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

LIMDD: A Decision Diagram for Simulation of Quantum Computing Including Stabilizer States