Tapestry of dualities in decohered quantum error correction codes and a bound on error threshold

Quantum error correction (QEC) codes protect quantum information from noises and errors that inevitably occur in near-term quantum computing platforms. An important measure of the performance of the QEC codes is the error threshold. We argue that the error threshold belongs to a family of decoherence-induced phase transitions (DIPT) of the QEC codes, which are signaled by the singularities of the Rényi entropies of the code after decoherence caused by the errors. For a large class of QEC codes, namely the Calderbank-Shor-Steane (CSS) codes, we find a mapping between the Rényi entropies caused by the Pauli errors and a pair of disordered classical ℤ₂ spin models. The two classical models are related to one another by a Kramers-Wannier-like duality. The DIPTs correspond to critical points of these classical models. Moreover, we find an additional duality between bit-flip (X) and phase (Z) errors for the Rényi entropies with Rényi index R = 2, 3. An intricate tapestry of dualities is thus woven with the aforementioned elements. For CSS codes with a symmetry between the X- and Z-stabilizers, we show that the R = 2, 3 dualities become self-dualities, which strongly constrain the critical points of the DIPTs with R = 2, 3. Utilizing these self-dualities, we further obtain a generic upper bound on the error threshold for this class of CSS symmetric codes, which includes the 2d toric code, Haah’s code, etc.