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arXiv · 2309.05253

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
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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.

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BibTeXRIS

Rahul Arvind, Kishor Bharti, Jun Yong Khoo, Dax Enshan Koh, Jian Feng Kong. 2023-09-11. A quantum tug of war between randomness and symmetries on homogeneous spaces. https://doi.org/10.1103/physrevresearch.7.013105

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