Computational complexity of residual matrix-function contributions

Characterize the computational complexity of the matrix-function contributions represented by the non-determinant, non-permanent coefficients and the residual component in the orthogonal decomposition of generalized hybrid fermion–boson immanants.

Background

The paper decomposes generalized matrix functions arising from hybrid fermion–boson sampling into determinant, permanent, standard-immanant, and residual components. The determinant and permanent provide familiar complexity benchmarks, and the authors relate complexity to overlaps with the corresponding coefficient vectors. However, the computational complexity associated with the remaining coefficients and the residual component is not established because these generalized functions have not been studied in the literature identified by the authors. This leaves open the task of determining whether these contributions are computationally comparable to determinants, permanents, standard immanants, or potentially harder.