A strict positivity of the ground-state energy is a necessary and sufficient
condition for spontaneous supersymmetry breaking. This ground-state energy may
be directly determined from the expectation value of the Hamiltonian in the
functional integral, defined with an \emph{antiperiodic} temporal boundary
condition for all fermionic variables. We propose to use this fact to observe
the dynamical spontaneous supersymmetry breaking in Euclidean lattice
simulations. If a lattice formulation possesses a manifestly preserved
fermionic symmetry, there exists a natural choice of a Hamiltonian operator
that is consistent with a topological nature of the Witten index. We
numerically confirm the validity of our idea in models of supersymmetric
quantum mechanics. We further examine the possibility of dynamical
supersymmetry breaking in the two-dimensional $\mathcal{N}=(2,2)$ super
Yang-Mills theory with the gauge group SU(2), for which the Witten index is
unknown. Although statistical errors are still large, we do not observe
positive ground-state energy, at least within one standard deviation. This
prompts us to draw a different conclusion from a recent conjectural claim that
supersymmetry is dynamically broken in this system.