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：摘要：量子态与量子操作的高效表示及仿真方法对量子电路的优化至关重要。决策图（DDs）作为一种最初用于表示布尔函数的经典数据结构，已被证明能够捕捉量子系统的关键特征，但其局限性尚不明确。本文研究并弥合了现有基于决策图的结构与稳定子形式体系（一种在可处理范围内仿真量子电路的重要工具）之间的差距。我们首先证明，尽管DDs曾被提议用于简洁表示重要量子态，但实际上对于某些稳定子态需要指数级空间。为解决这一问题，我们引入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，由多项式大小的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可对其进行高效仿真，而其输出态却无法被两种最先进的仿真范式（用于Clifford + $T$电路的稳定子分解技术与矩阵乘积态）简洁表示。通过融合两种成功方法，LIMDD为量子计算的仿真与分析开辟了更具根本性优势的解决方案。