  A notion of up and down Grover walks on simplicial complexes are proposed and
their properties are investigated. These are abstract Szegedy walks, which is a
special kind of unitary operators on a Hilbert space. The operators introduced
in the present paper are usual Grover walks on graphs defined by using
combinatorial structures of simplicial complexes. But the shift operators are
modified so that it can contain information of orientations of each simplex in
the simplicial complex. It is well-known that the spectral structures of this
kind of unitary operators are completely determined by its discriminant
operators. It has strong relationship with combinatorial Laplacian on
simplicial complexes and geometry, even topology, of simplicial complexes. In
particular, theorems on a relation between spectrum of up and down
discriminants and orientability, on a relation between symmetry of spectrum of
discriminants and combinatorial structure of simplicial complex are given. Some
examples, both of finite and infinite simplicial complexes, are also given.
Finally, some aspects of finding probability and stationary measures are
discussed.
