A new paper, a new paper! If you love the theory of the addition of angular momentum, and don’t we all just love the theory of the addition of angular momentum, then you will really love the new paper we (Isaac Chuang and Aram Harrow) just put on the arXiv. Unfortunately my spell check changed the title to Clench-Gordon and I didn’t notice. So I expect a lot of nasy emails complaining about the title. Doh. Well that’s what the replace button is for, I guess. Here is the paper:
Efficient Quantum Circuits for Schur and Clebsch-Gordon Transforms
The Schur basis on n d-dimensional quantum systems is a generalization of the total angular momentum basis that is useful for exploiting symmetry under permutations or collective unitary rotations. We present efficient (size poly(n,d,log(1/epsilon)) for accuracy epsilon) quantum circuits for the Schur transform, which is the change of basis between the computational and the Schur bases. These circuits are based on efficient circuits for the Clebsch-Gordon transformation. We also present an efficient circuit for a limited version of the Schur transform in which one needs only to project onto different Schur subspaces. This second circuit is based on a generalization of phase estimation to any nonabelian finite group for which there exists a fast quantum Fourier transform.