  The boundary operator is a linear operator that acts on a collection of
high-dimensional binary points (simplices) and maps them to their boundaries.
This boundary map is one of the key components in numerous applications,
including differential equations, machine learning, computational geometry,
machine vision and control systems. We consider the problem of representing the
full boundary operator on a quantum computer. We first prove that the boundary
operator has a special structure in the form of a complete sum of fermionic
creation and annihilation operators. We then use the fact that these operators
pairwise anticommute to produce an $O(n)$-depth circuit that exactly implements
the boundary operator without any Trotterization or Taylor series approximation
errors. Having fewer errors reduces the number of shots required to obtain
desired accuracies.
