Minimal nonorthogonal gate decomposition for qubits with limited control

In quantum control theory, a question of fundamental and practical interest is how an arbitrary unitary transformation can be decomposed into a minimum number of elementary rotations for implementation, subject to various physical constraints. Examples include the singlet-triplet (ST) and exchange-only (EO) qubits in quantum-dot systems, and gate construction in the Solovay-Kitaev algorithm. For two important scenarios, we present complete solutions to the problems of optimal decomposition of single-qubit unitary gates with non-orthogonal rotations. For each unitary gate, the criteria for determining the minimal number of pieces is given, the explicit gate construction procedure, as well as a computer code for practical uses. Our results include an analytic explanation to the four-gate decomposition of EO qubits, previously determined numerically by Divincenzo et al. [Nature (London) 408, 339 (2000)]. Furthermore, compared with the approaches of Ramon sequence and its variant [Phys. Rev. Lett. 118, 216802 (2017)], our method can reduce about 50% of gate time for ST qubits. Finally, our approach can be extended to solve the problem of optimal control of topological qubits, where gate construction is achieved through the braiding operations.