  Improving the simulation of quantum circuits on classical computers is
important for understanding quantum advantage and increasing development speed.
In this paper, we explore a new way to express stabilizer states and further
improve the speed of simulating stabilizer circuits with a current existing
approach. First, we discover a unique and elegant canonical form for stabilizer
states based on graph states to better represent stabilizer states and show how
to efficiently simplify stabilizer states to canonical form. Second, we develop
an improved algorithm for graph state stabilizer simulation and establish
limitations on reducing the quadratic runtime of applying controlled-Pauli $Z$
gates. We do so by creating a simpler formula for combining two Pauli-related
stabilizer states into one. Third, to better understand the linear dependence
of stabilizer states, we characterize all linearly dependent triplets,
revealing symmetries in the inner products. Using our novel controlled-Pauli
$Z$ algorithm, we improve runtime for inner product computation from $O(n^3)$
to $O(nd^2)$ where $d$ is the maximum degree of the graph.
