Universality of a standard two-qubit gate by catalytic embedding

We study the resources required to achieve universal quantum computing via the gate sets that provide the fundamental instructions from which quantum algorithms are built. While single-gate universal sets are known, they rely on precisely tuned irrational rotations, making them difficult to realize on near-term devices. We find that the controlled-$V$ gate, an elementary two-qubit interaction directly implementable on leading hardware, is universal and capable of simulating standard universal gate sets with minimal overhead. Specifically, we use catalytic embeddings to develop a constant-overhead algorithm that simulates standard universal gate sets, including Clifford$+T$ and Clifford$+$Toffoli. We combine this simulation algorithm with existing synthesis results to yield exact and approximate synthesis algorithms for unitaries with and without number-theoretic restrictions. The results highlight how full quantum computational power, complete with algorithms for synthesis and simulation, can emerge from unexpectedly simple ingredients.