Neural Network Quantum State

This is a Python implementation of neural network quantum state introduced in the paper "Solving the quantum many-body problem with artificial neural networks".

Required library: Numpy, Numba, MPI4Py, Matplotlib

Hamiltonian

$$ \hat{H}=\sum_{i=1}^{N-1} \hat{S}^z_i \hat{S}^z_{i+1} - \frac{1}{2}\left[ \hat{S}^+i \hat{S}^-{i+1} + \hat{S}^-i \hat{S}^+{i+1} \right]$$

Trial wave function

$$|\psi_t(\alpha)\rangle=\sum_x\langle x|\psi_t(\alpha)\rangle|x\rangle=\sum_x J_x(\alpha)|x\rangle$$

Variational energy

$$\langle E\rangle_t=\frac{\langle\psi_t|H|\psi_t\rangle}{\langle \psi_t|\psi_t\rangle}=\frac{\sum_x\langle\psi_t|x\rangle\langle x||H|\psi_t\rangle}{\sum_x\langle\psi_t|x\rangle\langle x||\psi_t\rangle} $$

Jastrow wave function with single parameter

$$ C(\alpha, \vec{s})=e^{-\alpha\sum_{i < j}\frac{\hat{S}^z_i \hat{S}^z_j}{|j-i|}}$$

On a 4 sites chain, the variational energy can be calculated analytically. Here is the comparison of the exact results with variational Monte Carlo results.

Optimize the sigle parameter Jastrow wave function

Gradient descent:

$$ \alpha = \alpha-\lambda\frac{\partial E}{\partial\alpha}$$

$$ \frac{\partial E}{\partial\alpha}=2\frac{\langle\frac{\partial\psi}{\partial\alpha}|H|\psi\rangle}{\langle\psi|\psi\rangle}-2E\frac{\langle\frac{\partial\psi}{\partial\alpha}|\psi\rangle}{\langle\psi|\psi\rangle}$$

Define

$$ O = \frac{1}{|\psi\rangle} |\frac{\partial\psi}{\partial\alpha}\rangle$$

Then

$$ \frac{\partial E}{\partial\alpha}=2\frac{\langle\psi|OH|\psi\rangle}{\langle\psi|\psi\rangle}-2\langle E\rangle\frac{\langle\psi|O|\psi\rangle}{\langle\psi|\psi\rangle}=2\langle EO\rangle-2\langle E\rangle\langle O\rangle$$

More complicated wave function

Jastrow wave function

$$J(\alpha_{ij}, \vec{s})=e^{-\sum_{ij}\alpha_{ij}{\hat{S}^z_i \hat{S}^z_j}} $$

Restricted Boltzman machine wave function

$$ \psi(\vec{s}, a, b ,W) = e^{ \sum_{i=1} a_i s^z_i } \prod_{i=1}^M 2\cosh{(\sum_{j=1}^N W_{ij}s^z_j + b_i)} $$

All the parameters in the wave function are complex numbers.

Optimize the Restricted Boltzman Machine wave function

In the modified natural gradient descent method, the Fubini-study metric, which is the complex-valued form of Fisher information, is used to measure the "distance" between wave functions |ψ〉 and |φ〉. NGD can greatly improve the convengence speed as shown below.

$$ \gamma(\psi,\phi)=\arccos{ \sqrt{ \frac{\langle\psi|\phi\rangle\langle\phi|\psi\rangle}{\langle\psi|\psi\rangle\langle\phi|\phi\rangle }} }$$