Phase transition and evidence of fast-scrambling phase in measurement-only quantum circuits

Information scrambling is nowadays one of the most important topics in various fields of research. Measurement-only circuits (MoCs) exhibit specific information scrambling dynamics, depending on the types of projective measurements and their mutual anticommutativity. The spatial range of the projective measurements in MoCs gives significant influences on circuit dynamics. In this work, we introduce and study long-range MoCs, which exhibit an interesting behavior in their dynamics. In particular, the long-range measurements can induce volume-law phases in MoCs without unitary time evolution, which come from anticommutative frustration of measurements specific to the long-range MoCs. This phenomenon occurs even in MoCs composed of solely two-body measurements, and it accompanies an entanglement phase transition. Crucially, our numerics find evidences that MoCs can be a fast scrambler. Interplay of high anticommutativity among measurements and their long-range properties generates fast entanglement growth in the whole system beyond linear-light-cone spreading.

Figure 1

Schematic image of circuit setting for LR2BM (a) and LR3BM (b). Unit of time (one-time step) includes $L$-projective measurement blocks. Green, gray, and yellow boxes represent measurement operator Pauli operators, $XY,Z$, respectively. (c) Schematic image of circuit system and four subsystems ($A,B,C,D$) on one-dimensional qubits for calculation of physical observables. Each subsystem includes $L/4$ qubits. In particular, for the TMI we focus on correlations $A$, $B$, and $C$. For the negativity and the entanglement entropy, we focus on $A$ and $B$. There is no overlap site between these subsystems. Periodic boundary conditions are considered.

Figure 2

Late-time dynamics for TMI ${\tilde{I}}_3$ (a) and negativity $N_{AB}$ (b) for LR2BM with system size $L=256$. For each data, we averaged over $O\left({10}^2\right)$ samples.

Figure 3

System-size dependence of TMI (a) and half-cut entanglement entropy (b) for LR2BM with various values of $\gamma$. We averaged over $O\left({10}^2\right)\text{–}O\left({10}^3\right)$ samples for each system size $L$.

Figure 4

Obtained results for LR3BM. (a) Late-time dynamics for TMI ${\tilde{I}}_3$. For each set of data, we averaged over $O\left({10}^3\right)$ samples. (b), (c) System-size dependence of TMI and half-cut entanglement entropy. For the data (b) and (c), we averaged over $O\left({10}^2\right)\text{–}O\left({10}^3\right)$ samples for each system size $L$.

Figure 5

(a) Different system-size data of TMI for LR2BM. (b) Heat map of $R^2$ value. At its maximum, ${\gamma }_c=-2.21\left(0\right)$, $\nu =0.67\left(6\right)$. (c) Scaling function for ${\gamma }_c=-2.21\left(0\right)$, $\nu =0.67\left(6\right)$. We averaged over ${10}^3,600,400$ samples for $L=128$, 256, 512, respectively.

Figure 6

Steady-state entanglement phase diagrams for LR2BM and LR3BM. The phase of LR2BM is clearly separated into volume and subvolume law phases. For the neighboring three-body cluster limit in the lower phase diagram for LR3BM, the volume-law phase remains, consistent with the prediction of the previous study [33].

Figure 7

Intermediate dynamics of entanglement entropy for half-subsystem for LR2BM. The time evolution is up to $t=2$. We plot three long-range cases: (a) $\gamma =0$, (b) $\gamma =-2$, (c) $\gamma =-3$. Cases (a) and (b) are in volume-law phase. The other (c) is in subvolume-law phase. For all data, we averaged over $O\left({10}^2\right)\text{–}O\left({10}^3\right)$ samples for each system size $L$.

Figure 8

Entanglement contour $s_{AB}\left(j\right)$ with $j=0,\cdots ,L_{AB}/2-1$ for $\gamma =-0.5$ (a), $\gamma =-1.5$ (b), and $\gamma =-2.5$ (c). We set $L=256$, $L_{AB}=128$. All data are obtained by averaging over $O\left({10}^4\right)$ samples. (d) Wave front $d_{0.1}\left(t\right)$ for various $\gamma$ and fitting curve is set $t=a_0{d}_{0.1}^{a_1}\left(t\right)+a_2$, where $a_{\ell }^{\prime }\mathrm{s}$ ($\ell =0,1,2$) are fitting coefficients.

Figure 9

Saturation time $t_{\mathrm{sat}}$ vs system size $L$ for half-cut entanglement entropy for LR2BM: (a) $\gamma =0$ data with $\alpha =0.8$, (b) $\gamma =-0.5$ with $\alpha =0.78$. For all data, we averaged over $O\left({10}^2\right)\text{–}O\left({10}^3\right)$ samples for each system size $L$.

Figure 10

Intermediate dynamics of entanglement entropy for half-subsystem for LR3BM. The time evolution is up to $t=2$. We plot three long-range cases, (a) $\gamma =0$, (b) $\gamma =-2$, (c) $\gamma =-3$. All cases are in volume-law phase. For all data, we averaged over $O\left({10}^2\right)\text{–}O\left({10}^3\right)$ samples for each system size $L$.

Figure 11

Entanglement contour $s_{AB}\left(j\right)$ with $j=0,\cdots ,L_{AB}/2-1$ for $\gamma =-1$ (a), $\gamma =-1.5$ (b), and $\gamma =-2$ (c). We set $L=256$, $L_{AB}=128$. All data are obtained by averaging over $O\left({10}^4\right)$ samples. (d) Wave front $d_{0.1}\left(t\right)$ for various $\gamma$ and the fitting curve for early time dynamics is set $t=a_0{d}_{0.1}^{a_1}\left(t\right)+a_2$ where $a_{\ell }^{\prime }\mathrm{s}$ ($\ell =0,1,2$) are fitting coefficients.

Figure 12

Saturation time $t_{\mathrm{sat}}$ vs system size $L$ for half-cut entanglement entropy $S_{AB}$. (a) $\gamma =0$ data with $\alpha =0.9$, (b) $\gamma =-1$ with $\alpha =0.9$. For all data, we averaged over $O\left({10}^2\right)\text{–}O\left({10}^3\right)$ samples for each system size $L$.

Figure 13

Saturation value of $\langle N_{AB}\rangle$ as a function of $\gamma$ for $L=64$, 128, and 256 with 6000, 4000, and 2000 samples, respectively.

Figure 14

Behavior of TMI $I_3$ for negative large $\gamma$ (strong short-range cases). We averaged over $1000,600,400$ samples for $L=128$, 256, 512, respectively..