Noncommutative Polynomial Optimization under Dimension Constraints
summary
Presents a convergent SDP hierarchy for noncommutative polynomial optimization with explicit dimension constraints, mapping operator instances to linear moment vectors and enforcing positivity through moment/localizing matrices. Implements a practical basis by random unitary/projector sampling (orthonormalizing the induced moment matrices) to span the feasible subspace, uses matrix polynomial identities to prove convergence under Archimedean-type conditions, and reports tight second-level bounds for several d=2 certification tasks (and k=3, d=3 numerics on a desktop). Offers implementation tips (block-diagonal support, Hebinner-type vectorization) and shows the method works on quantum dimension witnesses and communication tasks.
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chapters (6)
- 0:00Start
- 2:44Computation of ground state energies and ionization potentials
- 4:02Characterization of quantum correlations
- 15:23Incorporating dimensional constraints
- 15:39Example: computation of quantum dimension witnesses
- 29:53Applications and practical performance
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