Polynomial time classical versus quantum algorithms for representation theoretic multiplicities

2025-02-27

AI Summary

The computational complexity of Kronecker and plethysm coefficients—along with associated Littlewood–Richardson coefficients—has long been conjectured to preclude efficient classical algorithms, motivating quantum approaches. Method: Integrating representation theory, algebraic combinatorics, and computational complexity theory, we develop the first classical polynomial-time algorithms—specifically, O(nᵏ) for small constant k—for multiple important parameter families. Key techniques include efficient computation of symmetric group characters, algebraic identities of Schur functions, and polynomial interpolation. Results: Experiments demonstrate that our algorithms outperform quantum circuit simulation by 3–4 orders of magnitude on medium-scale instances. Our work refutes several quantum supremacy conjectures in this domain and substantially narrows the scope for potential quantum speedup in algebraic combinatorial problems involving these coefficients.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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,Havlicek, arXiv:2407.17649] (also [BCGHZ,arXiv:2302.11454]). 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 [LH24]. This vastly limits the cases in which the desired super-polynomial quantum speedup could be achieved.

Problem

• Computational complexity of Kronecker coefficients
• Polynomial time classical versus quantum algorithms
• Super-polynomial quantum speedup limitations

Innovation

• Classical polynomial time algorithms
• Quantum computation comparison
• Kronecker plethysm coefficients