Abstract
We consider the problem of deciding if a set of quantum one-qudit gates \(\mathcal {S}=\{g_1,\ldots ,g_n\}\subset G\) is universal, i.e. if \({<}\mathcal {S}{>}\) is dense in G, where G is either the special unitary or the special orthogonal group. To every gate g in \(\mathcal {S}\) we assign the orthogonal matrix \(\mathrm {Ad}_g\) that is image of g under the adjoint representation \(\mathrm {Ad}:G\rightarrow SO(\mathfrak {g})\) and \(\mathfrak {g}\) is the Lie algebra of G. The necessary condition for the universality of \(\mathcal {S}\) is that the only matrices that commute with all \(\mathrm {Ad}_{g_i}\)’s are proportional to the identity. If in addition there is an element in \({<}\mathcal {S}{>}\) whose Hilbert–Schmidt distance from the centre of G belongs to \(]0,\frac{1}{\sqrt{2}}[\), then \(\mathcal {S}\) is universal. Using these we provide a simple algorithm that allows deciding the universality of any set of d-dimensional gates in a finite number of steps and formulate a general classification theorem.
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Communicated by David Pérez-García.
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Sawicki, A., Karnas, K. Universality of Single-Qudit Gates. Ann. Henri Poincaré 18, 3515–3552 (2017). https://doi.org/10.1007/s00023-017-0604-z
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DOI: https://doi.org/10.1007/s00023-017-0604-z