Neural-network quantum state tomography in a two-qubit experiment

We study the performance of efficient quantum state tomography methods based on neural-network quantum states using measured data from a two-photon experiment. Machine-learning-inspired variational methods provide a promising route towards scalable state characterization for quantum simulators. While the power of these methods has been demonstrated on synthetic data, applications to real experimental data remain scarce. We benchmark and compare several such approaches by applying them to measured data from an experiment producing two-qubit entangled states. We find that in the presence of experimental imperfections and noise, confining the variational manifold to physical states, i.e., to positive semidefinite density matrices, greatly improves the quality of the reconstructed states but renders the learning procedure more demanding. Including additional, possibly unjustified, constraints, such as assuming pure states, facilitates learning, but also biases the estimator.

Figure 1

Neural-network quantum state tomography scheme: (a) Entangled photonic Bell pairs from an SPDC process are measured by coincidence count rates. Arbitrary single-qubit projectors can be selected by adjusting the angles ${\theta }_A,{\phi }_A,{\theta }_B,{\phi }_B$ in $\frac{\lambda }{4}$-plates and polarizers $P$. (b) This is repeated for all elements of the POVM ${\left\{M_{\mathbf{a}}\right\}}_{\mathbf{a}}$ yielding a probability distribution $P\left(\mathbf{a}\right)$ describing the state. (c) This forms the training data for different neural-network quantum state architectures based on restricted Boltzmann machines (figure adapted from Ref. [24]). (d) After training, the network-encoded states are used to reconstruct the density matrix or to predict observables.

Figure 2

Comparison of the predicted Bell parameter from the different NQS tomography schemes to direct measurements and maximum likelihood estimation. (a) Training with measurements in the Pauli product bases applicable to all considered NQS tomography schemes (b) Training with measurements of the tetrahedral POVM applicable to the POVM ansatz only. Gray regions indicate results forbidden by local realism, red regions denote results forbidden for physical quantum states. Error regions for linear reconstruction and MLE are the standard deviations after repeating the evaluation with data resampled from the measured distribution.

Figure 3

Bell parameter estimations when applying the NQS tomography schemes to synthetic data in the Pauli bases sampled from the maximum likelihood density matrix from Fig. 2.

Figure 4

Average KL divergence in the Pauli bases between measurements and reconstructed states for all employed tomography schemes. The complexity of the methods increases left to right, expressing the scaling of computational resources consumed (see text for details).