Universality Classes of Stabilizer Code Hamiltonians

Stabilizer code quantum Hamiltonians have been introduced with the intention of physically realizing a quantum memory because of their resilience to decoherence. In order to analyze their finite temperature thermodynamics, we show how to generically solve their partition function using duality techniques. By unveiling each model’s universality class and effective dimension, insights may be gained on their finite temperature dynamics and robustness. Our technique is demonstrated in particular on the 4D toric code and Haah’s code; we find that the former falls into the 4D Ising universality class, whereas Haah’s code exhibits dimensional reduction and falls into the 1D Ising universality class.

Figure 1

A 2D cross section of a 4D lattice, with classical Ising spins at each site. Each red dotted link corresponds to a broken bond, and each blue loop is the 1D cross section of a 3D hypersurface domain wall. In the 4D lattice, the product of $A_{\ell }$ over all red links and the product of $B_c$ over all blue cubes correspond to two independent constraints.

Figure 2

Haah’s code: the two operators of Eqs. (14) and (15).