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

arXiv: https://arxiv.org/abs/2309.05253 · HTML: https://mlsanity.com/paper/2309.05253/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 36 references that are in the map.
- First posted on arXiv 11 Sep 2023.

## Builds on (most cited first, 13 of 13)

- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)
- [Group-Invariant Quantum Machine Learning](https://mlsanity.com/paper/2205.02261/) (2022)
- [Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.09974/) (2022)
- [Representation Theory for Geometric Quantum Machine Learning](https://mlsanity.com/paper/2210.07980/) (2022)
- [Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ans"atze](https://mlsanity.com/paper/2112.07611/) (2021)
- [Classical Shadows With Noise](https://mlsanity.com/paper/2011.11580/) (2020)
- [On the universality of Sn-equivariant k-body gates](https://mlsanity.com/paper/2303.00728/) (2023)
- [Cryptography from Pseudorandom Quantum States](https://mlsanity.com/paper/2112.10020/) (2021)
- [Pseudorandom unitaries are neither real nor sparse nor noise-robust](https://mlsanity.com/paper/2306.11677/) (2023)
- [Quantum Pseudorandomness and Classical Complexity](https://mlsanity.com/paper/2103.09320/) (2021)
- [Pseudorandom Strings from Pseudorandom Quantum States](https://mlsanity.com/paper/2306.05613/) (2023)

## Abstract (arXiv)

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 mathbbU/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 mathbbU/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.