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arXiv · 1712.09798

Trading inverses for an irrep in the Solovay-Kitaev theorem

Adam Bouland·Maris Ozols
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Abstract

The Solovay-Kitaev theorem states that universal quantum gate sets can be exchanged with low overhead. More specifically, any gate on a fixed number of qudits can be simulated with error $ε$ using merely $\mathrm{polylog}(1/ε)$ gates from any finite universal quantum gate set $\mathcal{G}$. One drawback to the theorem is that it requires the gate set $\mathcal{G}$ to be closed under inversion. Here we show that this restriction can be traded for the assumption that $\mathcal{G}$ contains an irreducible representation of any finite group $G$. This extends recent work of Sardharwalla et al. [arXiv:1602.07963], and applies also to gates from the special linear group. Our work can be seen as partial progress towards the long-standing open problem of proving an inverse-free Solovay-Kitaev theorem [arXiv:quant-ph/0505030, arXiv:0908.0512].

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Adam Bouland, Maris Ozols. 2018-05-28. Trading inverses for an irrep in the Solovay-Kitaev theorem. https://doi.org/10.4230/lipics.tqc.2018.6

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