Measurement-induced entanglement in noisy 2D random Clifford circuits

TL;DR

This work investigates measurement-induced entanglement (MIE) in noisy 2D random Clifford circuits under column-by-column sampling, focusing on the boundary-state operator entanglement $S_{ m op}$. By combining stabilizer-based numerics for mixed Clifford states with an MPO-SEBD boundary-state framework and a simplified stabilizer-length distribution model, the authors show that any constant depolarizing noise rate $p>0$ enforces area-law scaling $S_{ m op}^{\max}\sim \frac{T}{p}$ and exponentially localizes stabilizer generators, leading to exponential decay of conditional mutual information. In the noiseless limit, a finite-depth area-to-volume transition is recovered, consistent with prior work. The results imply that noisy 2D random Clifford circuits are classically simulable via MPO-SEBD in sub-logarithmic depths (for $p=\\Omega(1)$) or constant depths (for $p=\\Omega(1/\log N)$), shedding light on the boundary between quantum advantage and classical simulability in realistic noisy settings. The study also discusses extensions to Haar-random circuits and connections to patching algorithms and L1-norm error control, highlighting the broader relevance to quantum simulation and complexity of noisy quantum dynamics.

Abstract

We study measurement-induced entanglement generated by column-by-column sampling of noisy 2D random Clifford circuits of size $N$ and depth $T$. Focusing on the operator entanglement $S_{\rm op}$ of the sampling-induced boundary state, first, we reproduce in the noiseless limit a finite-depth transition from area- to volume-law scaling. With on-site probablistic trace noise at any constant rate $p>0$, the maximal $S_{\rm op}$ attained along the sampling trajectory obeys an area law in the boundary length and scales approximately linearly with $T/p$. By analyzing the spatial distribution of stabilizer generators, we observe exponential localization of stabilizer generators; this both accounts for the scaling of the maximal $S_{\rm op}$ and implies an exponential decay of conditional mutual information across buffered tripartitions, which we also confirm numerically. Together, these results indicate that constant local noise destroys long-range, volume-law measurement-induced entanglement in 2D random Clifford circuits. Finally, based on the observed scaling, we conjecture that a tensor-network-based algorithm can efficiently sample from noisy 2D random Clifford circuits (i) at sub-logarithmic depths $T = o(\log N)$ for any constant noise rate $p = Ω(1)$, and (ii) at constant depths $T = O(1)$ for noise rates $p = Ω(\log^{-1}N)$.

Figures (4)

• Figure 1: Setup and conjectured phase diagram.(a) Brickwork layer order. Edges of the same color denote two-qubit gates applied within a single layer; the four-layer cycle (green $\rightarrow$ yellow $\rightarrow$ red $\rightarrow$ blue) is repeated periodically to entangle the lattice. One cycle corresponds to a circuit depth of $4$. (b) Column-by-column sampling process napp2022. After every circuit layer, an on-site noise channel of rate $p$ acts independently on each qubit. Sampling proceeds along the width direction. To sample the $t$-th column ($t=3$ illustrated here), we include the gates and qubits within its past light cone (blue region) in the boundary state $\rho_t$ during the sampling procedure, and then measure the $t$-th column. The boundary state on the unmeasured region has length $L$ and width $W_b$vftn_fn. We diagnose its half-space operator entanglement $S_{\rm op}$ across the cut along the direction of the width (green line), and also study conditional mutual information (CMI) in \ref{['sec_clif_string']}. (c) Conjectured entanglement phase diagram of the maximal measurement-induced operator entanglement $S_{\rm op}^{\max}$, plotted versus noise rate $p$ and depth $T$. For $p=0$ (noiseless), a finite-depth transition at $T_c$ [cf. \ref{['fig2']}(b)] separates volume-law and area-law behavior napp2022. For any constant noise rate $p>0$, our numerical and analytical analysis indicate area-law scaling of $S_{\rm op}^{\max}\sim T/p$.
• Figure 2: Dynamics and scaling of measurement-induced operator entanglement.(a) Evolution of $S_{\rm op}(t)$ during sampling for noiseless and noisy circuits, with circuit depth $T=8$ and length $L=40$. For the noiseless case ($p=0$), $S_{\rm op}(t)$ increases monotonically with $t$ and saturates to its volume-law value. For noisy case ($p>0$), $S_{\rm op}(t)$ grows, reaches a peak value $S_{\rm op}^{\rm peak}$, and then decreases toward a steady-state value. (b) Finite-depth transition in the noiseless case. Plotted is $S_{\rm op}^{\rm peak}$ versus $L$ for depths $T\in4,5,6,7,8$ with linear fits $S_{\rm op}^{\rm peak}=\alpha_T L+c_T$. The inset shows the fitted slope $\alpha_T$ as a function of $T$, identifying a critical depth $T_c= 6$ where $\alpha_T$ turns nonzero. (c) Area-law scaling in the noisy case at fixed depth $T=8$. For each noise rate $p$, $S_{\rm op}^{\rm peak}$ exhibits a nearly logarithmic growth with $L$ at small $L$ and then saturates to the asymptotic maximal value $S_{\rm op}^{\rm max}$, indicated by the dashed lines.
• Figure 3: Saturated area-law value $S_{\rm op}^{\rm max}$ versus circuit depth $T$ for several fixed noise rates $p$. The dashed lines are linear fits $S_{\rm op}^{\rm max}(T,p)=\beta_p T+c_p$. Inset: fitted slopes $\beta_p$ plotted against $p$. The line is a power-law fit $\beta_p\sim p^{-0.9}$ to the data.
• Figure 4: Distribution of stabilizer generators and the behavior of conditional mutual information (CMI) at $t_{\rm peak}$. (a) 1D labeling of the qubit sites for the boundary state $\rho_{t_{\rm peak}}$ (here $L=6$, $W_b=3$). The green line denotes the horizontal half-space bipartition for computing $S_{\rm op}$ [cf. \ref{['fig1']}(b)]. Colored dashed segments are examples of stabilizer generators, with circles marking their left endpoints $x_l$ and right endpoints $x_r$. The center locations of the stabilizer generators, com($g$)$=[x_l(g)+x_r(g)]/2$, are marked with crosses. (b) The spatial center distribution ${\mathcal{C}}(x_c)$ of stabilizer generators. Here $p=0.01,L=80,T=8$. The labeling of qubit sites follows the convention from panel (a). Since $\mathrm{com}(g)$ can take half-integer values, we place each half-integer site $x + 1/2$ immediately to the right of its corresponding integer site $x$. (c) Normalized length distribution $\mathcal{D}(\ell)$ of stabilizer generators versus $\ell$ for fixed depth $T=8$, system length $L=80$, and several noise rates $p$. The dashed lines are exponential fits to the data for noisy cases. (d) The extracted exponential scaling exponent $\gamma_{p,T}^{\rm len}$ [cf. \ref{['clif_len_scale']}] of the stabilizer generator length distribution, as well as that of the CMI, $\gamma_{p,T}^{\rm CMI}$. For fixed depth $T=8$ (circle markers), we varied the noise rate $p\in{0.005,0.01,0.02,0.04}$. For fixed $p=0.02$ (plus markers), we studied $T\in{8,12,16}$. The line shows a single linear fit to the combined data for $\gamma_{p,T}^{\rm len}$ and $\gamma_{p,T}^{\rm CMI}$. The plotted exponents for $T=8$ are extracted from the dashed lines in panels (c) and (f), while the corresponding plots of the length distribution and CMI for $T=12,16$ are omitted. (e) Schematic of the bipartition $AB$ and tripartition $ACB$ with buffer region $C$ of length $d_C$. For the bipartition, only the stabilizer generators (illustrated as colored circles) that cross the cut contribute to $S_{\rm op}$ [cf. \ref{['eq:string_ent']}]. For the tripartition, only the stabilizer generators that cross the buffer region $C$ contribute to the CMI $I(A:B|C)$. (f) The scaling of CMI $I(A:B|C)$ with the length $d_C$ of the buffer region $C$. The markers and the corresponding parameters ($T=8,L=80$) are the same as in panel (c). The dashed lines are exponential fits to the data for noisy cases.