Isometric tensor network optimization for extensive Hamiltonians is free of barren plateaus

Abstract: We explain why and numerically confirm that there are no barren plateaus in the energy optimization of isometric tensor network states (TNS) for extensive Hamiltonians with finite-range interactions which are, for example, typical in condensed matter physics. Specifically, we consider matrix product states (MPS) with open boundary conditions, tree tensor network states (TTNS), and the multiscale entanglement renormalization ansatz (MERA). MERA are isometric by construction and, for the MPS and TTNS, the tensor network gauge freedom allows us to choose all tensors as partial isometries. The variance of the energy gradient, evaluated by taking the Haar average over the TNS tensors, has a leading system-size independent term and decreases according to a power law in the bond dimension. For a hierarchical TNS (TTNS and MERA) with branching ratio $b$, the variance of the gradient with respect to a tensor in layer $\tau$ scales as $(b\eta)^\tau$, where $\eta$ is the second largest eigenvalue of a Haar-average doubled layer-transition channel and decreases algebraically with increasing bond dimension. The absence of barren plateaus substantiates that isometric TNS are a promising route for an efficient quantum-computation-based investigation of strongly-correlated quantum matter. The observed scaling properties of the gradient amplitudes bear implications for efficient TNS initialization procedures.