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We show how the integrators used for the molecular dynamics step of theHybrid Monte Carlo algorithm can be further improved. These integrators notonly approximately conserve some Hamiltonian $H$ but conserve exactly a nearbyshadow Hamiltonian $\tilde{H}$. This property allows for a new tuning method ofthe molecular dynamics integrator and also allows for a new class ofintegrators (force-gradient integrators) which is expected to reducesignificantly the computational cost of future large-scale gauge field ensemblegeneration.

Improving dynamical lattice QCD simulations through integrator tuning using Poisson brackets and a force-gradient integrator