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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$$5{{\rm{log}}}_{2}(\frac{1}{\epsilon })+O(1)$$$5{\log }_2\left(\frac{1}{ϵ}\right)+O\left(1\right)$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.

We consider the use of quantum-limited mechanical force sensors to detect ultralight (sub-meV) dark matter (DM) candidates which are weakly coupled to the standard model. We show that mechanical sensors with masses around or below the milligram scale, operating around the standard quantum limit, would enable novel searches for DM with natural frequencies around the kHz scale. This would complement existing strategies based on torsion balances, atom interferometers, and atomic clock systems.