Measurement-Induced Transition in Long-Range Interacting Quantum Circuits

The competition between scrambling unitary evolution and projective measurements leads to a phase transition in the dynamics of quantum entanglement. Here, we demonstrate that the nature of this transition is fundamentally altered by the presence of long-range, power-law interactions. For sufficiently weak power laws, the measurement-induced transition is described by conformal field theory, analogous to short-range-interacting hybrid circuits. However, beyond a critical power law, we demonstrate that long-range interactions give rise to a continuum of nonconformal universality classes, with continuously varying critical exponents. We numerically determine the phase diagram for a one-dimensional, long-range-interacting hybrid circuit model as a function of the power-law exponent and the measurement rate. Finally, by using an analytic mapping to a long-range quantum Ising model, we provide a theoretical understanding for the critical power law.

Figure 1

(a) Schematic of our long-range interacting hybrid circuit, which consists of layers of unitary evolution and randomly placed projective measurements. Two-qubit gates separated by distance $r$ occur with a probability $P(r)\sim 1/r^{\alpha }$. (b) Phase diagram as a function of the measurement rate $p$ and the power-law exponent $\alpha$. For $\alpha \gtrsim 3$, the measurement-induced phase transition is described by conformal field theory (purple), while for $\alpha \lesssim 3$, the universality changes continuously (purple-red gradient). For $\alpha <2$, area-law entropy scaling crosses over to subvolume law scaling, where half-chain entanglement entropy ($S_{L/2}$) scales as $L^{2-\alpha }$. Despite this different scaling behavior, both the area and subvolume law regimes are in the purifying phase.

Figure 2

The correlation length critical exponent, $\nu$ (red), and the dynamical critical exponent, $z$ (teal), extracted from a finite-size scaling analysis of the purification time. For $\alpha \gtrsim 3$, one finds $z\approx 1$ corresponding to CFT. For $\alpha \lesssim 3$, both critical exponents vary continuously, indicating a continuum of non-CFT universality classes in this regime. The dotted line (and shaded grey region) is consistent with a critical power law, ${\alpha }_{\mathrm{c}}=3-\eta$, where $\eta \sim 0.2$ is the anomalous dimension.

Figure 3

(a)–(c) The antipodal mutual information $I_{\mathrm{AB}}$, purification time ${\tau }_p$, and global entropy dynamics $S(t)$ as a function of the measurement rate $p$, for power law $\alpha =3.5$. Insets depict the corresponding finite-size scaling collapse with $x=(p-p_c)L^{1/\nu }$ (the $x$-axis tick denotes $x=0$). Different system sizes ($L=[32,64,128,256,512]$) are indicated via increasing opacity. (d)–(f) Depict analogous plots for $\alpha =2.25$. Both the peak heights of $I_{\mathrm{AB}}$ [d] and the crossing points of ${\tau }_p/L$ [e] exhibit marked $L$ dependence. This immediately indicates that the measurement-induced transition is no longer conformal. (c),(f) Colors indicate different time slices of $S(t)$ (see legend). Finite-size collapses (insets) are obtained by rescaling $t=cL\to c{L}^z$, with $c=\{1/2,2/3,2\}$ depending on the time slice. (g) Circuit schematic for $I_{\mathrm{AB}}$. The system is initialized in a product state and the mutual information is measured between antipodal regions $A$ and $B$. (h) Circuit schematic for ${\tau }_p$. The system is initialized in a product state with a single maximally mixed qubit. To avoid early-time finite-size effects, we apply a global scrambling Clifford, $U_s$, before evolving with our hybrid circuit. (i) Circuit schematic for $S(t)$. The system is initialized in a maximally mixed state and slowly purifies under hybrid dynamics.