Polynomial time classical versus quantum algorithms for representation theoretic multiplicities

Abstract Littlewood-Richardson, Kronecker and plethysm coefficients are fundamental multiplicities of interest in Representation Theory and Algebraic Combinatorics. Determining a combinatorial interpretation for the Kronecker and plethysm coefficients is a major open problem and prompts the consideration of their computational complexity. Recently, it was shown that they behave relatively well with respect to quantum computation, and for some large families there are polynomial-time quantum algorithms (Larocca and Havlicek in Quantum algorithms for representation-theoretic multiplicities, 2024. arXiv:2407.17649 ) (also Bravyi et al. in PRX Quantum 5(1):010329, 2024). In this paper, we show that for many of those cases the Kronecker and plethysm coefficients can also be computed in polynomial time via classical algorithms, thereby refuting some of the conjectures in Larocca and Havlicek (2024). This vastly limits the cases in which the desired superpolynomial quantum speedup could be achieved.