Projective quantum Monte Carlo simulations guided by unrestricted neural network states

We investigate the use of variational wave functions that mimic stochastic recurrent neural networks, specifically, unrestricted Boltzmann machines, as guiding functions in projective quantum Monte Carlo (PQMC) simulations of quantum spin models. As a preliminary step, we investigate the accuracy of such unrestricted neural network states as variational Ansätze for the ground state of the ferromagnetic quantum Ising chain. We find that by optimizing just three variational parameters, independently on the system size, accurate ground-state energies are obtained, comparable to those previously obtained using restricted Boltzmann machines with few variational parameters per spin. Chiefly, we show that if one uses optimized unrestricted neural network states as guiding functions for importance sampling, the efficiency of the PQMC algorithms is greatly enhanced, drastically reducing the most relevant systematic bias, namely, the one due to the finite random-walker population. The scaling of the computational cost with the system size changes from the exponential scaling characteristic of PQMC simulations performed without importance sampling, to a polynomial scaling, apparently even at the ferromagnetic quantum critical point. The important role of the protocol chosen to sample hidden-spin configurations, in particular at the critical point, is analyzed. We discuss the implications of these findings for what concerns the problem of simulating adiabatic quantum optimization using stochastic algorithms on classical computers.

Figure 1

Structure of the unrestricted Boltzmann machine. The lower (yellow) nodes depict visible spins, the upper (magenta) nodes depict the hidden spins. The horizontal segments indicate intralayer visible-visible and hidden-hidden correlations. The vertical (blue) segments represent the interlayer correlations between the corresponding visible and hidden spins. The green lines allude to a possible extension to deep layers architectures.

Figure 2

Relative error $e_{\mathrm{rel}}$ in the variational estimates of the ground-state energy, see Eq. (8), as a function of the transverse field $\mathrm{\Gamma }$, obtained using the simple Boltzmann wave function and for the unrestricted Boltzmann machine (uRBM) Ansatz. The system size is $N=80$. For comparison, we also show the data corresponding to the restricted Boltzmann machine (RBM) obtained using the code from Ref. [15], where $\alpha$ indicates the hidden-spin density. The thin lines are guides to the eyes.

Figure 3

Number of random walkers $N_w$ required to determine, using the PQMC algorithm without importance sampling, the ground-state energy with a relative error $e_{\mathrm{rel}}$, see Eq. (8), as a function of the system size $N$. Different data sets correspond to different transverse field intensities $\mathrm{\Gamma }$ and different relative errors. The lines represent exponential fitting functions.

Figure 4

Number of random walkers $N_w$ required to determine, using the optimized Boltzmann-type wave function to guide importance sampling in the PQMC simulation, the ground-state energy with a relative error $e_{\mathrm{rel}}$, see Eq. (8), as a function of the system size $N$. Different data sets correspond to different transverse field intensities $\mathrm{\Gamma }$. The (red) dotted and (blue) dot-dashed lines represent exponential fitting functions, while the (green) dashed line represents a power-law fit with power $b=0.54\left(5\right)$.

Figure 5

Number of random walkers $N_w$ required to determine, using the optimized uRBM Ansatz to guide importance sampling in the PQMC simulation, the ground-state energy with a relative error $e_{\mathrm{rel}}$, see Eq. (8), as a function of the system size $N$. The number of single-spin Metropolis updates of the hidden spins per CTGFMC hidden-spin update is $k=0.1N$. The (red) dotted line represents and exponential fit, while the (blue) dot-dashed line represents a linear fit.

Figure 6

Number of random walkers $N_w$ required to determine, using the optimized uRBM to guide importance sampling in the PQMC simulation, the ground-state energy with a relative error $e_{\mathrm{rel}}$, see Eq. (8), as a function of the system size $N$. The transverse field intensity is set at the ferromagnetic quantum critical point $\mathrm{\Gamma }=1$. Different data sets correspond to different values of the number of single-spin Metropolis updates $k$. The (red) dotted line represents an exponential fit, while the (black) dot-dashed line represents a power-law fit, with power $b=0.55\left(1\right)$.