Optimal two-qubit circuits for universal fault-tolerant quantum computation

Abstract We study two-qubit circuits over the Clifford+CS gate set, which consists of Clifford gates together with controlled-phase CS = diag(1, 1, i ). The set is universal for quantum computation and its elements can be implemented fault-tolerantly in most error-correcting schemes through magic state distillation. Since non-Clifford are typically more expensive to perform a fault-tolerant manner, it often desirable construct that use few gates. In present paper, we introduce an efficient optimal synthesis algorithm operators. Our inputs operator U outputs circuit , uses least possible number Because deterministic, associates viewed as normal form operator. give explicit description these forms this derive worst-case lower bound $$5{{\rm{log}}}_{2}(\frac{1}{\epsilon })+O(1)$$ 5 log 2 ( 1 ? ) + O on required ? -approximate SU(4). work leverages wide variety mathematical tools may find further applications circuits.