Neural-network quantum states at finite temperature

We propose a method to obtain the thermal-equilibrium density matrix of a many-body quantum system using artificial neural networks. The variational function of the many-body density matrix is represented by a convolutional neural network with two input channels. We first prepare an infinite-temperature state, and the temperature is lowered by imaginary-time evolution. We apply this method to the one-dimensional Bose-Hubbard model and compare the results with those obtained by exact diagonalization.

Figure 1

Schematic illustration of the CNN to represent the density matrix $\rho (\mathit{n},{\mathit{n}}^{\prime })=\langle \mathit{n}|\widehat{\rho }|{\mathit{n}}^{\prime }\rangle$. The one-dimensional configurations of bosons $\mathit{n}$ and ${\mathit{n}}^{\prime }$ on $M$ sites are input into the two input channels. Successive $N_L$ convolutional layers are followed by a fully connected layer, which gives output $u^{\left(\mathrm{out}\right)}$. The matrix element of the density matrix $\rho (\mathit{n},{\mathit{n}}^{\prime })$ is given by $e^{u^{\left(\mathrm{out}\right)}}$.

Figure 2

Imaginary-time evolution of the density matrix represented by a CNN for $U=1$, 4, and 10. (a) Expectation value of the Hamiltonian $\langle \widehat{H}\rangle$. The lines of the energies $E_{\beta }^{\mathrm{exact}}$ obtained by exact diagonalization (ED) are also drawn, which however almost overlap with the lines obtained by our method and cannot be seen. The horizontal lines represent the exact energies of the ground states. The inset shows the error in the energy $\delta E=\langle \widehat{H}\rangle -E_{\beta }^{\mathrm{exact}}$. (b) Mean-squared error $\delta \rho$ defined in Eq. (12).

Figure 3

Dependence of the accuracy on various conditions for $U=4$, where (i) the same as in Fig. 2, (ii) the cutoff order in Eq. (5) is reduced to $K=1$, (iii) the number of convolutional layers is reduced to $N_L=2$, and (iv) the asymmetric expansion in Eq. (13) is used with $K=2$. (a) Expectation value of the Hamiltonian $\langle \widehat{H}\rangle$. The inset shows the error in the energy $\delta E=\langle \widehat{H}\rangle -E_{\beta }^{\mathrm{exact}}$. (b) Mean-squared error $\delta \rho$ defined in Eq. (12).