Universal Effectiveness of High-Depth Circuits in Variational Eigenproblems

We explore the effectiveness of high-depth, noiseless, parameteric quantum circuits by challenging their capability to simulate the ground states of quantum many-body Hamiltonians. Even a generic layered circuit Ansatz can approximate the ground state with high precision, as long as the circuit depth exceeds a certain threshold level that exponentially scales with the number of qubits, despite the abundance of the barren plateaus. This success is due to the fact that the energy landscape in the high-depth regime has a suitable structure for the gradient-based optimization, i.e., the presence of local extrema -- near any random initial points -- reaching the ground level energy. We check if these advantages are preserved across different Hamiltonians, by working out two variational eigensolver problems for the transverse field Ising model as well as for the Sachdev-Ye-Kitaev model. We expect that the contributing factors to the universal success of the high-depth circuits may also serve as the evaluation guidelines for more realistic circuit designs under hybrid quantum-classical algorithms.