  Efficient methods for the representation and simulation of quantum states and
quantum operations are crucial for the optimization of quantum circuits.
Decision diagrams (DDs), a well-studied data structure originally used to
represent Boolean functions, have proven capable of capturing relevant aspects
of quantum systems, but their limits are not well understood. In this work, we
investigate and bridge the gap between existing DD-based structures and the
stabilizer formalism, an important tool for simulating quantum circuits in the
tractable regime. We first show that although DDs were suggested to succinctly
represent important quantum states, they actually require exponential space for
certain stabilizer states. To remedy this, we introduce a more powerful
decision diagram variant, called Local Invertible Map-DD (LIMDD). We prove that
the set of quantum states represented by poly-sized LIMDDs strictly contains
the union of stabilizer states and other decision diagram variants. Finally,
there exist circuits which LIMDDs can efficiently simulate, while their output
states cannot be succinctly represented by two state-of-the-art simulation
paradigms: the stabilizer decomposition techniques for Clifford + $T$ circuits
and Matrix-Product States. By uniting two successful approaches, LIMDDs thus
pave the way for fundamentally more powerful solutions for simulation and
analysis of quantum computing.
