Many natural computational problems in computer science, mathematics, physics, and other sciences amount to deciding if two objects are equivalent. Often this equivalence is defined in terms of group actions. A natural question is to ask when two objects can be distinguished by polynomial functions that are invariant under the group action. For finite groups, this is the usual notion of equivalence, but for continuous groups like the general linear groups it gives rise to a new notion, called orbit closure intersection. It captures, among others, the graph isomorphism problem, noncommutative PIT, null cone problems in invariant theory, equivalence problems for tensor networks, and the classification of multiparty quantum states. Despite recent algorithmic progress in celebrated special cases, the computational complexity of general orbit closure intersection problems is currently quite unclear. In particular, tensors seem to give rise to the most difficult problems. In this work we start a systematic study of orbit closure intersection from the complexity-theoretic viewpoint. To this end, we define a complexity class TOCI that captures the power of orbit closure intersection problems for general tensor actions, give an appropriate notion of algebraic reductions that imply polynomial-time reductions in the usual sense, but are amenable to invariant-theoretic techniques, identify natural tensor problems that are complete for TOCI, including the equivalence of 2D tensor networks with constant physical dimension, and show that the graph isomorphism problem can be reduced to these complete problems, hence GI$\subseteq$TOCI. As such, our work establishes the first lower bound on the computational complexity of orbit closure intersection problems, and it explains the difficulty of finding unconditional polynomial-time algorithms beyond special cases, as has been observed in the literature.

翻译：在计算机科学、数学、物理学及其他科学领域中，许多自然计算问题可归结为判定两个对象是否等价。这种等价性通常由群作用定义。一个自然的问题是：何时两个对象能够被群作用下不变的多项式函数所区分？对于有限群，这是通常的等价概念；但对于连续群（如一般线性群），则产生了一种称为轨道闭包交集的新概念。它涵盖了图同构问题、非交换多项式恒等式检验、不变量理论中的零锥问题、张量网络的等价性问题，以及多方量子态的分类等。尽管近期在若干著名特例中取得了算法进展，但一般轨道闭包交集问题的计算复杂性目前仍不甚明确。特别地，张量似乎引出了最困难的问题。在本工作中，我们从复杂性理论的视角系统性地研究轨道闭包交集问题。为此，我们定义了一个复杂性类 TOCI，用于刻画一般张量作用下轨道闭包交集问题的计算能力；提出了一种代数归约的适当概念，该概念蕴含通常意义上的多项式时间归约，同时适用于不变量理论的技术方法；识别了 TOCI 的若干自然张量完备问题，包括具有恒定物理维度的二维张量网络等价性问题；并证明了图同构问题可归约至这些完备问题，从而得到 GI$\subseteq$TOCI。因此，我们的工作首次为轨道闭包交集问题的计算复杂性建立了下界，并解释了为何在文献中观察到，超越特例寻找无条件多项式时间算法如此困难。