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Polynomial time classical versus quantum algorithms for representation theoretic multiplicities

Greta Panova

TL;DR

The paper investigates the boundary between polynomial-time classical and quantum algorithms for fundamental representation-theoretic multiplicities, showing that for broad families of Kronecker and plethysm coefficients, classical algorithms run in polynomial time. By analyzing dimension-growth regimes and leveraging polyhedral counting (GT/hive polytopes) and Barvinok’s algorithm, the authors refute several LH24 conjectures about inevitable quantum speedups and significantly constrain the parameter ranges where quantum advantages could be super-polynomial. They provide concrete polynomial-time classical algorithms for Kronecker and plethysm coefficients in many regimes (including when f^ν is polynomial in n) and extend classical tractability results for Kostka and LR coefficients via polytope methods. The findings recalibrate expectations for quantum speedups in this domain and highlight a deep link between asymptotic representation theory and computational complexity.

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.

Polynomial time classical versus quantum algorithms for representation theoretic multiplicities

TL;DR

The paper investigates the boundary between polynomial-time classical and quantum algorithms for fundamental representation-theoretic multiplicities, showing that for broad families of Kronecker and plethysm coefficients, classical algorithms run in polynomial time. By analyzing dimension-growth regimes and leveraging polyhedral counting (GT/hive polytopes) and Barvinok’s algorithm, the authors refute several LH24 conjectures about inevitable quantum speedups and significantly constrain the parameter ranges where quantum advantages could be super-polynomial. They provide concrete polynomial-time classical algorithms for Kronecker and plethysm coefficients in many regimes (including when f^ν is polynomial in n) and extend classical tractability results for Kostka and LR coefficients via polytope methods. The findings recalibrate expectations for quantum speedups in this domain and highlight a deep link between asymptotic representation theory and computational complexity.

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.

Paper Structure

This paper contains 12 sections, 16 theorems, 53 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background and definitions
  3. Young tableaux
  4. Symmetric functions
  5. Multiplicities
  6. Computational Complexity
  7. Useful inequalities and notation
  8. Dimension growth
  9. Computing Kostka and Littlewood--Richardson coefficients
  10. Kronecker coefficients
  11. Plethysm coefficients
  12. Additional remarks

Key Result

Theorem 1.1

Let $\lambda,\mu,\nu \vdash n$ and suppose that $f^\nu \leq n^k$ for some $k$. Then $g(\lambda,\mu,\nu)$ can be computed in time $O(D(k) n^{4k^2+1}\log(n))$, where $D(k) = (4k)^{8(8k^4+k^2)}$. In particular, if $\lambda^{(n)}, \mu^{(n)},\nu^{(n)}$ are families of partitions of $n$, such that $f^{\nu

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3: regev
  • Theorem 3.4: GM
  • Proposition 3.5
  • ...and 27 more