Robust detection of an entanglement transition in the projective transverse field Ising model

TL;DR

This work addresses the challenge of observing entanglement transitions in projective quantum circuits where postselection and noise obscure direct measurements. The authors introduce a dual strategy: decoding-based lower bounds via MWPM-enabled error correction to recover encoded information, and shadow tomography upper bounds augmented by the same error-correction framework to bound the ancilla entanglement entropy from above. By mapping PTIM trajectories to a 1+1D grid and employing extended colored cluster models, they show robust finite-size crossings at the critical point $p_c=0.5$, with a bound interval $\delta$ that scales with the noise rate $\eta$ and serves as a noise diagnostic. The approach yields experimentally accessible, frame-worked bounds on the entanglement transition without full state tomography, offering practical routes to characterize non-equilibrium entanglement transitions in large, noisy quantum systems.

Abstract

We propose a scalable and noise-resilient protocol for the detection of the entanglement transition in a projective version of the transverse field Ising model. Entanglement transitions are experimentally difficult to observe due to the inherent randomness of projective measurements and noise in large-scale experimental settings. Our approach combines error correction algorithms with classical shadow tomography to overcome both problems. This allows for experimentally accessible upper and lower bounds on the entanglement transition without postselection or full state tomography. These bounds remain robust under noise and their sharpness is a measure of the noise rate.

Figures (6)

• Figure 1: Entanglement transition of the PTIM. (a) Exemplary trajectory of a PTIM. Circles (boxes) correspond to projective $E_i\sim X$ ($S_e\sim ZZ$) measurements applied with probability $p$ ($1-p$) per site and time step. Noise manifests as unrecorded measurements (gray circles and boxes). (b) The PTIM features a transition in the trajectory-averaged half-system EE $\mathcal{S}(L/2)$ in the steady state for $T\to\infty$ (here $T\gtrsim 10L$) at $p_c=0.5$ between two area-law phases. (c) Alternative protocol to detect the transition utilizing an intially fully-entangled ancilla qubit. (d) The transition in the survival probability of an initial Bell cluster is measured by the EE $\mathcal{S}_\text{a}$ of the ancilla for $L=T$. All simulations in this paper are based on $10^5$ trajectories.
• Figure 2: Lower bound via decoding. (a) Protocol to measure the decoding transition of the PTIM as described in the text. (b) Sample-averaged correlation as a function of $p$ for system sizes $L=10,\ldots,40$ with (green) and without (black) noise. In a noisy circuit, the transition is shifted to the left and provides a lower bound on the entanglement transition.
• Figure 3: Upper and lower bound via shadow tomography. (a) Protocol to measure the shadow of the entanglement entropy of a subsystem $A$ on the noisy PTIM, based on a decoding algorithm for classical state prediction (see text). (b) Optimization over the regularization $0<\varepsilon<1$ yields an upper (dashed blue) and lower bound (dashed red) on the ancilla EE $\mathcal{S}_\text{a}$ (solid black), shown for a finite noise rate $\eta=0.2$ and $L=T=30$. (c) Comparison of the lower bounds from decoding (green) and ST (red), and the upper bound from ST (blue). For $L=T=10,\ldots,40$ we find clear crossings which provide experimentally accessible lower and upper bounds on the entanglement transition at $p_c=0.5$ (black). The simulations suggest that the two lower bounds share the same threshold. The interval $\delta$ between upper and lower bounds can be used to measure the noise rate $\eta$ (inset).
• Figure S1: MWPM algorithm for the error correction $\mathcal{E}_E$. (a) Sample PTIM trajectory $M^\text{p}=(E^\text{p},S^\text{p})$ with noise (unseen measurements: gray circles and boxes). (b) The pattern $(E^\text{p},S^\text{p})$ is re-formatted into a grid and measurement results $S^\text{r}$ are included (red and blue vertical edges). This data is the input of the algorithm $\mathcal{E}_E$. The pattern is augmented by nodes at the endpoints of strings of $S_e$-measurements with result $-1$ (blue discs). (c) The grid is reduced and converted into a weighted graph by deleting all solid vertical edges and adding auxiliary nodes on the boundaries; the edge weights count the number of traversed horizontal solid edges in the grid. For this graph, a minimum-weight perfect matching is computed (green edges) and the matched edges are associated with their corresponding weights. Boundary nodes are used to correctly handle the system boundaries. (d) Matched edges with positive weight correspond to additional $E_i$-measurements which augment the original trajectory $E^\text{p}\mapsto\tilde{E}^\text{p}$ (green discs). Note that these predictions rarely match the actual missed measurements (cf. green and gray discs).
• Figure S2: MWPM algorithm for the error correction $\mathcal{E}_S$. (a) Sample PTIM trajectory $M^\text{p}=(E^\text{p},S^\text{p})$ with noise (unseen measurements: gray circles and boxes). For demonstration purposes we use a slightly different trajectory than in \ref{['fig:DecodingE']} (a). (b) The grid encodes the measurement pattern together with the measurement outcomes $E^\text{r}$ (red and blue vertical edges). Note that in comparison to \ref{['fig:DecodingE']} (b), here solid vertical edges correspond to $E_i$-measurements and dashed horizontal edges encode $S_e$-measurements (without results). Nodes are assigned to endpoints of strings of $E_i$-measurements with outcome $-1$ (blue discs). (c) To construct the reduced grid, all $E_i$-measurements (solid vertical edges) are removed. The resulting picture is transformed into a weighted graph of which a minimum-weight perfect matching is computed (every solid horizontal edge contributes a weight of one). One such matching is drawn on the grid (green edges). (d) Finally, matched edges with non-zero weight are identified with additional $S_e$-measurements (green boxes). Note that again the predictions typically do not match the actual missed measurements (cf. green and gray boxes).