Topological order and entanglement dynamics in the measurement-only XZZX quantum code

• URL: http://arxiv.org/abs/2204.08489v3
• Date: Tue, 9 Aug 2022 09:47:11 GMT
• Title: Topological order and entanglement dynamics in the measurement-only XZZX quantum code
• Authors: Kai Klocke, Michael Buchhold
• Abstract summary: We study the dynamics of a $(1+1)$-dimensional measurement-only circuit defined by the stabilizers of the quantum error correcting code. The code corrects arbitrary single-qubit errors and it stabilizes an area law entangled with a $D = mathbbZ times mathbbZ$ symmetry protected topological (SPT) order. The Pauli measurements break the topological order and induce a phase transition into a trivial area law phase.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $D_2 = \mathbb{Z}_2 \times \mathbb{Z}_2$ 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 $\nu=\tfrac43$ and the dynamical critical exponent $z=1$ are consistent with bond percolation, the prefactor of the logarithmic entanglement growth may take non-integer 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 $z^*\neq z$, which is observable up to times $t\sim L^{z^*}$ and is reminiscent of the relaxation of critical systems into a prethermal state.

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