Statement
Problem 4.2 of Goto et al.: toy models of QG routinely assume dynamics are approximately Haar-random. What properties of physical (local, time-independent) many-body Hamiltonian dynamics are necessary and sufficient to reproduce near-Haar behavior (unitary designs) relevant for scrambling and encoding?
Why It Matters
Peripheral to the group; a quantum-information / many-body-chaos question. Nearest group contact is the many-body-theory interest of one of the PhD theses, but not a current line.
What We Know
- Haar-random unitaries and unitary -designs reproduce scrambling; local random circuits form designs in polynomial depth.
- Recent work: constant-local time-independent Hamiltonian ensembles cannot form accurate unitary 2-designs; polylog-local ones can approach design behavior.
Possible Approaches
- Characterize the Hamiltonian-locality thresholds for approximate-design formation.
- Relate design-formation time to scrambling / OTOC decay.