Extension beyond strongly chaotic Hamiltonians and quench-count scaling

Extend the quenched temporal-ensemble framework for generating unitary k-designs—specifically the multi-quench protocols that randomize only evolution times under fixed Hamiltonians—beyond strongly chaotic Hamiltonians to Hamiltonians that are weakly chaotic or near-integrable, and determine how the minimal number of quenches required to achieve unitary k-design behavior scales with system size and the degree of chaoticity.

Background

The paper proves that a three-step protocol (3SP) with time-randomized evolution under three fixed, independently drawn chaotic Hamiltonians achieves the Haar value of the k-th frame potential in the long-time limit, establishing unitary k-design behavior for strongly chaotic dynamics (e.g., GUE). In contrast, a two-step protocol (2SP) cannot realize higher-order k-designs (k>1).

These results rely on chaotic eigenbasis overlaps and time-filtering arguments that enforce stringent energy-index matching, which may not hold in weakly chaotic or near-integrable systems. The authors explicitly highlight that extending their construction beyond strongly chaotic regimes and understanding how many quenches are minimally needed is unresolved, particularly as a function of system size and chaoticity.