2,105 research outputs found
A paradox in bosonic energy computations via semidefinite programming relaxations
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.Comment: 22 pages, 4 figure
Positivity results for the Hecke algebras of non-crystallographic finite Coxeter groups
This paper is a report on a computer check of some positivity properties of
the Hecke algebra in type H4, including the non-negativity of the coefficients
of the structure constants in the Kazhdan-Lusztig basis. This answers a
long-standing question of Lusztig's. The same algorithm, carried out by hand,
also allows us to deal with the case of dihedral Coxeter groups.Comment: septembre 2005; 13 pages; cet article s'accompagne d'un logicie
Quantum Query Algorithms are Completely Bounded Forms
We prove a characterization of -query quantum algorithms in terms of the
unit ball of a space of degree- polynomials. Based on this, we obtain a
refined notion of approximate polynomial degree that equals the quantum query
complexity, answering a question of Aaronson et al. (CCC'16). Our proof is
based on a fundamental result of Christensen and Sinclair (J. Funct. Anal.,
1987) that generalizes the well-known Stinespring representation for quantum
channels to multilinear forms. Using our characterization, we show that many
polynomials of degree four are far from those coming from two-query quantum
algorithms. We also give a simple and short proof of one of the results of
Aaronson et al. showing an equivalence between one-query quantum algorithms and
bounded quadratic polynomials.Comment: 24 pages, 3 figures. v2: 27 pages, minor changes in response to
referee comment
- …