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 ${D}_{2}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry protected topological (SPT) order, as well as a symmetry breaking (SB) order from a twofold 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 $\ensuremath{\nu}=\frac{4}{3}$ and the dynamical critical exponent $z=1$ are consistent with bond percolation, the prefactor of the logarithmic entanglement growth may take noninteger multiples of the percolation value. Remarkably, we identify a robust transient scaling regime for the purification dynamics of $L$ qubits. It reveals a modified dynamical critical exponent ${z}^{*}\ensuremath{\ne}z$, which is observable up to times $t\ensuremath{\sim}{L}^{{z}^{*}}$ and is reminiscent of the relaxation of critical systems into a prethermal state.