A t-design for quantum states is a finite set of quantum states with the property of simulating the Haar-measure on quantum states w.r.t. any test that uses at most t copies of a state. We give efficient constructions for approximate quantum t-designs for arbitrary t.

We then show that an approximate 4-design provides a derandomization of the state-distinction problem considered by Sen (Complexity'05), which is relevant to solving certain instances of the hidden subgroup problem.