A paradox in bosonic energy computations via semidefinite programming relaxations

TL;DR

This paper investigates a paradox in semidefinite programming relaxations for bosonic energy calculations, revealing that numerical convergence is due to rounding errors, despite theoretical collapse after the first step.

Contribution

It demonstrates the discrepancy between theoretical hierarchy collapse and numerical convergence, attributing the latter to computational rounding errors in semidefinite relaxations.

Findings

Numerical implementations show convergence due to rounding errors.

Hierarchy collapses after the first step theoretically, but not in practice.

Convergence is lost with increased computer precision.

Abstract

We show that the recent hierarchy of semidefinite programming relaxations based on non-commutative polynomial optimization and reduced density matrix variational methods exhibits an interesting paradox when applied to the bosonic case: even though it can be rigorously proven that the hierarchy collapses after the first step, numerical implementations of higher order steps generate a sequence of improving lower bounds that converges to the optimal solution. We analyze this effect and compare it with similar behavior observed in implementations of semidefinite programming relaxations for commutative polynomial minimization. We conclude that the method converges due to the rounding errors occurring during the execution of the numerical program, and show that convergence is lost as soon as computer precision is incremented. We support this conclusion by proving that for any element p of a Weyl algebra which is non-negative in the Schrodinger representation there exists another element p' arbitrarily close to p that admits a sum of squares decomposition.