Gauge invariants of eigenspace and eigenvalue anholonomies: Examples in   hierarchical quantum circuits
  A set of gauge invariants are identified for the gauge theory of quantum anholonomies, which comprise both the Berry phase and an exotic anholonomy in eigenspaces. We examine these invariants for hierarchical families of quantum circuits whose qubit size can be arbitrarily large. It is also found that a hierarchical family of quantum circuits generally involves an NP-complete problem. 
Atushi Tanaka, Taksu Cheon and Sang Wook Kim
quant-ph