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

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 gates. In the present paper, we introduce an efficient and optimal synthesis algorithm for two-qubit Clifford+CS operators. Our algorithm inputs a Clifford+CS operatorUand outputs a Clifford+CS circuit forU, which uses the least possible number of CS gates. Because the algorithm is deterministic, the circuit it associates to a Clifford+CS operator can be viewed as a normal form for that operator. We give an explicit description of these normal forms and use this description to derive a worst-case lower bound of$$5{{\rm{log}}}_{2}(\frac{1}{\epsilon })+O(1)$$5log2(1ϵ)+O(1)on the number of CS gates required toϵ-approximate elements of SU(4). Our work leverages a wide variety of mathematical tools that may find further applications in the study of fault-tolerant quantum circuits.

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Publication Date:
Journal Name:
npj Quantum Information
Nature Publishing Group
Sponsoring Org:
National Science Foundation
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