From algebra and symmetries to quantum computing

How can better algebraic and Lie-algebraic methods help to more efficiently analyze, design, control, and program large quantum computers? Symmetries will be the key! High-dimensional quantum systems and their dynamics in emerging quantum technologies are a key application of tensors and tensor-product structures. We have developed symmetry methods [J. Math. Phys. 52(11):113510, 2011; J. Math. Phys. 56(8):081702, 2015; Phys. Rev. A 92(4):042309, 2015] for quantum control theory to answer simulability questions for quantum computing devices. In our approach, Lie algebras are characterized using so-called quadratic symmetries related to the tensor power of a representation. Symmetries are computed (and defined) as the linear space of all matrices commuting with a set of Lie-algebra generators using efficient sparse linear algebra in Magma. But we wonder if more efficient approaches might be possible following ideas of Wilson and Maglione. Capabilities of quantum computers are related to identifying the generated Lie algebra (as in the work of de Graaf), preferably without constructing it explicitly. Algebraic methods such as the Meataxe algorithm can help to identify Lie algebras from symmetries. We close by outlining recent developments which include the use of the Weisfeiler-Leman algorithm to relate graph properties to symmetries as well as a Lie-algebra classification related to so-called variational quantum algorithms.