2020/01/05 by Yifan Zhang, Zhang, Yifan, Edgar Solomonik +1 · 5 citations
Computer Science · Mathematics · Physics and Astronomy · #Applied mathematics #Cartesian tensor #Computer science #Exact solutions in general relativity #FOS: Mathematics #Machine learning #Mathematical analysis #Mathematics #Numerical Analysis (math.NA) #Pure mathematics #Quantum Computing Algorithms and Architecture #Quantum many-body systems #Stability (learning theory) #Tensor (intrinsic definition) #Tensor decomposition and applications #Tensor density #Tensor field #Tensor product #cs.NA #math.NA
paper · pdf · doi:10.48550/arxiv.2001.01191
published in arXiv (Cornell University) (Cornell University) · 24 pages, 7 figures, comments welcome!
arxiv created 2020/01/05 · openalex publication_date 2020/01/05 · arxiv updated 2020/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.