Ground state preparation is classically intractable for general Hamiltonians. On quantum devices, shallow parametrized circuits can be effectively trained to obtain short-range entangled states under the paradigm of variational quantum eigensolver, while deep circuits are generally untrainable due to the barren plateau phenomenon. In this Letter, we give a general lower bound on the variance of circuit gradients for arbitrary quantum circuits composed of local 2-designs. Based on our unified framework, we prove the absence of barren plateaus in training finite local-depth circuits (FLDC) for the ground states of local Hamiltonians. FLDCs are allowed to be deep in the conventional circuit depth to generate long-range entangled ground states, such as topologically ordered states, but their local depths are finite, i.e., there is only a finite number of gates acting on individual qubits. This characteristic sets FLDC apart from shallow circuits: FLDC in general cannot be classically simulated to estimate local observables efficiently by existing tensor network methods in two and higher dimensions. We validate our analytical results with extensive numerical simulations and demonstrate the effectiveness of variational training using the generalized toric code model.