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An open problem of quantum information theory has been to determine underwhat conditions universal exchange-only computation is possible for quditsencoded on $d$-state systems for $d>2$. This problem can be posed in terms ofrepresentation theory by recognizing that each quantum mechanical swap,generated by exchange-interaction, can be identified with a transposition in asymmetric group, each $d$-state system can be identified with the fundamentalrepresentation of $SU(d)$, and each encoded qudit can be identified with anirreducible representation of a Lie algebra generated by transpositions.Towards this end we first give a mathematical definition of exchange-onlyuniversality in terms of a map from the special unitary algebra on the productof qudits into a representation of a Lie algebra generated by transpositions.We show that this definition is consistent with quantum computing requirements.We then proceed with the task of characterizing universal families of qudits,that is families of encoded qudits admitting exchange-only universality. Thisendeavor is aided by the fact that the irreducible representationscorresponding to qudits are canonically labeled by partitions. In particular wederive necessary and sufficient conditions for universality on one or two suchqudits, in terms of simple arithmetic conditions on the associated partitions.We also derive necessary and sufficient conditions for universality onarbitrarily many such qudits, in terms of Littlewood--Richardson coefficients.Among other results, we prove that universal families of multiple qudits areupward closed, that universality is guaranteed for sufficiently many qudits,and that any family that is not universal can be made so by simply adding atmost five ancillae. We also obtain results for 2-state systems as a specialcase.

Universality of swap for qudits: a representation theory approach