Figure 1

(a), (b) Closed timelike curves. Grandfather paradox (a) and unproved theorem paradox (b) tensor networks from [7], with value 0 (left) and $(1/\sqrt{2})(|00\rangle +|11\rangle )$ (right). The problem of determining whether a tensor network can be constructed from a set of generators such that it contains a grandfather-paradox-type contradiction (is the zero tensor) is undecidable. The filled dot denotes a copy tensor; in the qubit case, the copy tensor is cnot $\circ (\langle 0|\otimes \mathrm{id})$. (c) Quantum singular-value decomposition to implement an $n\times n$ integer matrix up to global rescaling. The wire ending in the measurement $\Psi$ has $n$ states and the measurement is $|\Psi \rangle ={\left({\sum }_i{\sigma }_i^2\right)}^{-1}{\sum }_i{\sigma }_i|i\rangle$ where ${\sigma }_i$ are the singular values of $M$.Reuse & Permissions