Deep Learning

The bosonic matrix model can be represented with a collection of complex variables \(x_i\) representing the elements of \(N \times N\) Hermitean matrices, for a SU(\(N\)) gauge group. We have studied the matrix model with 2 Hermitean matrices, so we will use 6 complex variables \(x_i\) to represent our quantum states. A vector in this 6-dimensional space is said to be in the coordinate basis of our system.

We represent each state using a quantum neural ansatz, where a neural network architecture is used to map the element of the basis into the corresponding quantum amplitude or wave function. If we denote by \(X\) the vector of \(x_i\) coordinates, we aim to find the function \(\psi_\theta(X)=\langle X|\psi_\theta\rangle\).

For simplicity, we choose to parametrize the wave function norm (separetely from the wave function phase) as \(|\psi_\theta(X)| = \sqrt{p_\theta(X)}\) where \(p_\theta(X)\), the wave function probability distribution, takes the following form:

\[ p_\theta(X) = p(x_1; F^0_\theta) p(x_2; F^1_\theta(x_1)) p(x_3; F^2_\theta(x_1, x_2)) \ldots \]

This is an autoregressive form: the parameters \(F^i_\theta\) of the distribution \(p(x_i; F^{i-1}_\theta(\ldots))\) of one coordinates \(x_i\) only depend on the previous coordinates \(x_{i-1}\), etc...

The parameters \(F^{i}_\theta(\ldots)\) are obtained from a fully-connected feed-forward neural network.

Sofware

The code used for the variational quantum monte carlo with quantum neural state based on autoregressive flows is in this repository.

Results

Under construction