Critical properties of the measurement-induced transition in random quantum circuits

We numerically study the measurement-driven quantum phase transition of Haar-random quantum circuits in $1+1$ dimensions. By analyzing the tripartite mutual information we are able to make a precise estimate of the critical measurement rate $p_c=0.17\left(1\right)$. We extract estimates for the associated bulk critical exponents that are consistent with the values for percolation, as well as those for stabilizer circuits, but differ from previous estimates for the Haar-random case. Our estimates of the surface order parameter exponent appear different from those for stabilizer circuits or percolation, but we cannot definitively rule out the scenario where all exponents in the three cases match. Moreover, in the Haar case the prefactor for the entanglement entropies $S_n$ depends strongly on the Rényi index $n$; for stabilizer circuits and percolation this dependence is absent. Results on stabilizer circuits are used to guide our study and identify measures with weak finite-size effects. We discuss how our numerical estimates constrain theories of the transition.

Figure 1

(a) Phase diagram with $p_c^H$ and $p_c^P$ marking the separation between volume-law and area-law entanglement in the Rényi entropies $n\ge 1$ and $n=0$, respectively. (b) Depiction of the model: Blue rectangles represent two-site entangling gates and the green circles denote local projective measurements that are performed with a probability $p$. (c) The geometry used to compute the TMI, partitioning the system with periodic boundary conditions into four equal-length segments. (d) The setup to probe the order parameter correlation function through entangling the local system qubits at time $t=t_0=2L$ with two ancilla qubits separated by a distance $r-r^{\prime }$ and computing their mutual information at later times.

Figure 2

Tripartite mutual information (TMI) near the transition. TMI near the transition for a circuit with (a) Clifford and (b) Haar gates. Scaling collapse of the data for (c) Clifford and (d) Haar gates.

Figure 3

Scaling collapse of mutual information between two ancilla qubits (inset: unscaled data). Mutual information between ancilla entangled at $r-r^{\prime }=L/2$ for (a) Clifford and (b) Haar gates with periodic boundary conditions. Mutual information between ancilla entangled at $r=1, r^{\prime }=L$ for (c) Clifford and (d) Haar gates with open boundary conditions.

Figure 4

Properties of the Rényi entropies at criticality. (a) The Rényi entropies show a $lnL$ dependence near the critical point estimated by ${\mathcal{I}}_{3,n}$ for $n=0.7,1,2,3,4,5,\infty$, with fits shown as solid lines. (b) The coefficient of the $lnL$ term has a strong Rényi index dependence that is well described by a functional form $a(1+1/n)+b$.