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Area of Science:

  • Representation Theory
  • Algebraic Combinatorics
  • Quantum Computing

Background:

  • Kostka, Littlewood-Richardson, Plethysm, and Kronecker coefficients are crucial in representation theory, geometric complexity, and algebraic combinatorics.
  • These coefficients represent multiplicities in the decomposition of symmetric group representations.

Purpose of the Study:

  • To develop quantum algorithms for computing these important coefficients.
  • To investigate the existence of efficient classical algorithms for these computations.

Main Methods:

  • Development of quantum algorithms for coefficient computation under specific dimension ratio conditions.
  • Analysis of classical algorithm feasibility for different coefficient types.

Main Results:

  • Quantum algorithms are presented for computing coefficients when representation dimension ratios are polynomial.
  • An efficient classical algorithm for Kostka numbers is demonstrated.
  • Conjectures are made regarding classical algorithms for Littlewood-Richardson coefficients and quantum speedups for Plethysm and Kronecker coefficients.

Conclusions:

  • The study provides new quantum algorithmic approaches for key representation-theoretic computations.
  • While classical algorithms exist for some coefficients (e.g., Kostka), quantum algorithms offer potential advantages for others.
  • Recent work has disproven conjectures about classical vs. quantum complexity for Kronecker coefficients, highlighting a significant polynomial gap.