A quantum tug of war between randomness and symmetries on homogeneous
  spaces
Rahul Arvind Kishor Bharti Jun Yong Khoo Dax Enshan Koh Jian Feng Kong
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.