Complexity theory of orbit closure intersection for tensors: reductions, completeness, and graph isomorphism hardness

TL;DR

The paper analyzes the complexity of orbit closure intersection (OCI) under reductive group actions, focusing on tensors and tensor tuples to model a broad class of equivalence problems. It defines a complexity class TOCI capturing polynomial-time reductions to OCI problems for general linear groups, and proves TOCI-completeness for naturally motivated tensor formats, including 2D PEPS gauge equivalence, while showing GI-hardness by GI ⊆ TOCI. The authors develop an invariant-theoretic reduction toolkit based on contraction invariants, balanced tensor spaces, and Mumford–GIT, establishing rigorous preservation and reflection criteria to ensure reductions preserve closure equivalence. They extend the framework to orthogonal, symplectic, and unitary groups, showing TOCI equivalence and GI-hardness in broad settings, and connect OCI complexity to longstanding questions in invariant theory and geometric complexity theory. The results explain why unconditional polynomial-time algorithms are elusive beyond special cases, and they map a path for future work on the exact boundaries of OCI’s computational hardness.

Abstract

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.

Figures (4)

• Figure 1: Diagrammatic representation of the invariant $F\in\mathbb{C}[V^{\otimes 2}]^{\mathsf{S}_n}$. We use different numbers of small dots to indicate the two tensor factors (indices) of the matrix $x$. The other tensors are symmetric.
• Figure 2: Diagrammatic representation of the invariant $F'\in\mathbb{C}[V^{\otimes 2} \oplus V^{\otimes 3} \oplus (V^*)^{\otimes 2}]^{\mathsf{GL}(V)}$ corresponding to the invariant $F$ in the previous diagram. Different numbers of small dots indicate different tensor factors.
• Figure 3: Construction of the invariant $F"\in\mathbb{C}[V^{\otimes 6} \otimes (V^*)^{\otimes 6} \oplus V^{\otimes 2} \otimes (V^*)^{\otimes 2}]^{\mathsf{GL}(V)}$ corresponding to the invariant $F'$ in the previous diagram. Again, we use different numbers of small dots to indicate different tensor factors.
• Figure 4: Diagrammatic representations of the tensors $r$, $s$, and $t$ from the proof of \ref{['thm:o 3 2']}

Theorems & Definitions (119)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6: Mumford-GITDerksen-Kemper
• Definition 1.7
• Lemma 1.8
• Lemma 1.9
• Theorem 1.10