (d) Phase diagram. For $\mathbb{Z}_{N<4}$, there exists only two phases: truly fuzzy (deconfined) and sharp (trivial), where sharpening time scales as $\exp(L)$ and ${\cal O} (1)$, respectively. However, for $\mathbb{Z}_{N\geq4}$ an intermediate Coulomb (QLRO) phase opens up, where the purification time scales as ${\cal O} (L)$. In the limit $N \rightarrow \infty$, where the symmetry becomes U(1), the deconfined phase collapses into a point.
• Figure 2: Random Symmetric Circuits: We consider a random symmetric circuit with bricklayer geometry as depicted. The symmetry group $G$ acts by means of some unitary representation $V(\cdot):G \rightarrow \mathcal{U}(\mathcal{H}_{\text{global}})$. Each gate is individually chosen to commute with the group action but are otherwise random.
• Figure 3: Symmetry Decomposition I. Here we depict the symmetry decomposition of a trivalent vertex with two incoming legs and one outgoing leg.
• Figure 4: Symmetry Decomposition II. Here we depict the symmetry decomposition of a four-legged vertex with two incoming legs and two outgoing legs.
• Figure 5: $F$-move. Local move relating different trivalent decompositions of a four-legged invariant tensor via F-symbols.