Unitary $k$-designs without Hamiltonian quenches

• URL: http://arxiv.org/abs/2607.20640v2
• Date: Thu, 30 Jul 2026 07:46:27 GMT
• Title: Unitary $k$-designs without Hamiltonian quenches
• Abstract summary: Unitary designs provide a resource-efficient framework for emulating Haar randomness up to a desired order.<n>We introduce a two-Pauli-kick (2PK) protocol, in which unitary evolution under a fixed Hamiltonian is interspersed with two Pauli operator insertions.<n>We verify this protocol for single realizations of Gaussian random matrices, the Majorana and Spin Sachdev-Ye-Kitaev models, and deterministic local quantum spin chains.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Unitary designs provide a resource-efficient framework for emulating Haar randomness up to a desired order. Quench-based protocols have recently been shown to generate such designs, but achieving this typically requires multiple Hamiltonian realizations, even when using temporal ensembles. Here, we show that no Hamiltonian quenches are required: a single chaotic Hamiltonian is sufficient to generate approximate unitary $k$-designs, even when that Hamiltonian is spatially local. We introduce a two-Pauli-kick (2PK) protocol, in which 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 approaches the Haar value. We verify this protocol for single realizations of Gaussian random matrices, the Majorana and Spin Sachdev-Ye-Kitaev models, and deterministic local quantum spin chains. Remarkably, in these deterministic spin systems, the 2PK protocol generates approximate unitary designs in regimes where conventional quench-based protocols are either inapplicable or fail to converge. Furthermore, our protocol provides a finite-temperature extension of the frame potential and establishes an analytic bound in terms of the equilibrium partition function. We discuss a holographic perspective on this mechanism.

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