random states.

              $D$   $\lambda_c(D)$   
  --- ------- ----- ---------------- --
        $q=2$ 2     3.28             
              6     3.235            
              8     3.23             
        $q=3$ 3     2.620            
              6     2.616            
              9     2.616            
        $q=4$ 4     2.430            
              8     2.428            
              10    2.426            
        $q=5$ 5     2.330            
              8     2.326            
              10    2.326            

  : \[table\] Estimates of critical points $\lambda_c$ for the $q$-state quantum Potts model on the square lattice. The values for $\lambda_c(D)$ follow from the behavior of the local order parameter and the GE per site, where $D$ is the iPEPS truncation dimension.

2D $q$-state quantum Potts model
--------------------------------

The quantum Ising model discussed above is the special $q=2$ case of the more general $q$-state quantum Potts model defined on the square lattice. For a regular lattice, classical mean-field solutions [@Wu] and extensive computations (see, e.g., Refs. [@Wu; @dyw] and references therein) have suggested that the 3D classical $q$-state Potts model, and thus the 2D $q$-state quantum version, undergo a continuous phase transition for $q \le 2$ and a discontinuous phase transition for $q > 2$. We now turn to this 2D $q$-state quantum model and examine the GE per site and local order parameters in the vicinity of the phase transition points for the values $q=3,4$ and $5$.

On the square lattice the 