Quantum Algorithms for Representation-Theoretic Multiplicities

Kostka, Littlewood-Richardson, Plethysm and Kronecker coefficients are multiplicities of irreducible representations (irreps) of the symmetric group in restrictions and products of irreps. They play an important role in representation theory and are notoriously hard to compute. We give quantum algorithms that efficiently compute these coefficients whenever the ratio of dimensions of the representations is polynomial. Using that the Kostka numbers admit combinatorial interpretation, we show that there is an efficient classical algorithm for polynomially-bounded Kostka numbers and conjecture existence of a similar algorithm for the Littlewood-Richardson coefficients. We argue why the same classical algorithm does not straightforwardly work for the Plethysm and Kronecker coefficients, give evidence on how our quantum algorithm may avoid some hardness obstructions in their computation, and conjecture that the problem could lead to superpolynomial quantum speedups on some inputs. We finally use Frobenius reciprocity to derive another quantum algorithm that estimates these coefficients using induction and has a different cost-to-input dependence.