Optimization On Tensor Network Varieties
summary
Defines tensor network varieties as closures of images of linear-evaluation maps on graph tensors and proves: acyclic graphs have no boundary while graphs with cycles do, so boundary tensors cause ill-conditioned numerical fits. Uses degenerations characterized by an approximation degree A and error degree E, geometric osculating-space arguments to show any degeneration with error E becomes a restriction after Kroneckering by a small auxiliary weight state and increasing bond dimension by at most factor (E+1), exploits E+1 points span degree-E curve and implements modified TNS code that empirically outperforms naive bond-dimension inflation.
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chapters (6)
- 0:00Start
- 18:49Matrix Product States
- 26:04Definition of Restriction Degeneration
- 28:40Quantum Teleportation
- 37:36Weight State
- 40:34Recap
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