Learning Quantum State Representations

We reconstruct a quantum state from measurements with shallow neural networks. We apply both a supervised approach with feedforward networks and an unsupervised approach with autoencoders (shown below).

An Encoder $E$ maps the measurement probabilities $\mathbf{p}$ to a lower dimensional representation $\ell$ . If the decoder $D$ successfully reconstructs the probabilities $\tilde{\mathbf{p}} \approx \mathbf{p}$ , $\ell$ holds all necessary information of the measured state and is thus a valid state representation. State reconstruction for a single qubit system with three degrees of freedom is shown.

Dataset Generation

Single and two qubit systems are considered. One datapoint ($\rho$, $\mathbf{p}$) consists of a mixed quantum state $\rho$ and its measurement probabilities $\mathbf{p}$ for a set of projective valued measurements (PVMs) called $\hat{P_i}$. Here, $p_i = \mathrm{Tr}(\rho \hat{P_i})$ is the probability of measuring state $\rho$ along the projector $\hat{P_i}$ . All states in a dataset are measured with the same set of PVMs. I sample mixed quantum states and a sets of PVMs from a uniform distribution.

Supervised State Reconstruction

In the supervised approach a feedforward network reconstructs a given state representation $\rho$ from measurement data $\mathbf{p}$ . This model can reconstruct unseen $\rho$ measured with the same set of PVMs used in the training dataset. A system of $N$ qubits has $4N-1$ degrees of freedom. I see that measurements with $4N-1$ random PVMs are sufficient for state reconstruction, as expected for linear independent measurements.

Unsupervised State Reconstruction

An encoder maps the measurement data $\mathbf{p}$ to a latent layer $\mathbf{l}$ with $\mathrm{dim} \mathbf{l} \le \mathrm{dim} \mathbf{p}$ . From the latent representation $\mathbf{l}$ a decoder reconstructs the measurement data $\mathbf{p}$ . As expected, the reconstruction error for significantly after for $\mathrm{dim} \mathbf{l} \le 4N -1$. We further tune single parameters of the quantum state rho and observe mostly linear behavior.

Bachelor's degree in physics at the University of Innsbruck
Supervised by Prof. Dr. Hans J. Briegel and Hendrik Poulsen Nautrup, PhD