🤖 AI Summary
This work investigates the computational complexity of the Tensor Orbit-Closure Intersection (TOCI) problem under group actions, which underlies fundamental questions in group-action equivalence testing and separability of invariant polynomials.
Method: We introduce the algebraic complexity class TOCI and develop a novel algebraic reduction framework grounded in geometric invariant theory and representation theory.
Contributions: We establish that tensor network equivalence is TOCI-complete—the first completeness result for TOCI. We further construct a polynomial-time many-one reduction from Graph Isomorphism (GI) to TOCI, proving GI ⊆ TOCI and thereby providing the first rigorous lower bound for TOCI. Moreover, our framework unifies and explains the complexity of longstanding open problems—including noncommutative Polynomial Identity Testing (PIT) and quantum state classification—by reducing them to TOCI. This work constitutes the first systematic complexity-theoretic treatment of orbit-closure intersection problems.
📝 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.