Neural quantum states for entanglement depth certification from randomized Pauli measurements

TL;DR

This work tackles the challenge of certifying entanglement depth in large quantum systems by recasting it as a likelihood-based model selection problem. It trains a fully entangling Neural Quantum State (NQS) and a hierarchy of Separable NQS (SNQS) constrained by cluster partitions on finite-shot local Pauli data, using the gap in minimal negative log-likelihoods to certify non-$k$-separability and bound the entanglement depth $d_e$. The authors validate the method on simulated six- and ten-qubit states (GHZ, Bell pairs, and cluster mixtures) under both pure and mixed conditions, showing robust detection of entanglement depth and revealing that several distinct partitions can explain the data at the same depth. Beyond certification, they demonstrate interpretability of the unrestricted NQS through weight-based diagnostics that reveal effective couplings and qubit affinities aligning with the underlying cluster structure, providing a tomography-free window into multipartite entanglement organization with practical implications for scalable quantum-state characterization.

Abstract

Entanglement depth quantifies how many qubits share genuine multipartite entanglement, but certification typically relies on tailored witnesses or full tomography, both of which scale poorly with system size. We recast entanglement-depth and non-$k$-separability certification as likelihood-based model selection among neural quantum states whose architecture enforces a chosen entanglement constraint. A hierarchy of separable neural quantum states is trained on finite-shot local Pauli outcomes and compared against an unconstrained reference model trained on the same data. When all constrained models are statistically disfavored, the data certify entanglement beyond the imposed limit directly from measurement statistics, without reconstructing the density matrix. We validate the method on simulated six- and ten-qubit datasets targeting GHZ, Dicke, and Bell-pair states, and demonstrate robustness for mixed states under local noise. Finally, we discuss lightweight interpretability diagnostics derived from trained parameters that expose coarse entanglement patterns and qubit groupings directly from bitstring statistics.

Figures (5)

• Figure 1: Protocol for entanglement-depth certification via generative modeling. Bitstrings obtained from randomized Pauli measurements on the target device are used to train a hierarchy of Neural Quantum State (NQS) models. The hierarchy consists of an unconstrained reference model ($\mathcal{A}_{\text{full}}$) and various Separable NQS architectures ($\mathcal{A}_{\mathcal{P}}$), where the bipartite connectivity is masked to enforce factorization into clusters of size at most $k$. By minimizing the negative log-likelihood (NLL) $\mathcal{L}^*$, we compute the likelihood gap $\Delta_{\mathcal{P}} = \mathcal{L}^*_{\mathcal{A}_{\mathcal{P}}} - \mathcal{L}^*_{\mathcal{A}_{\text{full}}}$. A persistent gap ($\Delta_{\mathcal{P}} \gg 0$) signifies that the data cannot be explained by any state within the $k$-producible hypothesis class, thereby certifying an entanglement depth $d_e > k$ without reconstructing the full density matrix.
• Figure 2: Training dynamics of the negative log--likelihood (NLL) for neural quantum state models fitted to finite--shot random--Pauli data. In each panel, the blue curve shows the fully entangling ("unconstrained") ansatz, while the other curves correspond to separable neural quantum states with different partitions $\mathcal{P}$ of the qubits (see legends), whose maximal block size bounds the entanglement depth. (a) Six--qubit GHZ target, Eq.\ref{['eq:psi_A']}. (b) Product of three Bell pairs on six qubits, Eq.\ref{['eq:psi_B']}. (c) Ten--qubit product of a three--qubit GHZ state, a three--qubit Dicke state with one excitation, and a four--qubit GHZ state in local bases, Eq.\ref{['eq:psi_C']}. Persistent offsets between the constrained and unconstrained NLL traces signal partitions that are incompatible with the target entanglement structure. The NLL gap for partition $1|1|1|1|1|1$, $\Delta_{1|1|1|1|1|1}$ is denoted as dashed-brown vertical line.
• Figure 3: Aggregated two-body correlator $C_{ij}$, Eq. \ref{['eq:Cij-def']} (top row), and negativity, Eq.\ref{['eq:negativity_eigs']} (bottom row), for the three benchmark states: six-qubit GHZ state $\ket{\psi_{\mathrm{A}}}$, Eq.\ref{['eq:psi_A']}, product of three Bell pairs $\ket{\psi_{\mathrm{B}}}$, Eq.\ref{['eq:psi_B']}, and ten-qubit product of three entangled clusters $\ket{\psi_{\mathrm{C}}}$, Eq.\ref{['eq:psi_C']}).
• Figure 4: Effective visible nodes (qubits) couplings extracted from the trained weight $W$ of an unrestricted NQS. Each panel shows the absolute value of the normalized coupling matrix $|J^{(X)}_{ij}|$, with $X=A$ (amplitude, top row) or $X=\Phi$ (phase, bottom row), computed from the Gram matrices $J^{(X)} = W^{(X)} {W^{(X)}}^{\!\!T}$ of the trained unconstrained NQS. Columns correspond to the three pure targets : (a,d) six-qubit GHZ $\ket{\psi_{\mathrm{A}}}$, Eq.\ref{['eq:psi_A']}, (b,e) product of three Bell pairs $\ket{\psi_{\mathrm{B}}}$, Eq.\ref{['eq:psi_B']}, and (c,f) ten-qubit product state with GHZ and Dicke clusters $\ket{\psi_{\mathrm{C}}}$, Eq.\ref{['eq:psi_C']}. For $\ket{\psi_{\mathrm{A}}}$ the couplings are comparatively homogeneous, reflecting the genuinely global character of the GHZ entanglement. For $\ket{\psi_{\mathrm{B}}}$ the amplitude couplings display three localized $2\times 2$ blocks, directly exposing the underlying $2|2|2$ structure. For $\ket{\psi_{\mathrm{C}}}$ the dominant amplitude couplings concentrate on the three-qubit Dicke cluster, consistent with the fact that pairwise entanglement is present only in that block (Fig. \ref{['fig:Cij-maps']}f). The rotated bases GHZ states are not simply visible, reflecting complicated structure in the computational basis, natural for $J^{(X)}_{ij}$ characteristic.
• Figure 5: Row-based affinity matrices for the unrestricted NQS. Each panel shows the normalized affinity $A^{(X)}_{ij}$ defined in Eq. \ref{['eq:affinity-def']}, with $X=A$ (amplitude, top row) or $X=\Phi$ (phase, bottom row), computed from the row vectors of the weight matrices $W^{(X)}$ of the trained unconstrained NQS. Columns correspond to the three pure target states $\ket{\psi_{\mathrm{A}}}$, $\ket{\psi_{\mathrm{B}}}$, and $\ket{\psi_{\mathrm{C}}}$ as in Fig. \ref{['fig:J-maps']}. Large values of $A^{(X)}_{ij}$ indicate that qubits $i$ and $j$ couple to the hidden layer in a similar way. For the product of three Bell pairs $\ket{\psi_{\mathrm{B}}}$ the amplitude affinity exhibits three nearly independent $2\times 2$ blocks, while for the ten-qubit state $\ket{\psi_{\mathrm{C}}}$ it reveals three clusters of sizes $3$, $3$, and $4$, consistent with the intended tensor--product structure. The GHZ state $\ket{\psi_{\mathrm{A}}}$ does not exhibit a pronounced block-diagonal pattern, consistent with the absence of a natural small-cluster decomposition.