Topological Order And Entanglement Dynamics In The Measurement-only XZZX Quantum Code

Abstract

We examine the dynamics of a $(1+1)$-dimensional measurement-only circuit defined by the stabilizers of the [[5,1,3]] quantum error correcting code interrupted by single-qubit Pauli measurements. The code corrects arbitrary single-qubit errors and it stabilizes an area law entangled state with a $D_2=\{Z{\}}_2\times \{Z{\}}_2$ symmetry protected topological (SPT) order, as well as a symmetry breaking (SB) order from a two-fold bulk degeneracy. The Pauli measurements break the topological order and induce a phase transition into a trivial area law phase. Allowing more than one type of Pauli measurement increases the measurement-induced frustration, and the SPT and SB order can be broken either simultaneously or separately at nonzero measurement rate. This yields a rich phase diagram and unanticipated critical behavior at the phase transitions. Although the correlation length exponent $\nu =\frac{4}{3}$ and the dynamical critical exponent $z=1$ are consistent with bond p