  Nonadiabatic holonomic quantum computation (NHQC) provides a method to
implement error resilient gates and that has attracted considerable attention
recently. Since it was proposed, three-level {\Lambda} systems have become the
typical building block for NHQC and a number of NHQC schemes have been
developed based on such systems. In this paper, we investigate the realization
of NHQC beyond the standard three-level setting. The central idea of our
proposal is to improve NHQC by enlarging the Hilbert space of the building
block system and letting it have a bipartite graph structure in order to ensure
purely holonomic evolution. Our proposal not only improves conventional
qubit-based NHQC by efficiently reducing its duration, but also provides
implementations of qudit-based NHQC. Therefore, our proposal provides a further
development of NHQC that can contribute significantly to the physical
realization of efficient quantum information processors.
