Circuit Optimization for Universality Transformation

Abstract

It is known that a computationally universal gate set $\{H,CCZ\}$ can be transformed to a strictly universal one $\{H,\hat{I}›(S)\}$ using one maximally imaginary state $|+i⟩$ 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 $|0⟩|+i⟩$.

2 pages, 1 figure. Corrected statements and proofs of Lemma 1 and Theorem 2 regarding non-imaginary ancillary qubits, and updated related discussions