Circuit Optimization for Universality Transformation

It is known that a computationally universal gate set $\{H,CCZ\}$ can be transformed to a strictly universal one $\{H,\mathrm{\Lambda }(S)\}$ using one maximally imaginary state $\mathrm{\mid }+i⟩$ and non-imaginary ancillary qubits. We succeed this transformation with a shorter circuit that eliminates non-imaginary ancillary qubits. We further extend this to the continuous gate-set setting, showing that any multi-qubit unitary can be exactly generated by real single-qubit unitary gates, $CCZ$ gates and $\mathrm{\mid }0⟩\mathrm{\mid }+i⟩$.