Unitary Designs from Two Chaotic Hamiltonians and a Random Pauli Operation

TL;DR

This paper demonstrates that applying two different chaotic Hamiltonians with an intermediate random Pauli operation can generate unitary designs in qubit systems, offering a new approach to quantum randomness.

Contribution

It introduces a novel protocol using two chaotic Hamiltonians and a random Pauli operation to produce unitary designs, expanding the understanding of quantum chaos and randomness generation.

Findings

Numerical verification with Gaussian and spin model Hamiltonians confirms the protocol's effectiveness.

Theoretical analysis shows the universal Pauli spectrum underpins the design emergence.

Finite-time and finite-size effects are characterized, indicating practical feasibility.

Abstract

The realization of unitary designs is of fundamental interest in quantum science and typically requires the ability to implement structured quantum circuits. Recent developments have explored the possibility of generating unitary designs using only a small number of quantum quenches, in which the evolution during each interval is governed by a static Hamiltonian. In particular, it has been established that at least three chaotic Hamiltonians are required when only Hamiltonian evolutions are employed. In this work, we propose the emergence of unitary designs in the temporal ensemble of qubit systems evolved under two distinct chaotic Hamiltonians for sufficiently long times, supplemented by an intermediate random Pauli operation inserted between them. This result follows from the universal Pauli spectrum of chaotic Hamiltonians, a central concept in the study of non-stabilizerness. Our theoretical predictions are verified numerically using explicit examples, including Gaussian unitary ensemble Hamiltonians and random spin models. We further investigate finite-time and finite-size corrections to the protocol. Our results provide new insights into the dynamical generation of quantum randomness and offer a new route toward realizing unitary designs in chaotic systems.