Quantum neural networks to simulate many-body quantum systems

We conduct experimental simulations of many-body quantum systems using a hybrid classical-quantum algorithm. In our setup, the wave function of the transverse field quantum Ising model is represented by a restricted Boltzmann machine. This neural network is then trained using variational Monte Carlo assisted by a $D$-wave quantum sampler to find the ground-state energy. Our results clearly demonstrate that already the first generation of quantum computers can be harnessed to tackle nontrivial problems concerning physics of many-body quantum systems.

Figure 1

D-wave processor specification. (a) An example of the chimera architecture comprising a $2\times 2$ grid consisting of clusters (units cells) of eight qubits each. (b) A typical annealing schedule where the annealing time reads $\tau$.

Figure 2

Restricted Boltzmann machine embedded on a $D$ wave. (a) Graphical representation of the neural network (2) with eight visible and eight hidden neurons. (b) The same network embedded on the chimera graph $\mathcal{G}=(\mathcal{E},\mathcal{V})$ of size $2\times 2\times 8$. Strong ferromagnetic couplings (thick lines) “glue” qubits in different unit cells to represent single neurons.

Figure 3

The ground-state energy per spin $E$ for the transverse field quantum Ising model (3) in 1D obtained for different system sizes. The dashed lines indicate the exact values. Sampling was carried out using Monte Carlo with $\gamma =0.2$ in panels (a) and tensor networks algorithm with $\gamma =0.05$ in (b). They serve as a reference point for the results in Fig. 4. In panels (c), we show the influence of incorrect inverse temperature estimation $x=\beta /{\beta }_x$ on the results.

Figure 4

The ground-state energy per spin $E$ for the transverse field quantum Ising models (3) in 1D and 2D. The dashed lines indicate the exact values. Sampling was executed using $D$-wave annealers: $2000Q$ in (a) and (b) and DW2X in (c). $\delta E$ shows the relative energy error reached. We set $h=0.5$ for the 1D system in panels (a) and (c), and $h=0.5$, 1, and $3.044\simeq h_c$ for the 2D system in (b), where $\tau =20$. The annealing time $\tau$ is measured in microseconds. Besides, $\alpha =1,\gamma =0.2$, and $N_{\mathrm{s}}={10}^4$.

Figure 5

The inverse temperature ${\beta }_x$ estimated during the learning stage. See Fig. 4 for comparison and the text for a discussion.