  We provide a new hierarchy of semidefinite programming relaxations, called
NCTSSOS, to solve large-scale sparse noncommutative polynomial optimization
problems. This hierarchy features the exploitation of term sparsity hidden in
the input data for eigenvalue and trace optimization problems. NCTSSOS
complements the recent work that exploits correlative sparsity for
noncommutative optimization problems by Klep, Magron and Povh in
arXiv:1909.00569, and is the noncommutative analogue of the TSSOS framework by
Wang, Magron and Lasserre in arXiv:1912.08899. We also propose an extension
exploiting simultaneously correlative and term sparsity, as done previously in
the commutative case arXiv:2005.02828. Under certain conditions, we prove that
the optimums of the NCTSSOS hierarchy converge to the optimum of the
corresponding dense SDP relaxation. We illustrate the efficiency and
scalability of NCTSSOS by solving eigenvalue/trace optimization problems from
the literature as well as randomly generated examples involving up to several
thousands of variables.
