How a Clebsch-Gordan transform helps to solve the Heisenberg hidden subgroup problem

It has recently been shown that quantum computers can efficiently solve the Heisenberg hidden subgroup problem, a problem whose classical query complexity is exponential. This quantum algorithm was discovered within the framework of using pretty-good measurements for obtaining optimal measurements in the hidden subgroup problem. Here we show how to solve the Heisenberg hidden subgroup problem using arguments based instead on the symmetry of certain hidden subgroup states. The symmetry we consider leads naturally to a unitary transform known as the Clebsch-Gordan transform over the Heisenberg group. This gives a new representation theoretic explanation for the pretty-good measurement derived algorithm for efficiently solving the Heisenberg hidden subgroup problem and provides evidence that Clebsch-Gordan transforms over finite groups are a vital new primitive in quantum algorithm design.