  Traditionally, lattice QCD computations of generalized parton distributions
(GPDs) have been carried out in a symmetric frame, where the transferred
momentum is symmetrically distributed between the incoming and outgoing
hadrons. However, such frames are inconvenient since they require a separate
calculation for each value of the momentum transfer, increasing significantly
the computational cost. In this work, by focusing on the quasi-distribution
approach, we lay the foundation for faster and more effective lattice QCD
calculations of GPDs exploiting asymmetric frames, with freedom in the
transferred momentum distribution. An important ingredient of our approach is
the Lorentz covariant parameterization of the matrix elements in terms of
Lorentz-invariant amplitudes, which allows one to relate matrix elements in
different frames. We also use this amplitude approach to propose a new
definition of quasi-GPDs that is frame-independent and, more importantly, may
lead to smaller power corrections in the matching relations to the light-cone
GPDs. We demonstrate the efficacy of the formalism through numerical
calculations using one ensemble of $N_f$=2+1+1 twisted mass fermions with a
clover improvement. The value of the light-quark masses lead to a pion mass of
about 260 MeV. Concentrating on the proton, and limiting ourselves to a
vanishing longitudinal momentum transfer to the target, we extract the
invariant amplitudes from matrix element calculations in both the symmetric and
asymmetric frame, and obtain results for the twist-2 light-cone GPDs for
unpolarized quarks, that is, $H$ and $E$.
