  Tensor networks such as matrix product states (MPS) and projected entangled
pair states (PEPS) are commonly used to approximate quantum systems. These
networks are optimized in methods such as DMRG or evolved by local operators.
We provide bounds on the conditioning of tensor network representations to
sitewise perturbations. These bounds characterize the extent to which local
approximation error in the tensor sites of a tensor network can be amplified to
error in the tensor it represents. In known tensor network methods, canonical
forms of tensor network are used to minimize such error amplification. However,
canonical forms are difficult to obtain for many tensor networks of interest.
We quantify the extent to which error can be amplified in general tensor
networks, yielding estimates of the benefit of the use of canonical forms. For
the MPS and PEPS tensor networks, we provide simple forms on the worst-case
error amplification. Beyond theoretical error bounds, we experimentally study
the dependence of the error on the size of the network for perturbed random MPS
tensor networks.
