Topological Order and Criticality in $(2+1)\mathrm{D}$ Monitored Random Quantum Circuits

It has recently been discovered that random quantum circuits provide an avenue to realize rich entanglement phase diagrams, which are hidden to standard expectation values of operators. Here we study $(2+1)\mathrm{D}$ random circuits with random Clifford unitary gates and measurements designed to stabilize trivial area law and topologically ordered phases. With competing single qubit Pauli-Z and toric code stabilizer measurements, in addition to random Clifford unitaries, we find a phase diagram involving a tricritical point that maps to $(2+1)\mathrm{D}$ percolation, a possibly stable critical phase, topologically ordered, trivial, and volume law phases, and lines of critical points separating them. With Pauli-Y single qubit measurements instead, we find an anisotropic self-dual tricritical point, with dynamical exponent $z\approx 1.46$, exhibiting logarithmic violation of the area law and an anomalous exponent for the topological entanglement entropy, which thus appears distinct from any known percolation fixed point. The phase diagram also hosts a measurement-induced volume law entangled phase in the absence of unitary dynamics.

Figure 1

(a) A typical measurement-only random circuit. (b) Phase diagram of $(2+1)\mathrm{D}$ measurement-only random circuits. (c) A typical hybrid random circuit. (d) Phase diagram of $(2+1)\mathrm{D}$ hybrid random circuits. (e) Entanglement dynamics at the $p_y=0$ line of measurement-only random circuits (as well as the $p_u=0$ line of hybrid random circuits) maps to a classical bond percolation problem on a cubic lattice.

Figure 2

Phase transitions across the $p_y=0$ (top row) and $p_z=0$ (bottom row) lines of the phase diagram for the measurement-only circuit: (a) $S_{\mathrm{topo}}$ and (b) $S_a$ measured at $t=4L$ versus $p_z$ for fixed $p_y=0$. Insets show the corresponding data collapse. (c) $S_R(x)$ for system size $L=64$ at the percolation critical point $(p_z,p_y)=(0.188,0)$, with the best fit of scaling functions $S^{\mathrm{qlm}}(x)$ (solid line) and $S^{\mathrm{q}1\mathrm{D}}$ (dashed line). The inset is the best fit value of the $b$ parameter in Eq. (5) as a function of $L$. (d) $S_{\mathrm{topo}}$ and (e) $S_a$ measured at $t=0.6L^{1.46}$ versus $p_y$ for fixed $p_z=0$. Insets show the corresponding data collapse. (f) $S_R(x)$ for system size $L=64$ at the self-dual critical point $(p_z,p_y)=(0,0.5)$, with the best fit of scaling functions $S^{\mathrm{q}1\mathrm{D}}$ (solid line) and $S^{\mathrm{qlm}}(x)$ (dashed line). The inset shows the linear dependence of the best fit value of the $a$ parameter in Eq. (4).

Figure 3

(a) The ancilla EE $S_a$ measured at $t=L$ as a function of $p_z$ for fixed $p_u=0.01$ in the hybrid random circuit. (b) $S_a$ as a function of time at $(p_z,p_u)=(0.17,0.01)$ in the hybrid random circuit. The inset is the same, plotted as a function of $t/L$.