Inferring Effective Temperature from Noisy Quantum Simulations

Quantum processors are inherently noisy, so the prepared quantum states are mixed and deviate from the pure ground state of the target model. Z₂ gauge models provide a natural testing ground for understanding how such noisy quantum ensembles can still encode thermodynamic information. We show that this deviation can be characterized by an effective temperature T_eff, inferred from measured stabilizer (plaquette) violations and the average energy of the prepared state. As a paradigmatic example, we consider the Kitaev honeycomb model relevant to candidate materials such as RuCl₃, which maps to free Majorana fermions coupled to a static Z₂ gauge field. Within this free-fermion framework, we compute the thermal energy of the canonical ensemble across flux sectors and compare it to the measured energy of the quantum-prepared state to extract Teff. A noisy ground-state preparation of such a model exhibits low plaquette violations, consistent with low-T behavior of the Kitaev model, and an average energy consistent with a finite, low effective temperature matching the expected flux excitation statistics. This approach provides a route for connecting noisy quantum data to finite-temperature behavior in Z₂ gauge and quantum spin-liquid materials.