  Ground state preparation is classically intractable for general Hamiltonians.
On quantum devices, shallow parameterized 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.
