We have developed a semiclassical theory of short periodic orbits to obtain all quantum information of a bounded chaotic Hamiltonian system. If T₁ is the period of the shortest periodic orbit, T₂ the period of the next one and so on, the number N_p,o of periodic orbits required in the calculation is such that T₁+⋯+T_Np,o ≃ T_H, with T_H the Heisenberg time. As a result N_po ≃ hT_H/ln(hT_H), where h is the topological entropy. For methods related to the trace formula N_po ≃ exp(hT_H) / (hT_H).