Dephasing threshold of the critically deformed Z3 toric code

Determine the dephasing-noise threshold of the critically deformed Z3 toric code at the Potts critical point [?] = [?]c, including its relationship to the measurement-induced threshold.

Background

The paper maps the Bayesian learning problem for the three-state Potts model to weak measurements of a deformed Z3 toric-code state. When the classical measurement record is discarded, the resulting quantum channel is a generalized dephasing channel with a threshold for loss of logical information. The authors determine the threshold for the undeformed code and bound the threshold for the critically deformed code by the corresponding measurement threshold, but do not determine the dephasing threshold itself.

References

Note that the dephasing-noise threshold of the critically deformed ($\beta=\beta_c$) $\mathbb Z_3$ toric code is {\it unknown}.

— Learning Potts Models and $Z_3$ Toric Codes: Higher and Ordinary Nishimori Criticality  (2608.20268 - Patil et al., 20 Aug 2026) in Appendix A, “Noise Channel obtained by Tracing out the Measurement Record”