All you need is controlled-V: universality of a standard two-qubit gate by catalytic embedding

Abstract: We present an encoding that renders the controlled-V gate (also known as controlled-$\sqrt{X}$) computationally universal in isolation. Specifically, we show that this gate can simulate the universal Clifford+Toffoli gate set with at most two clean auxiliary qubits and a constant overhead in gate count, and that an additional auxiliary qubit suffices to simulate Clifford+T. Our result settles an open question on the expressiveness of De Vos' gate set based on Negators, and shows that the two-qubit gate $SU<sup>{(\tau)}$ due to Sleator and Weinfurter is capable of universal quantum computation even for rational choices of $\tau$.