Averaging symmetric Z_N quantum circuits over random noise produces a noisy surface code whose logical information is protected against symmetric errors up to a threshold, with charge-sharpening transitions coinciding with bulk confinement transitions that differ for N≤4 versus N>4.
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A frustrated Z6 clock chain has decoupled Ising-Potts multicritical points and a terminating Z6 parafermion multicritical point far from integrability.
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Holographically Emergent Gauge Theory in Symmetric Quantum Circuits
Averaging symmetric Z_N quantum circuits over random noise produces a noisy surface code whose logical information is protected against symmetric errors up to a threshold, with charge-sharpening transitions coinciding with bulk confinement transitions that differ for N≤4 versus N>4.
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Parafermionic and decoupled multicritical points in a frustrated $\mathbb{Z}_6$ clock chain
A frustrated Z6 clock chain has decoupled Ising-Potts multicritical points and a terminating Z6 parafermion multicritical point far from integrability.