Three Hamiltonians are Sufficient for Unitary $k$-Design in Temporal Ensemble

TL;DR

This paper demonstrates that a three-step Hamiltonian protocol with fixed Hamiltonians and random evolution times can generate unitary $k$-designs, outperforming simpler two-step protocols, with rigorous proofs for Gaussian ensembles.

Contribution

It introduces a three-step protocol (3SP) for generating unitary $k$-designs from a quenched temporal ensemble, showing its superiority over two-step protocols (2SP) in theory and numerics.

Findings

3SP can realize arbitrary $k$-designs, unlike 2SP.

3SP's additional phase constraints eliminate permutation degrees of freedom.

For Gaussian Hamiltonians, 3SP achieves high accuracy with narrower time windows.

Abstract

Unitary $k$-designs are central to quantum information and quantum many-body physics as efficient proxies for Haar-random dynamics. We study how chaotic Hamiltonian evolution can generate unitary $k$-designs. Standard approaches typically rely on many independent Hamiltonian realizations or fine-tuning evolution times. Here we show that unitary designs can instead arise from a quenched temporal ensemble, where Hamiltonians are sampled once and held fixed, while randomness enters only through the evolution times. We analyze a two-step protocol (2SP), applying $H_1$ for time $t_1$ and $H_2$ for time $t_2$, and a three-step protocol (3SP) with an additional quench, with all times randomly drawn from a prescribed distribution. Time averaging imposes energy-index matching in the frame potential (FP), which quantifies the distance to Haar random. Analytically and numerically, we show that 2SP cannot realize a general unitary $k$-design, whereas 3SP can do so for arbitrary $k$. The advantage of 3SP is that the additional random phases impose stronger constraints, eliminating independent permutation degrees of freedom in the FP. For Gaussian unitary ensemble Hamiltonians, we prove these results rigorously and show that under imperfect time averaging, 3SP achieves the same accuracy as 2SP with a parametrically narrower time window.