Unitary $k$-designs without Hamiltonian quenches

Abstract:Unitary designs provide an efficient means of emulating Haar randomness without the requirement of exponential resources. Hamiltonian quench protocols have recently been shown to generate such designs, but achieving this typically requires multiple independent Hamiltonian quenches, even via temporal ensembles. Here, we show that no Hamiltonian quenches are required: a single chaotic Hamiltonian is sufficient to generate approximate unitary $k$-designs. We introduce a two-Pauli-kick (2PK) protocol, where unitary evolution under a fixed Hamiltonian is interspersed with two Pauli operator insertions (kicks). By evaluating the frame potential, we demonstrate that the resulting ensemble converges to the Haar value, indicating the emergence of approximate unitary designs. We verify this protocol for Gaussian random matrix ensembles, the Sachdev-Ye-Kitaev (SYK) model, and simplified spin (bosonic) SYK models. Furthermore, our protocol admits a canonical definition of finite-temperature frame potentials and establishes the relation with the partition function. We outline a plausible holographic interpretation of our protocol from the dual gravity picture.

Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th)

arXiv:2607.20640 [quant-ph]

(or arXiv:2607.20640v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.20640 arXiv-issued DOI via DataCite (pending registration)