  Structure in quantum entanglement entropy is often leveraged to focus on a
small corner of the exponentially large Hilbert space and efficiently
parameterize the problem of finding ground states. A typical example is the use
of matrix product states for local and gapped Hamiltonians. We study the
structure of entanglement entropy using persistent homology, a relatively new
method from the field of topological data analysis. The inverse quantum mutual
information between pairs of sites is used as a distance metric to form a
filtered simplicial complex. Both ground states and excited states of common
spin models are analyzed as an example. Furthermore, the effect of homology
with different coefficients and boundary conditions is also explored. Beyond
these basic examples, we also discuss the promising future applications of this
modern computational approach, including its connection to the question of how
spacetime could emerge from entanglement.
