A quantum tug of war between randomness and symmetries on homogeneous spaces

TL;DR

This paper investigates the relationship between symmetry and randomness in quantum states using a geometric approach involving homogeneous spaces, introducing new methods to characterize and approximate true randomness in symmetric quantum systems.

Contribution

It is the first to apply homogeneous space mathematics to characterize symmetry and randomness in quantum information, including new definitions of t-designs and pseudorandomness in this context.

Findings

Introduces Haar measure on homogeneous spaces for quantum states

Defines t-designs and pseudorandom unitaries in homogeneous spaces

Analyzes expressibility of quantum machine learning models in symmetric spaces

Abstract

We explore the interplay between symmetry and randomness in quantum information. Adopting a geometric approach, we consider states as $H$-equivalent if related by a symmetry transformation characterized by the group $H$. We then introduce the Haar measure on the homogeneous space $\mathbb{U}/H$, characterizing true randomness for $H$-equivalent systems. While this mathematical machinery is well-studied by mathematicians, it has seen limited application in quantum information: we believe our work to be the first instance of utilizing homogeneous spaces to characterize symmetry in quantum information. This is followed by a discussion of approximations of true randomness, commencing with $t$-wise independent approximations and defining $t$-designs on $\mathbb{U}/H$ and $H$-equivalent states. Transitioning further, we explore pseudorandomness, defining pseudorandom unitaries and states within homogeneous spaces. Finally, as a practical demonstration of our findings, we study the expressibility of quantum machine learning ansatze in homogeneous spaces. Our work provides a fresh perspective on the relationship between randomness and symmetry in the quantum world.