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

@article{Glaudell2020OptimalTC,
title={Optimal two-qubit circuits for universal fault-tolerant quantum computation},
author={Andrew N. Glaudell and Neil J. Ross and Jacob M. Taylor},
journal={npj Quantum Information},
year={2020},
volume={7},
pages={1-11}
}
• Published 16 January 2020
• Computer Science
• npj Quantum Information
We study two-qubit circuits over the Clifford+CS gate set, which consists of the Clifford gates together with the controlled-phase gate CS = diag(1, 1, 1,  i ). The Clifford+CS gate 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 gates are typically more expensive to perform in a fault-tolerant manner, it is often desirable to construct circuits that use few CS…
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Efficient Clifford+T approximation of single-qubit operators
• P. Selinger
• Computer Science, Mathematics
Quantum Inf. Comput.
• 2015
An efficient randomized algorithm for approximating an arbitrary element of SU(2) by a product of Clifford+T operators, up to any given error threshold e > 0.05, which is within an additive constant of optimal for certain z-rotations.