Tunneling in projective quantum Monte Carlo simulations with guiding wave functions

Quantum tunneling is a valuable resource exploited by quantum annealers to solve complex optimization problems. Tunneling events also occur during projective quantum Monte Carlo (PQMC) simulations, and in a class of problems characterized by a double-well energy landscape their rate was found to scale linearly with the first energy gap, i.e., even more favorably than in physical quantum annealers, where the rate scales with the gap squared. Here we investigate how a guiding wave function—which is essential to make many-body PQMC simulations computationally feasible—affects the tunneling rate. The chosen test beds are a continuous-space double-well problem, the ferromagnetic quantum Ising chain, and the recently introduced shamrock model. As guiding wave function, we consider an approximate Boltzmann-type ansatz, the numerically exact ground state of the double-well model, and a neural-network wave function based on a Boltzmann machine. Remarkably, for each ansatz we find the same asymptotic linear scaling of the tunneling rate that was previously found in the PQMC simulations performed without a guiding wave function. We also provide a semiclassical theory for the double-well with exact guiding wave function that explains the observed linear scaling. These findings suggest that PQMC simulations guided by an accurate ansatz represent a valuable benchmark for physical quantum annealers and a potentially competitive quantum-inspired optimization technique.

Figure 1

Profile of the quartic double-well potential Eq. (2) (dotted blue line), shifted for better comparison with the wave functions of the ground state ${\mathrm{\Psi }}_0\left(x\right)$ and of the first excited state ${\mathrm{\Psi }}_1\left(x\right)$ (orange and green solid lines, respectively).

Figure 2

DMC tunneling time $\xi$ for the quartic double-well potential (2) as a function of the inverse energy gap ${\mathrm{\Delta }}^{-1}$. Three different DMC protocols are shown: The simple DMC algorithm without a GWF (red circles), the DMC algorithm guided by a Boltzmann ansatz (green empty squares), and the one guided by the numerically exact representation of the ground state ${\mathrm{\Psi }}_0\left(x\right)$ (blue empty diamonds). The dashed line represents the scaling $\xi \propto {\mathrm{\Delta }}^{-1}$. Here and in all plots, if not visible, the error bars are smaller than the symbol size.

Figure 3

Tunneling time $\xi$ in PQMC simulations performed with the Boltzmann GWF (open symbols) as a function of the number of spins $N$ in the ferromagnetic Ising chain. Different data sets correspond to different transverse field intensities $\mathrm{\Gamma }$, with the coupling parameter $J=1$. The dashed lines represent exponential fitting functions valid in the large-$N$ regime. The closed symbols represent the inverse gap values ${\mathrm{\Delta }}^{-1}$ computed with the exact formula obtained from the free fermion representation of the quantum Ising chain and rescaled by an appropriate prefactor $\alpha =O\left(1\right)$.

Figure 4

Tunneling times $\xi$ of the PQMC algorithm implemented without GWF (blue pentagons), with the uRBM GWF (red triangles), and with the Boltzmann GWF (green diamonds), as a function of the inverse energy gap ${\mathrm{\Delta }}^{-1}$, for the quantum Ising chain at $\mathrm{\Gamma }=0.6$. The dashed lines represent the fitting function $\xi \left(\mathrm{\Delta }\right)=\alpha {\mathrm{\Delta }}^{-b}$, valid in the large ${\mathrm{\Delta }}^{-1}$ regime. In all three cases, the fitted exponent is $b\simeq 1$ (see Table 2).

Figure 5

The shamrock, a model of $N$ frustrated spins in a transverse field. It is made of $K=(N-1)/2$ leaves, each having three spins. The solid dark-green lines represent ferromagnetic interactions (with interaction strength $J$) between the central spin and all the other $N-1$ spins. The dashed light-green lines indicate the antiferromagnetic interactions (with interaction strength $J-\epsilon$) between the outer spins of the same leaf. The overall effect is to create $2^K$ tunneling paths between the degenerate classical ground states.

Figure 6

Tunneling time $\xi$ in the shamrock model as a functions of the system size $N$. The PQMC results obtained with the Boltzmann GWF (green diamonds) and without GWF (blue pentagons) are compared with the scaling of the incoherent quantum tunneling time $1/{\mathrm{\Delta }}^2$ (black triangles), and with the scaling of the finite-temperature PIMC tunneling time $\xi =2^K/{\mathrm{\Delta }}^2$ [29], where $K$ is the number of leaves in the shamrock. The values of the gap $\mathrm{\Delta }$ are obtained from exact diagonalization. The model parameters are $\mathrm{\Gamma }=0.5, J=6$, and $\epsilon =0.2$.