Breaking the cubic barrier in the Solovay-Kitaev algorithm

06/22/2023

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by   Greg Kuperberg, et al.

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We improve the Solovay-Kitaev theorem and algorithm for a general finite, inverse-closed generating set acting on a qudit. Prior versions of the algorithm can efficiently find a word of length O((log 1/ϵ)^3+δ) to approximate an arbitrary target gate to within ϵ. Using two new ideas, each of which reduces the exponent separately, our new bound on the world length is O((log 1/ϵ)^1.44042…+δ). Our result holds more generally for any finite set that densely generates any connected, semisimple real Lie group, with an extra length term in the non-compact case to reach group elements far away from the identity.