UBC Undergraduate Research

Robustness of an AKLT State on a Honeycomb Lattice. Mar, Philip Allen

Abstract

Cluster states used in measurement based quantum computation do not exist as nondegenerate ground states of two-body Hamiltonians, making them difficult to realize physically [8]. Using Affleck-Kennedy-Lieb-Tasaki (AKLT) states on a honeycomb lattice, a universal resource can be obtained for performing MBQC, by transforming these states into cluster states via two transformations. The first transformation turns the AKLT states into a graph state by using a three-element positive operator value measures and analyzing the effects of these operators in yielded encoded qubits on groups of vertices (domains). The graph state is then transformed to a cluster state as described in [8]. The transformation to a cluster state requires that the graph state contain a connected path from the left side of the original lattice to the right side. Since POVMs are measurements, they may sometimes, in a physical implementation of MBQC, contain errors. Instead of a three- element POVM, we might have a nine-element POVM, with various effects on the groups of vertices they are applied to. This can cause entire groups of vertices to be rendered useless for quantum computation, in particular, they may prevent the existence of a connected path, so that the graph state cannot be transformed to a cluster state. We must make sure the AKLT state is robust, that is, even with certain errors in the POVM, it still can be transformed to a cluster state. We ran simulations to con rm that for certain errors in the POVM (dependent on a parameter pdelete), there always exists a connected path. The main result in this study was the formulation of an algorithm that helped drastically reduce the time to run these simulations. Since the AKLT states are large entangled systems, it is time consuming to produce `typical lattices, but the introduction of this algorithm should help in running further simulations for other POVM outcomes in the future.

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