Efficient methods for the representation of relevant quantum states and quantum operations are crucial for the simulation and optimization of quantum circuits. Decision diagrams (DDs), a well-studied data structure originally used to represent Boolean functions, have proven capable of capturing interesting 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, a well-studied method 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 a subset of 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. We also provide evidence that LIMDD-based simulation is capable of efficiently simulating some circuits for which both stabilizer-based and other DD-based methods require exponential time. By uniting two successful approaches, LIMDDs thus pave the way for fundamentally more powerful solutions for simulation and analysis of quantum computing.

翻译：用于代表相关量子状态和量子运行的有效方法对于量子电路的模拟和优化至关重要。决定图(DDs)是一个研究周密的数据结构,最初用来代表布林函数,但已证明能够捕捉量子系统的有趣方面,但其限度却不十分清楚。在这项工作中,我们调查并弥合现有基于DD结构与稳定器形式学之间的差距,稳定器形式学是研究周密的模拟量子电路的方法。我们首先表明,虽然建议DDDs简明代表重要的量子国家,但实际上它们需要一组稳定器状态的指数空间。为了纠正这一点,我们引入了一个更强大的决定图变体,称为“本地不可逆地图-DDD(LIMDD) ” 。我们证明,多尺寸LIMDDs所代表的量子组严格包含稳定器状态和其他决策图变体的结合。我们还提供证据表明,基于LIMDD的模拟能够有效地模拟某些电路,而稳定器和其他以DD为基础的方法都需要指数化时间。通过合并两种基础性强的模拟方法,从而计算出更成功的量子模拟方法。