Minimal non-orthogonal gate decomposition for qubits with limited control

How an arbitrary unitary transformation can be decomposed into minimum number of elementary rotations, subject to particular physical constraints, is a fundamental and practical question. Here, we consider two important scenarios. The first one is when rotation axes are allowed to vary in a range of a plane, which corresponds to the Singlet-Triplet (ST) qubit in quantum-dot systems; the second one is when rotation axes can only along two fixed directions, corresponding to the Exchange-Only (EO) qubits. For both scenarios, we provide the criteria for determining the minimal number of pieces and explicit gate construction procedure for each unitary gate. Our results analytically explain the four-gate decomposition of EO qubits, previously determined numerically by Divincenzo et al. Moreover, our approaches can reduce both gate time and gate fidelity dramatically for ST qubits, compared with Ramon sequence and its variant.