Infinitely fast critical dynamics: Teleportation through temporal rare regions in monitored quantum circuits

TL;DR

This work analyzes measurement-induced phase transitions in monitored quantum circuits with temporally fluctuating measurement rates, introducing a spacetime-rotated infinite-randomness perspective and an ultrafast scaling framework where $t^{\psi_{\tau}} \sim \log L$. Using ancilla-based entanglement probes and mutual information metrics, it identifies a sub-volume law entangling phase, an area-law phase, and a critical point characterized by vanishing dynamical exponent $z\to0$ and critical exponents $\nu_{\tau}\approx1.9(2)$ and $\psi_{\tau}\approx0.30(2)$, with a critical rate $\bar{p}_c \approx 0.150$. The results show temporal Griffiths effects arising from rare high-measurement-time events and reveal teleportation-driven ultrafast information propagation, including abrupt teleportation-induced jumps in information distance. These findings establish a new dynamical universality class for monitored quantum circuits with potential implications for fast state preparation and quantum information processing leveraging measurement-based teleportation.

Abstract

We consider measurement-induced phase transitions in monitored quantum circuits with a measurement rate that fluctuates in time. The spatially correlated fluctuations in the measurement rate disrupt the volume-law phase for low measurement rates; at a critical measurement rate, they give rise to an entanglement phase transition with ``ultrafast'' dynamics, i.e., spacetime ($x,t$) scaling $\log x \sim t^{ψ_τ}$. The ultrafast dynamics at the critical point can be viewed as a spacetime-rotated version of an infinite-randomness critical point; despite the spatial locality of the dynamics, ultrafast information propagation is possible because of measurement-induced quantum teleportation. We identify temporal Griffiths phases on either side of this critical point. We provide a physical interpretation of these phases, and support it with extensive numerical simulations of information propagation and entanglement dynamics in stabilizer circuits.

Figures (22)

• Figure 1: (a) Graphical depiction of a brickwork quantum circuit with temporal randomness in the measurement rate. The blue lines mark the two-qubit Clifford gates, while the projective measurements operating between them follow a time-dependent measurement rate $p(t)$ denoted by the different colors. (b) The steady-state half-cut bipartite entanglement entropy for chains of $L$ qubits with periodic boundary conditions at different values of $\bar{p}$ grows as $S\sim L^{\zeta}$ with $\zeta(\bar{p})<1$ for $0<\bar{p}<\bar{p}_c$, obeying a sub-volume-law behavior. (c) The phase diagram of the temporally random quantum circuit.
• Figure 2: Schematic description of the ancilla probes that we use. In both cases, we first let the circuit evolve until reaching a stationary state. In (a), we measure a qubit in the system and then arrange it in a Bell state with an ancilla qubit. We then let the system evolve for a time $\tau$, and repeat this process with the same site and a second ancilla. We follow by calculating the mutual information between the two ancillas $\mathcal{I}_2(Q_1,Q_2)$ over time. In (b), we use a system with open boundaries. At the initial time we entangle the ancilla $Q$ with the qubit at $x=0$. At later times, we calculate the ancilla's mutual information with a symmetric segment $A_x=[-x,x]$. The information propagation is evaluated using the smallest $x$ for which $\mathcal{I}_2(Q,A_x)>0$.
• Figure 3: (a) The exponent $\zeta(\bar{p})$ (solid black line), controlling the saturated entanglement entropy scaling $S\sim L^\zeta$, continuously decreases with an increase in $\bar{p}$ in the entangling phase. Our finite-size numerical estimates suggest $\zeta<1$ even for the smallest positive $\bar{p}$, see the inset. For $\bar{p}>\bar{p}_c$, we depict the dynamical exponent $z$ (dashed black line), evaluated via the entanglement time, $t_{\text{ent}}$, which decreases upon approach to the critical point. (b) Temporal sawtooth behavior of the half-cut bipartite entanglement entropy $S$ for a specific circuit realization. Here $L=256$ and $\bar{p}=0.005$. The green line marks the measurement rate at time $t$, $p(t)$. Each sharp drop in $S$ coincides with a significant peak of $p(t)$ corresponding to a high measurement rate. The sawtooth shape gradually disappears upon approaching criticality. (c) Numerical estimates of the dynamical exponent $z$ in the area-law phase, calculated using finite-size scaling of the entanglement time, defined as the first time for which $\mathcal{I}_3(A,B,C)\neq 0$ for any possible partition of the system in to adjacent segments $\{A,B,C\}$ each of length $L/4$; an example for such a partition is depicted in the inset.
• Figure 4: (a) Curve collapse analysis of $P\qty[\mathcal{I}_3=0]$, according to the ansatz in \ref{['eqn:ptmi_ansatz']}, yields a horizontal axis scaling with the values $\bar{p}_c=0.151$ and $\nu_{\tau}\psi_{\tau}=0.6$. The unscaled data is depicted in the inset. (b) The single ancilla entanglement $S_Q(t)$ and (c) the mutual information between two ancillas separated by $\tau=16$ scale as functions of $t^{\psi_{\tau}}/\log L$ with $\psi_{\tau}=0.3$. The inset in (b) shows that the same scaling applies to the average ancilla purification time, namely the average time it takes an ancilla to disentangle from the system completely.
• Figure 5: (a) The early time entanglement entropy before saturation grows at the critical point as $S\sim t^{1/2}$, a direct rotation of the infinite-randomness model. (b) Combined with the ultrafast scaling, we derive the system size dependence of the late-time saturated entanglement entropy at $\bar{p}_c$: $S\sim \qty(\log L)^{1/2\psi_{\tau}}$, with $\psi_{\tau}=0.32(3)$ over the range studied