Estimating entanglement entropy via variational quantum circuits with classical neural networks

Entropy plays a crucial role in both physics and information science, encompassing classical and quantum domains. In this paper, we present the quantum neural entropy estimator (QNEE), an approach that combines classical neural network (NN) with variational quantum circuits to estimate the von Neumann and Rényi entropies of a quantum state. QNEE provides accurate estimates of entropy while also yielding the eigenvalues and eigenstates of the input density matrix. Leveraging the capabilities of classical NN, QNEE can classify different phases of quantum systems that accompany the changes of entanglement entropy. Our numerical simulation demonstrates the effectiveness of QNEE by applying it to the 1D XXZ Heisenberg model. In particular, QNEE exhibits high sensitivity in estimating entanglement entropy near the phase transition point. We expect that QNEE will serve as a valuable tool for quantum entropy estimation and phase classification.

Figure 1

Architecture of QNEE. (a) QNEE requires three inputs: density matrix $\widehat{\rho }$, initial parameter ${\mathrm{\Theta }}_Q$ for a quantum circuit, and ${\mathrm{\Theta }}_N$ for a NN. (b) From the given ${\mathrm{\Theta }}_Q$ and density matrix $\widehat{\rho }$, two string data sets are generated: one for training and the other one for testing. (c) We update ${\mathrm{\Theta }}_N$ by minimizing the cost function evaluated with the train data, ${min}_{{\mathrm{\Theta }}_N}{C}^{\mathrm{train}}$. During the learning process, the cost function evaluated with the test data $C^{\mathrm{test}}$ is recorded. Then, the NN outputs the lowest value of $C^{\mathrm{test}}$ from the recorded values, denoted as $C^{\mathrm{NN}}$. The quantum circuit's parameter ${\mathrm{\Theta }}_Q$ is updated by minimizing the NN's output ${min}_{{\mathrm{\Theta }}_Q}{C}^{\mathrm{NN}}$. On successful training, QNEE finds the optimal parameters ${\mathrm{\Theta }}_N^{*}$ and ${\mathrm{\Theta }}_Q^{*}$, and QNEE generates three outputs. (d) The cost function itself becomes von Neumann entropy with optimal parameters. (e) The trained NN can generate the eigenvalues of $\widehat{\rho }$. (f) With a conjugated quantum circuit ${\widehat{V}}^{†}\left({\mathrm{\Theta }}_Q^{*}\right)$, eigenvectors can be generated.

Figure 2

Schematic diagrams of (a) layered hardware efficient ansatzes $\widehat{V}\left({\mathrm{\Theta }}_Q\right)$ and (b) its block gate $\overline{V}$. $\widehat{V}\left({\mathrm{\Theta }}_Q\right)$ consists of $N_l$ layers of $\overline{V}$ block gates. In this research, we choose $\overline{V}$ as the combination of four $R_Y$ gates and one controlled-$Z$ gate. For $n$ qubits, the dimension of ${\mathrm{\Theta }}_Q$ or the total number of parameters is given as $2n{N}_l$.

Figure 3

Plots of estimation results on the reduced density matrices of the XXZ chain. Total number of qubits is eight. The reduced-density matrix is prepared by partially tracing out (a) five qubits and (b) four qubits. The entanglement entropy is estimated with five different randomly initialized quantum circuits. The main figures show the lowest value of cost functions among five trials. Insets show the mean and standard deviations of five trials. $•$ symbols represent the results of QNEE. $\times$ symbols represent the results of VQSE. The exact value of entanglement entropy is denoted by a black line.

Figure 4

(a)–(c) Plots of estimated eigenvalues and (d)–(f) learning curves for the reduced density matrix with three qubits. These results are from three $\lambda$: (a), (d) $\lambda =0.5$; (b), (e) $\lambda =2.0$; and (c), (f) $\lambda =3.0$. In the top plots, $\circ$ ($\times$) symbol denotes the estimation results of QNEE (VQSE). $+$ represents the exact (analytic) result. In the bottom plots, solid line represents cost function $C^{\mathrm{NN}}$ trained with finite samples. The dashed line represents ideally trained $C({\overline{\mathrm{\Theta }}}_N,{\mathrm{\Theta }}_Q)$ without finite-sample effect. Here we utilize the same data that is used for Fig. 3.

Figure 5

Scatter plot of the absolute errors of QNEE ($\circ$ symbol) and VQSE ($\times$ symbol) as a function of entanglement entropy. For this plot, we take data from reduced density matrices of three and four qubits of the XXZ chain.