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.

翻译：量子态和量子操作的表示与模拟高效方法对量子电路优化至关重要。决策图作为一种研究成熟的、最初用于表示布尔函数的数据结构，已被证明能够捕捉量子系统的相关特性，但其局限性尚未被充分理解。本文研究并弥合了现有基于决策图的结构与稳定子形式主义（一种在可处理范围内模拟量子电路的重要工具）之间的差距。我们首先证明，尽管决策图被建议用于紧凑表示重要量子态，但对于某些稳定子态，它们实际上需要指数级空间。为解决这一问题，我们提出了一种更强大的决策图变体，称为局部可逆映射决策图（LIMDD）。我们证明，由多项式规模LIMDD表示的量子态集合严格包含稳定子态与其他决策图变体的并集。最后，存在某些电路，LIMDD能够高效模拟，而它们的输出态却无法被两种最先进的模拟范式（Clifford+$T$电路的稳定子分解技术和矩阵乘积态）紧凑表示。通过融合两种成功方法，LIMDD为量子计算的模拟与分析开辟了本质上更强大解决方案的路径。