Quantum Algorithms for Representation-Theoretic Multiplicities

• URL: http://arxiv.org/abs/2407.17649v3
• Date: Fri, 27 Sep 2024 21:46:07 GMT
• Title: Quantum Algorithms for Representation-Theoretic Multiplicities
• Authors: Martin Larocca, Vojtech Havlicek,
• Abstract summary: We give quantum algorithms for computing Kostka, Littlewood-Richardson, Plethysm and Kronecker coefficients. We show that there is an efficient classical algorithm for computing the Littlewood-Richardson coefficients. We conjecture that our quantum algorithms lead to superpolynomial speedups.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Kostka, Littlewood-Richardson, Plethysm and Kronecker coefficients are the multiplicities of irreducible representations in decomposition of representations of the symmetric group that play an important role in representation theory and algebraic combinatorics. We give quantum algorithms for computing these coefficients whenever the ratio of dimensions of the representations is polynomial and study the computational complexity of this problem. We show that there is an efficient classical algorithm for computing the Kostka numbers under this restriction and conjecture the existence of an analogous algorithm for the Littlewood-Richardson coefficients. We argue why such classical algorithm does not straightforwardly work for the Plethysm and Kronecker coefficients and conjecture that our quantum algorithms lead to superpolynomial speedups for this problem. We support this conjecture by showing how our quantum algorithm avoids some hardness obstructions in computation of these coefficients. We give another quantum algorithm that estimates the multiplicities using induction and has a different cost-to-input dependence.

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