The improvement problem of lattice gauge field theories is discussed within the Hamiltonian formulation. For a pure gauge theory, we derive an improved quantum Hamiltonian from a lattice Lagrangian free of O(a2) errors in the classical limit. We do this by the transfer matrix method, but we also show that the alternative via Legendre transformation gives identical results. The resulting color-electric energy is an infinite series, which is expected to be rapidly convergent. For the purpose of practical calculations, we construct a simpler improved Hamiltonian, which includes only nearest-neighbor interactions. We also consider tadpole improvement and the structure of Lu