Faster pulse sequences for performing arbitrary rotations in singlet-triplet qubits

We present new composite pulses that perform universal single-qubit operations in singlet-triplet spin qubits faster than existing methods. We introduce two types of composite pulses: one that generalizes the standard Hadamard-$x$-Hadamard sequence used to perform rotations about the $z$ axis, and another that generalizes a sequence proposed by Guy Ramon (G. Ramon, Phys. Rev. B {\bf 84}, 155329 (2011)). We determine how much time it takes to perform each set of pulses and find that our ``generalized Hadamard-$x$-Hadamard'' sequence can be made faster than any of the other sequences. We also present composite pulses for performing $x$ rotations and show that a generalization of the Hadamard-$z$-Hadamard sequence is faster than other existing sequences, as well as faster and more precise than performing $x$ rotations with single pulses. We present versions of these gates that also dynamically correct for noise-induced errors along the lines of SUPCODE (X.\ Wang {\it et. al.}, Phys. Rev. A {\bf 89}, 022310 (2014)).