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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.

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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