Quantum estimates for classical polynomial optimization

Finding global bounds (minima and maxima) for multivariate homogeneous polynomials is a computationally difficult problem due to the complex topography of the objective functions, which often leads to local extrema or failure to converge in standard iterative algorithms. This paper explores a novel, global strategy inspired by quantum-mechanical variational methods to estimate these bounds.

The author proposes replacing a classical polynomial with a quantum operator acting on a finite-dimensional Fock space. By choosing an appropriate operator, the original polynomial is recovered in the classical limit. The problem of finding bounds on the polynomial is then reduced to diagonalizing the resulting quantum operator—represented as a large matrix—and identifying its smallest and largest eigenvalues. This approach leverages the particle-number-conserving nature of specific Hamiltonians to simplify the diagonalization process into manageable sectors.

The author demonstrates that this quantum-inspired approach effectively sidesteps the convergence issues inherent in traditional nonlinear iterative methods. The technique is applied to standard test cases from tensor eigenvalue theory and used to derive new, nontrivial inequalities for resonant Hamiltonian systems, such as those appearing in the study of nonlinear wave equations and Strichartz norms. Specifically, the paper provides a new upper bound for a class of resonant Hamiltonians introduced by Biasi, showing that the quantum eigenvalue pattern can be used to conjecture sharp analytic inequalities that are otherwise difficult to derive.

This method provides a powerful heuristic tool for researchers in mathematical physics and optimization. By mapping difficult polynomial optimization problems onto quantum systems, it allows for the use of well-established numerical diagonalization techniques to guess sharp inequalities. These conjectures can then serve as a starting point for rigorous mathematical proofs, bridging the gap between numerical exploration and formal analysis in complex dynamical systems.