Testing and tuning symplectic integrators for Hybrid Monte Carlo algorithm in lattice QCD

TL;DR

This study evaluates new minimum-norm symplectic integrators for Hybrid Monte Carlo in lattice QCD, focusing on a 2nd-order MN (2MN) scheme that markedly reduces energy errors relative to the standard 2LF leapfrog, enabling larger time steps and yielding about 50% higher efficiency after cost considerations. The authors also examine 4th-order MN integrators, finding limited practical benefits for zero-temperature lattices but potential gains at finite temperature under certain conditions. They develop a tuning framework for 2MN by modeling the RMS Hamiltonian error with two independent error channels, showing the optimal parameter λ depends on simulation parameters and that the position-first variant often performs best. Overall, the work demonstrates that 2MN offers a robust, more efficient alternative for HMC in lattice QCD and may be advantageous in related quantum Monte Carlo contexts, especially when combined with other acceleration techniques.

Abstract

We examine a new 2nd order integrator recently found by Omelyan et al. The integration error of the new integrator measured in the root mean square of the energy difference, $\braΔH^2\ket^{1/2}$, is about 10 times smaller than that of the standard 2nd order leapfrog (2LF) integrator. As a result, the step size of the new integrator can be made about three times larger. Taking into account a factor 2 increase in cost, the new integrator is about 50% more efficient than the 2LF integrator. Integrating over positions first, then momenta, is slightly more advantageous than the reverse. Further parameter tuning is possible. We find that the optimal parameter for the new integrator is slightly different from the value obtained by Omelyan et al., and depends on the simulation parameters. This integrator could also be advantageous for the Trotter-Suzuki decomposition in Quantum Monte Carlo.

Figures (6)

• Figure 1: $\langle\Delta H^2\rangle^{1/2}$ as a function of $\Delta t$. Simulations are performed at $\beta=5.00$ and $\kappa = 0.160$ on $4^4$ lattices. The line proportional to $\Delta t^2$ is drawn to guide the eye.
• Figure 2: $C_{2LF}/C_{2MN}$ as a function of $\kappa$. Simulations at $\beta=5.00(5.60)$ are performed on $4^4(8^4)$ lattices.
• Figure 3: $\langle\Delta H^2\rangle^{1/2}$ as a function of $\Delta t$. Simulations are performed on $8^4$ lattices. 4RC stands for the 4th order integrator obtained by eq.(\ref{['eq:rec']}). The lines proportional to $\Delta t^n$ are drawn to guide the eye.
• Figure 4: $\langle\Delta H^2\rangle^{1/2}$ as a function of $\lambda$. The right figure is a zoom of the left. Simulations are performed at $\beta=5.00$ and $\kappa=0.160$ on $4^4$ lattices with $\Delta t=0.05$. The lines are determined from simulations of the position version integrator at $\lambda_1$ and $\lambda_2$. The position version has a small advantage over the velocity version, since it gives a slightly reduced minimum RMS error (right).
• Figure 5: $|f|\Delta t^2$, $|g|\Delta t^2$ and $|f|/|g|$ as a function of $1/\kappa$. Simulations are performed at $\beta=5.00$ on $4^4$ lattices with $\Delta t=0.05$. The dashed line indicates $\kappa_c =0.187$Ukawa.