Quantum computing is finding promising applications in optimization, machine learning and physics, leading to the development of various models for representing quantum information. Because these representations are often studied in different contexts (many-body physics, machine learning, formal verification, simulation), little is known about fundamental trade-offs between their succinctness and the runtime of operations to update them. We therefore analytically investigate three widely-used quantum state representations: matrix product states (MPS), decision diagrams (DDs), and restricted Boltzmann machines (RBMs). We map the relative succinctness of these data structures and provide the complexity for relevant query and manipulation operations. Further, to chart the balance between succinctness and operation efficiency, we extend the concept of rapidity with support for the non-canonical data structures studied in this work, showing in particular that MPS is at least as rapid as some DDs. By providing a knowledge compilation map for quantum state representations, this paper contributes to the understanding of the inherent time and space efficiency trade-offs in this area.

翻译：量子计算在优化、机器学习及物理学领域展现出广阔的应用前景，这促使了多种量子信息表示模型的发展。由于这些表示方法通常在不同研究背景（多体物理、机器学习、形式化验证、模拟）下被研究，人们对其简洁性与运算更新运行时间之间的基本权衡知之甚少。为此，我们解析性地研究了三种广泛使用的量子态表示方法：矩阵乘积态（MPS）、决策图（DDs）和受限玻尔兹曼机（RBMs）。我们绘制了这些数据结构的相对简洁性图谱，并给出了相关查询与操作的计算复杂度。进一步，为平衡简洁性与操作效率，我们扩展了"快速性"概念以支持本文研究的非标准数据结构，特别证明了MPS至少与某些DDs同样快速。通过构建量子态表示的知识编译图谱，本文深化了对该领域内在时空效率权衡机制的理解。