Absence of barren plateaus in finite local-depth circuits with long-range entanglement

Abstract

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 for the ground states of local Hamiltonians. These circuits are allowed to be deep in the conventional definition of circuit depth so that they can generate long-range entanglement, but their local depths are finite, i.e., there is only a finite number of non-commuting gates acting on individual qubits. This fact suggests that long-range entangled ground states, such as topologically ordered states, are in general possible to be prepared efficiently on quantum devices via variational methods. We validate our analytical results with extensive numerical simulations and demonstrate the effectiveness of variational training using the generalized toric code model.Comment: 28 pages, 7 figure