Exact universality from any entangling gate without inverses

This note proves that arbitrary local gates together with any entangling bipartite gate V are universal. Previously this was known only when access to both V and V † was given, or when approximate universality was demanded. A common situation in quantum computing is that we can apply only a limited set S ⊂ Ud of unitary gates to some d-dimensional system. The first question we want to ask in this situation is whether gates from S can (approximately) generate any gate in PUd = Ud/U1 (the set of all d × d unitary matrices up to an overall phase). When this is possible, we say that S is (approximately) universal. See [1, 3, 4, 7] for original work on this subject, or Sect 4.5 of [9] or Chapter 8 of [8] for reviews. Formally, S is universal (for PUd) if, for all W ∈ PUd, there exists U1, . . . , Uk ∈ S such that W = UkUk−1 · · ·U2U1, whereas U is approximately universal (for PUd) if, for all W ∈ PUd and all ǫ > 0, there exists U1, . . . , Uk ∈ S such that d(W,UkUk−1 · · ·U2U1) < ǫ. (1) Here d(·, ·) can be any metric, but for concreteness we will take it to be the PUd analogue of operator distance: d(U, V ) := 1 − inf |ψ〉 6=0 〈ψ|U V |ψ〉 〈ψ|ψ〉 . Similar definitions could also be made for Ud, other groups, or even semigroups. A natural way to understand universality is in terms of the group generated by S, which we denote 〈S〉, and define to be smallest subgroup of PUd that contains S. An alternate and more constructive definition is that 〈S〉 consists of all products of a finite number of elements of S or their inverses. When S contains its own inverses (i.e. S = S−1 := {x : x−1 ∈ S}) then 〈S〉 provides a concise way to understand universality: S is universal iff 〈S〉 = PUd and S is approximately universal iff 〈S〉 is dense in PUd. But what if S does not contain its own inverses? The equivalence between approximate universality and 〈S〉 being dense in PUd still holds. One direction remains trivial: if S is approximately universal then 〈S〉 is dense in PUd. The easiest way to prove the converse is with simultaneous Diophantine approximation, which implies that for any U ∈ PUd and for any ǫ > 0, there exists n ≥ 0 such that d(U, U−1) ≤ ǫ. The