Data-Driven Learnability Transition of Measurement-Induced Entanglement

TL;DR

This work reframes MIE detection as a data-driven learning problem that assumes no prior knowledge of state preparation, and trains a neural network in a self-supervised manner to predict the uncertainty metric for MIE--the gap between upper and lower bounds of the average post-measurement bipartite entanglement.

Abstract

Measurement-induced entanglement (MIE) captures how local measurements generate long-range quantum correlations and drive dynamical phase transitions in many-body systems. Yet estimating MIE experimentally remains challenging: direct evaluation requires extensive post-selection over measurement outcomes, raising the question of whether MIE is accessible with only polynomial resources. We address this challenge by reframing MIE detection as a data-driven learning problem that assumes no prior knowledge of state preparation. Using measurement records alone, we train a neural network in a self-supervised manner to predict the uncertainty metric for MIE--the gap between upper and lower bounds of the average post-measurement bipartite entanglement. Applied to random circuits with one-dimensional all-to-all connectivity and two-dimensional nearest-neighbor coupling, our method reveals a learnability transition with increasing circuit depth: below a threshold, the uncertainty is small and decreases with polynomial measurement data and model parameters, while above it the uncertainty remains large despite increasing resources. We further verify this transition experimentally on current noisy quantum devices, demonstrating its robustness to realistic noise. These results highlight the power of data-driven approaches for learning MIE and delineate the practical limits of its classical learnability.

Figures (4)

• Figure 1: Setup and learnability transition. (a) Schematic workflow. Given a quantum state, random single-qubit rotations $U_{A}^{s}$ and $U_{B}^{s}$ are applied to qubits $A$ and $B$, after which all qubits are measured projectively in the computational basis. Measurement outcomes on all qubits except $A$ and $B$ are fed into a transformer encoder that outputs $\rho_{AB,m}$, while the outcomes on $A$ and $B$ are used to construct the classical snapshot $\sigma_{AB,m}^{s}$. The loss $\mathcal{L}(\theta)$ and uncertainty $\Delta$ are computed from these quantities. Notably, no prior knowledge of the state-preparation procedure is required. (b) Structure of the 1D random all-to-all circuit. (c) $\mathcal{L}(\theta)$ as a function of circuit depth $t$ for different system size $L$. Error bars denote the standard error over $M=50$ different circuit realizations. (d) $\Delta$ as a function of $t$ for different $L$. The inset shows the MIE between $A$ and $B$. Each point is averaged over $5\times10^4$ circuit realizations; the associated standard error is smaller than the symbol size.
• Figure 2: Learnability transition in 1D random all-to-all circuits for $L=20$. (a) $\Delta$ versus depth $t$ for varying $N_{m}$ with $N_{p} = 7\times10^4$ fixed. (b) $\Delta$ versus depth $t$ for varying $N_{p}$ with $N_{m}=8\times10^4$ fixed. (c,d) $\Delta$ as a function of $N_{m}$ and $N_{p}$ for representative depths $t=1.25$ and $t=3.5$, respectively.
• Figure 3: Learnability transition in 2D nearest-neighbor circuits for $L=5\times 5$. (a) $\Delta$ versus $t$ for varying $N_{m}$ with $N_{p} = 7\times10^4$ fixed. (b) $\Delta$ versus $t$ for varying $N_{p}$ with $N_{m}=8\times10^4$ fixed. (c,d) $\Delta$ as a function of $N_{m}$ and $N_{p}$ at representative depths $t=1.6$ and $t=6.4$, respectively.
• Figure 4: Learnability transition in 1D all-to-all circuits with noise. (a,b) $\mathcal{L}(\theta)$ and $\Delta$ for $L=16$, obtained from noisy classical simulations using the Qiskit noise snapshot of the IBM QPU $\text{ibm\_brisbane}$. Here $N_{m}=2\times10^4$, $N_{p}=2\times10^4$, $N_{e}=5\times10^4$, and $M=50$. (c,d) $\mathcal{L}(\theta)$ and $\Delta$ for $L=20$, obtained from experiments on the IBM QPU $\text{ibm\_marrakesh}$. Here $N_{m}=4\times10^4$, $N_{p}=7\times10^4$, $N_{e}=5\times10^3$, and $M=5$.