Complexity of recognizing topological order in ground states of 2-local Hamiltonians

Determine the computational complexity of recognizing topological order in quantum states that are ground states of 2-local Hamiltonians, including whether this task remains quantumly hard or admits efficient algorithms under such locality constraints.

Background

Beyond hardness for general states produced by shallow circuits, the authors seek results closer to physically realistic scenarios, notably ground states of 2-local Hamiltonians. Clarifying the complexity landscape for recognizing topological order in this setting would bridge the gap between theoretical hardness and practical many-body systems.

References

Looking forward, several important open questions remain. And can we extend our results even closer to the topologically-ordered states that might arise in the physical world---e.g.~what is the complexity of recognizing topological order in the ground states of 2-local Hamiltonians?

— Random unitaries in extremely low depth (2407.07754 - Schuster et al., 10 Jul 2024) in Appendix, Literature review, Recognizing topological order