Designing single-qubit gates for a silicon three-qubit device with always-on exchange coupling

Lie algebra subspaces have already proven to be useful in designing robust pulses for coupled two-qubit systems, such as electron spin qubits in silicon double quantum dots. We apply similar techniques, after a Schreiffer-Wolff transformation and rotating wave approximation, to decompose a coupled three-qubit system (formed in a triple quantum dot) into independent subalgebras. We are then able to choose analytical composite pulses in the applied magnetic field to perform specific logical gates. We can choose the pulse parameters to construct single-qubit and two-qubit gates, including local X rotations of any qubit and nonlocal CZ gates, by making use of the Euler decomposition in su(2) subalgebras. The resulting pulses are simple and short, though they are not inherently robust to noise. To construct robust single-qubit gates, we can make use of existing techniques such as SUPCODE, with modifications to be more efficient for our specific system. The construction of robust two-qubit gates is the topic of the following talk.