A quantum tug of war between randomness and symmetries on homogeneous spaces
2023
Online
report
Zugriff:
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.
Comment: 9 + 1 pages, 3 figures
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A quantum tug of war between randomness and symmetries on homogeneous spaces
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Autor/in / Beteiligte Person: | Arvind, Rahul ; Bharti, Kishor ; Khoo, Jun Yong ; Koh, Dax Enshan ; Kong, Jian Feng |
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Veröffentlichung: | 2023 |
Medientyp: | report |
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