### Quantum Fisher Information: Natural Quantum Gradient

#### (supervisor: John Gough)

Nature of project: **theory**, software

Available to
students on full-time physics degree schemes only.

#### Project description and methodology

The project aims to understand the ideas behind Fisher information, the Cramer-Rao inequality and the generalization to quantum probabilities.

Here we wish to understand the relationship with entropy.

We also want to apply the theory to the recent notion of a quantum natural gradient.

*A successful project will develop beyond the above in one/some of the following directions:*

A full description of the technical details behind the theory.

When considering where to take your project, please bear in mind the time available. It is preferable to do fewer things well than to try many and not get conclusive results on any of them. However, sometimes it is useful to have a couple of strands of investigation in parallel to work on in case delays occur.

*Additional scope or challenge if taken as a Year-4 project:* Applying the ideas to understand recent work on quantum gradient descent algorithms.

Please speak to **John Gough** if you consider doing this project.

*Initial literature for students:*

- Neilsen & Chuang, Quantum Computation and Quantum Information,Cambridge University Press New York, NY, USA, ISBN:1107002176 9781107002173
- Quantum Natural Gradient, Stokes et al., arxiv:1909.02108
- Naoki Yamamoto, On the natural gradient for variational quantum eigensolver, https://arxiv.org/abs/1909.05074
- O E Barndorff-Nielsen and R D Gill, Fisher information in quantum statistics, 2000 J. Phys. A: Math. Gen. 33 4481,

#### Novelty, degree of difficulty and amount of assistance required

Fisher information is not part of the curriculum and will need to be researched. neither, of course is its quantum analogue. Understanding these concepts will be important.

Describing the eigensolver problem and the natural gradient method, and how they relate to quantum Fisher information will be the challenging part of the project: this will be based on references 2 and 3.

#### Project milestones and deliverables (including timescale)

milestone | to be completed by |

basic theory and references | end of October |

formulation of problem - outlook | end of November |

applications of Fisher information and its quantum analogue | Christmas |

First applications | mid-March |