Data-enhanced variational Monte Carlo simulations for Rydberg atom arrays

Rydberg atom arrays are programmable quantum simulators capable of preparing interacting qubit systems in a variety of quantum states. Due to long experimental preparation times, obtaining projective measurement data can be relatively slow for large arrays, which poses a challenge for state reconstruction methods such as tomography. Today, novel ground-state wave-function Ansätze like recurrent neural networks (RNNs) can be efficiently trained not only from projective measurement data, but also through Hamiltonian-guided variational Monte Carlo (VMC). In this paper, we demonstrate how pretraining modern RNNs on even small amounts of data significantly reduces the convergence time for a subsequent variational optimization of the wave function. This suggests that essentially any amount of measurements obtained from a state prepared in an experimental quantum simulator could provide significant values for neural-network-based VMC strategies.

Figure 1

(a) Square lattice of $6\times 6$ atoms in the striated phase, where atoms in the Rydberg state (blue) are separated by stripes of atoms in the ground state (white) due to the Rydberg blockade radius $R_{\mathrm{b}}$. Projective measurements of individual atoms in the occupation number basis provide information about the quantum state. (b) A recurrent neural network (RNN), with green boxes representing a cell with $N_{\mathrm{h}}$ hidden units and tunable parameters $\mathcal{W}$. An input sequence $\mathit{\sigma }$ is iteratively provided to the network, generating an output depending on the current input ${\sigma }_i$ and the hidden unit state ${\mathit{h}}_i$. A single qubit state is used as input at iteration $i$ and the network output is the probability distribution underlying the state of qubit $i+1$, from which the input ${\sigma }_{i+1}$ is sampled [23].

Figure 2

(a) Energy density difference of data-driven trained RNN states for different system sizes $N=L\times L$ as a function of the number of hidden neurons $N_{\mathrm{h}}$ per network cell. Results are averaged over iterations 1350–1400 (950–1000 for $N=16\times 16$) and error bars denote standard deviations. The RNN is trained on ${10}^5$ data points with learning rate $\eta ={10}^{-4}$. (b) Energy density difference of an RNN trained with the Hamiltonian-driven procedure using $\eta ={10}^{-3}$. Results are plotted vs the number of RNN samples $N_{\mathrm{s}}$ for different system sizes, with $N_{\mathrm{h}}=2L$. Data are averaged over iterations 9500–$10000$ with standard deviations as error bars. The QMC energy densities are $L=4$: $H_{\mathrm{QMC}}/16=-0.4534\left(1\right)$; $L=8$: $H_{\mathrm{QMC}}/64=-0.4052\left(2\right)$; $L=12$: $H_{\mathrm{QMC}}/144=-0.3885\left(2\right)$; $L=16$: $H_{\mathrm{QMC}}/256=-0.3805\left(2\right)$.

Figure 3

Quantum state reconstruction of an $N=16\times 16$ atom array. (a) Energy density $H_{\mathrm{RNN}}/N$ as a function of optimization iterations using the data-driven training on 1000 data points. Different shapes and colors show different $N_{\mathrm{h}}$ compared to the target energy density $H_{\mathrm{QMC}}/N$. (b) Energy density evolution during data-driven, Hamiltonian-driven, and hybrid training with different $t_{\mathrm{trans}}$ (darker green shows larger $t_{\mathrm{trans}}$) and $N_{\mathrm{h}}=32$. The inset emphasizes the convergence to the target solution after $\ge 3500$ iterations within a margin of 0.015 for all $t_{\mathrm{trans}}>0$. (c) Convergence time $t_{\mathrm{conv}}$ (see the main text for the definition) as a function of transition point $t_{\mathrm{trans}}$ in the hybrid training of an RNN with $N_{\mathrm{h}}=32$.