Realizing Unitary $k$-designs with a Single Quench

TL;DR

This paper introduces a single-quench protocol that efficiently generates unitary $k$-designs, simplifying the process of achieving Haar-like randomness and providing practical tools for quantum chaos diagnostics and applications.

Contribution

A novel single-quench method for realizing unitary $k$-designs that reduces control complexity and enhances practical applicability in quantum information processing.

Findings

Achieves unitary $k$-designs with minimal control using a single quench.

Provides a new operational definition of the Thouless time based on the protocol.

Enables extensions to symmetry-resolved, open-system, and circuit-level implementations.

Abstract

We present a single-quench protocol that generates unitary $k$-designs with minimal control. A system first evolves under a random Hamiltonian $H_1$; at a switch time $t_s \geq t_{\mathrm{Th}}$ (the Thouless time), it is quenched to an independently drawn $H_2$ from the same ensemble and then evolves under $H_2$. This single quench breaks residual spectral correlations that prevent strictly time-independent chaotic dynamics from forming higher-order designs. The resulting ensemble approaches a unitary $k$-design using only a single control operation -- far simpler than Brownian schemes with continuously randomized couplings or protocols that apply random quenches at short time intervals. Beyond offering a direct route to Haar-like randomness, the protocol yields an operational, measurement-friendly definition of $t_{\mathrm{Th}}$ and provides a quantitative diagnostic of chaoticity. It further enables symmetry-resolved and open-system extensions, circuit-level single-quench analogs, and immediate applications to randomized measurements, benchmarking, and tomography.