Learnability scaling of quantum states: Restricted Boltzmann machines

Generative modeling with machine learning has provided a new perspective on the data-driven task of reconstructing quantum states from a set of qubit measurements. As increasingly large experimental quantum devices are built in laboratories, the question of how these machine learning techniques scale with the number of qubits is becoming crucial. We empirically study the scaling of restricted Boltzmann machines (RBMs) applied to reconstruct ground-state wave functions of the one-dimensional transverse-field Ising model from projective measurement data. We define a learning criterion via a threshold on the relative error in the energy estimator of the machine. With this criterion, we observe that the number of RBM weight parameters required for accurate representation of the ground state in the worst case – near criticality – scales quadratically with the number of qubits. By pruning small parameters of the trained model, we find that the number of weights can be significantly reduced while still retaining an accurate reconstruction. This provides evidence that overparametrization of the RBM is required to facilitate the learning process.

Figure 1

The procedure used to determine the RBM expressiveness required to represent the TFIM wave function at $h/J=1$ with $N=50$ qubits. The number of hidden units $N_h$ is increased until the desired $\epsilon$ is achieved. The inset illustrates the number of hidden units required for convergence to $\epsilon \le 0.002$ for different values of $h/J$ near criticality. The position of the peak is discussed in the text.

Figure 2

Minimum number of hidden units $N_h$ required for $\epsilon \le 0.002$ for various values of $h/J$. Straight lines are fits to the data.

Figure 3

Weight magnitudes, sorted in descending order from left to right, for various transverse field values and $N=60$. Converged RBM models from the parameter study shown in Fig. 2 are used here.

Figure 4

The minimum number of training examples $M$ required for $\epsilon \le 0.002$ for the TFIM at the critical point $h/J=1$ for different ratios of the number of hidden to visible units $\alpha =N_h/N$.

Figure 5

The left and right panels show the values of the weight matrix from an RBM trained on $N=40$ at the quantum critical point to $\epsilon \le 0.002$, summed along its rows and columns, respectively. One can see that the ratios of the summed weights to biases fluctuate around $-2$, as predicted.