Neural-network quantum state tomography

We revisit the application of neural networks to quantum state tomography. We confirm that the positivity constraint can be successfully implemented with trained networks that convert outputs from standard feed-forward neural networks to valid descriptions of quantum states. Any standard neural-network architecture can be adapted with our method. Our results open possibilities to use state-of-the-art deep-learning methods for quantum state reconstruction under various types of noise.

Figure 1

Sketch of a NN for estimating the Bloch parameters or the elements of the Cholesky matrix from the sampled statistics $\mathbf{f}$. The filled blue ovals represent neurons in deep layers, representing about $2\times {10}^5$ trainable parameters. The output layer has a hyperbolic tangent as an activation function since it has the same bounds as the Bloch parameters and the elements of the Cholesky matrix. In the hidden layers, the rectified linear unit is used as an activation function. The structure of the NN is the same, independently of the dimension or specific parametric representation of quantum states.

Figure 2

Average Hilbert-Schmidt distance for different estimating strategies on the number of trials with which we sampled the true probability distribution $\mathbf{p}$ in dimensions 3 (upper left panel), 5 (upper right panel), 7 (lower left panel), and 9 (upper right panel). The insets indicate the corresponding estimators. The average errors for both NNs in the undersampled regime are of the same order for SDP and MLE in all dimensions. When using highly sampled statistics, number of trials $>{10}^5$, MLE starts to outperfonm both NN approaches. Confidence intervals of 80% are depicted in respective colors.

Figure 3

Average time per state estimation for different tomographical methods, as indicated in the inset. The NN approach is about three orders faster than SDP and about four orders faster than MLE.

Figure 4

Quality of estimators based on linear inversion and NN. In all the panels, red diamonds, blue squares, green triangles, and yellow circles represent the results for estimators in dimensions 3, 5, 7, and 9, respectively. The upper plots show the dependence of the most negative eigenvalue on the number of trials for those dimensions and for the LI estimator (left) and the Bloch NN estimator (right). The bottom panels depict the percentage of positive semidefinite states amongst all the reconstructed states with LI (left) and Bloch NN (right). The results are obtained from datasets containing ${10}^4$ statistics in each dimension.