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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

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.

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

Abstract

We explore the interplay between symmetry and randomness in quantum information. Adopting a geometric approach, we consider states as -equivalent if related by a symmetry transformation characterized by the group . We then introduce the Haar measure on the homogeneous space , characterizing true randomness for -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 -wise independent approximations and defining -designs on and -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.
Paper Structure (12 sections, 9 theorems, 16 equations, 3 figures, 1 table)

This paper contains 12 sections, 9 theorems, 16 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Haar measure on homogeneous spaces
  4. Sampling from Haar Measure on U/H
  5. t-Designs for U/H
  6. State t-designs
  7. Homogeneous space t-designs
  8. Pseudorandom states and unitaries
  9. Application: The expressibility of parameterized quantum circuits
  10. Numerical Results for U4, H=I,SWAP
  11. Discussion
  12. Limitations of existing approaches for quantifying expressibility

Key Result

Theorem 2.1

Let $G$ be a locally compact Hausdorff topological group and $H \leq G$. Let $\mathrm{d}^{l}(G)$ and $\mathrm{d}^{l}(H)$ be Haar measures on $G$ and $H$ respectively.

Figures (3)

  • Figure 1: Pictorial representation of a quantum tug of war between randomness and symmetries on homogeneous spaces.
  • Figure 2: (a) and (b) show the two circuits that we considered for the evaluation. The first pair of RX gates at the front represents the encoding.
  • Figure 3: The table shows the values of the expressibilities calculated using the homogeneous expressibility scheme as well as the expressibility in Sim_2019.

Theorems & Definitions (25)

  • Theorem 2.1: nachbin_1985.
  • Lemma 2.2
  • proof
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Definition 4.4
  • Theorem 4.5
  • Definition 4.6
  • Definition 4.7
  • ...and 15 more