Mitigating Green's function Monte Carlo signal-to-noise problems using contour deformations

The Green's function Monte Carlo (GFMC) method provides accurate solutions to the nuclear many-body problem and predicts properties of light nuclei starting from realistic two- and three-body interactions. Controlling the GFMC fermion sign problem is crucial, as the signal-to-noise ratio decreases exponentially with imaginary time, requiring significant computing resources. Inspired by similar scenarios in lattice quantum field theory and spin systems, in this work, we employ integration contour deformations to improve the GFMC signal-to-noise ratio. Machine learning techniques are used to select optimal contours with minimal variance from parametrized families of deformations. As a proof of principle, we consider the deuteron binding energies and Euclidean density response functions. We only observe mild signal-to-noise improvement for the binding energy case. On the other hand, we achieve an order of magnitude reduction of the variance for Euclidean density response functions, paving the way for computing electron- and neutrino-nucleus cross sections of larger nuclei.

Figure 1

Comparison of the numerically optimized shift parameter ${\kappa }_0^{\varphi }$ (“trained”) as a function of the imaginary time $\tau$ versus the manually selected linear ramp contour ${\kappa }_0^{\varphi }=(\tau -N\delta \tau /2)\ell$ (“ramp”). Results are shown for $N\delta \tau =0.05{\mathrm{MeV}}^{-1}$ (a) and $N\delta \tau =0.10{\mathrm{MeV}}^{-1}$ (b).

Figure 2

Comparison of the deuteron binding energy evaluated without contour deformation (“original”) versus evaluations using the numerically optimized spherical contour deformation described in the main text (“trained”) as well as a manually selected linear ramp contour (“ramp”), shown for $N\delta \tau =0.05{\mathrm{MeV}}^{-1}$ (a) and $N\delta \tau =0.10{\mathrm{MeV}}^{-1}$ (b). Expectation values are evaluated using $N=10000$ walkers in all cases.

Figure 3

Comparison of the standard error of the deuteron binding energy measurements, given as a ratio between the original measurements and the measurements using contour deformation. The results are compared for the numerically optimized spherical contour deformation (“trained”) and the manually selected linear ramp contour (“ramp”), shown for $N\delta \tau =0.05{\mathrm{MeV}}^{-1}$ (a) and $N\delta \tau =0.10{\mathrm{MeV}}^{-1}$ (b). The ratios are estimated by computing the ratio of uncertainties determined by a bootstrap resampling procedure applied to an ensemble of $N=10000$ walkers, with the uncertainty on the ratio estimated by resampling the ensemble itself in an outer bootstrap resampling step.

Figure 4

Loss curves from training the numerically optimized contours for each component of ${\rho }_{ij}$ shown in Fig. 5.

Figure 5

Comparison of the optimal $\lambda$ appearing in the ramp deform and the optimal $\lambda (\mathit{q},\tau )$ appearing in the trained deformation shown in Fig. 6 for $\mathit{q}=(0,0,600)$ MeV.

Figure 6

Comparison of GFMC results for Euclidean response functions for the two values of $\mathit{q}$ indicated and $\tau \le 0.03{\mathrm{MeV}}^{-1}$ using $N=5000$ walkers without contour deformation (“original,” blue), with the one-parameter ramp contour deformation shown in Eq. (58) with $\lambda =\pm 0.64{\text{MeV}}^{-1}$ (“ramp,” orange), and with a multiparameter contour deformation that has been numerically optimized as described in the main text (“trained,” purple).

Figure 7

Comparison of GFMC results for Euclidean response functions without contour deformation (blue) and with a one-parameter ramp contour in which the coordinate integration contours are shifted in the imaginary direction by Eq. (58) (orange) using $N=5000$ walkers. Shift magnitudes of $\lambda =\pm 0.64{\text{MeV}}^{-1}$ are used for all $\mathit{q}$ except for $\mathit{q}=(600,600,600)$ MeV, where $\lambda =\pm 0.56{\text{MeV}}^{-1}$ was found to give more optimal variance reduction. Higher-statistics results using $N=50000$ walkers without contour deformation are also shown for reference (gray).

Figure 8

Comparison of GFMC results for Euclidean response functions with and without contour deformation analogous to Fig. 7 but with $\tau \le 0.03{\text{MeV}}^{-1}$.

Figure 9

Comparison of GFMC results for Euclidean response functions with and without contour deformation analogous to Fig. 7 but with $\tau \le 0.05{\text{MeV}}^{-1}$.

Figure 10

Comparison of the average phase factors of Euclidean response function calculations without contour deformation using $N=5000$ walkers (blue) and $N=50000$ walkers (gray) with the average phase factors for contour-deformed calculations using the same one-parameter ramp contours used in Figs. 7, 8, 9 using $N=5000$ walkers (orange). Small average phase factors indicate the presence of a severe sign problem leading to a signal-to-noise problem. Results using $\tau \le 0.01{\text{MeV}}^{-1}$ and $\tau \le 0.03{\text{MeV}}^{-1}$ are overlayed to contrast the behavior of deformed results using contours optimized for different $\tau$.