Machine Learning approach to reconstruct Density Matrices from Quantum Marginals

TL;DR

This work tackles the Quantum Marginal Problem (QMP) by proposing a scalable machine learning pipeline that combines the Marginal Imposition Operator (MIO) with a convolutional denoising autoencoder (CDAE). The two-channel density-matrix representation (real and imaginary parts) enables CNN-based learning to produce globally valid states that match given marginals, and transfer learning allows extension from 3- to 8-qubit systems. Empirical results show high fidelity to marginals, robust success rates, and significant speedups relative to semidefinite programming solvers, with the CDAE–MIO hybrid achieving exact marginals in many cases. The approach also demonstrates potential for warm-starting SDP and offers avenues for interpretability and anomaly detection in the quantum marginal landscape.

Abstract

In this work, we propose a machine learning-based approach to address a specific aspect of the Quantum Marginal Problem: reconstructing a global density matrix compatible with a given set of quantum marginals. Our method integrates a quantum marginal imposition technique with convolutional denoising autoencoders. The loss function is carefully designed to enforce essential physical constraints, including Hermiticity, positivity, and normalization. Through extensive numerical simulations, we demonstrate the effectiveness of our approach, achieving high success rates and accuracy. Furthermore, we show that, in many cases, our model offers a faster alternative to state-of-the-art semidefinite programming solvers without compromising solution quality. These results highlight the potential of machine learning techniques for solving complex problems in quantum mechanics.

Figures (8)

• Figure 1: Illustration of the proposed CDAE architecture. The boxes represent tensors with dimensions specified as $(\textit{number of channels}) \times \textit{height} \times \textit{width}$, indicated below each box. The operation applied to the tensor on the left is written above the arrow between two tensors, with the resulting tensor shown on the right. Since the Tanh activation function does not modify tensor dimensions, we simplify the notation by combining convolutional operations ($\mathit{Conv2D}$ and $\mathit{TConv2D}$) with the $\mathit{Tanh}$ activation function into single operations, denoted as $\mathit{Tanh}\circ\mathit{Conv2D}(\ldots)$ and $\mathit{Tanh}\circ\mathit{TConv2D}(\ldots)$. All boxes and operations to the left (right) of the Latent Representation correspond to the encoder (decoder). The leftmost box represents the input $\mathbf{\tilde{x}}$ of the CDAE, while the rightmost box represents its output $\mathbf{z}$.
• Figure 2: Mean and standard deviation (colored cloud) of the fidelity, obtained over $10^4$ samples, versus rank for mode1 (purple squares), model2 (green dots) and random guessing (red triangles) for the cases (a) $N3k1$ and (b) $N3k2$.
• Figure 3: $F_{\mathit{mean}}\pm SD$, over $10^4$ samples, as a function of rank for model1 (purple dashed line) and for random guessing (solid red line) for the cases $N4k2$, $N4k3$, $N5k3$, $N5k4$, $N6k4$, $N6k5$, $N7k5$, $N7k6$, $N8k6$ and $N8k7$.
• Figure 4: Success rates of model1 for all the cases presented in Figure \ref{['fig:model-fidelities']}. model1 achieved a $100\%$ success rate for all the cases with $N>5$, with the lines overlapping at the $100\%$ mark.
• Figure 5: Comparison of the proportion of negative eigenvalues relative to the total number of eigenvalues. Each bar represents the average over a sample of $10^4$ tests per rank, with error bars indicating the standard deviation. Light blue bars correspond to the first pass through the MIO, while pink bars correspond to the model1+MIO configuration.