UBC Theses and Dissertations

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UBC Theses and Dissertations

Classification of quantum wire in tensor network states with a local Pauli symmetry Herringer, Paul


Certain symmetry-protected topological (SPT) phases also possess the property that every state in the phase is a universal resource for measurement-based quantum computation (MBQC). These phases are called computational phases of quantum matter. Computational phases have been classified in 1D, and several examples are known in 2D. 2D computational phases are of interest because we can simulate any quantum computation using a 2D phase. Therefore, the classification of 2D phases may lead to a better understanding of the relationship between symmetry and quantum computation. Many known 2D phases have a projected entangled pair state (PEPS) representation where the local tensor symmetries define a quantum cellular automaton (QCA) acting on correlation space. However, not every tensor symmetry defines a QCA, which motivates the definition and study of a larger class of PEPS tensors called stabilizer tensors. Stabilizer tensors have symmetries that act as a tensor product of Pauli operators on correlation space but do not necessarily define a QCA. In this thesis, I present a classification of stabilizer tensors by their channel capacity for quantum wire. In particular, I consider the quantum wire capacity of translationally invariant, cylindrical PEPS constructed from stabilizer tensors and find that it falls into one of 13 classes. Furthermore, I prove that no other types of channel capacity exist for stabilizer tensors, regardless of the size of the cylinder. I also discuss the significance of this result for the classification of 2D computational phases along with possible next steps.

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