PhD: Nonlinear Quantum Canonical Transformations and the Structure of Non-Gaussian States

Project Description Quantum states in continuous variables (CV) represent a promising approach in quantum computing and communication. They offer significant advantages over discrete variable states (such as single-photon states), including deterministic generation and efficient detection. In particular, multimode squeezed states are essential for synthesizing cluster states, which are considered one of the most promising architectures for all-optical measurement-based quantum computation.

However, non-classical states generated from squeezing sources belong to the class of Gaussian states, which are insufficient for universal quantum computation. As a result, various de-Gaussification methods are being explored. These methods, however, are generally probabilistic and suffer from significant limitations in terms of generation rate and efficiency. Recently, attention has shifted toward deterministic sources capable of generating non-Gaussian states in continuous variables. These approaches rely on the implementation of nonlinear interactions characterized by Hamiltonians of higher than quadratic order. So far, research has primarily focused on single-mode states, such as generalized squeezing and cubic-gate states.

This project aims to investigate the role of nonlinear quantum canonical transformations in the deterministic generation and characterization of non-Gaussian states. The goal is to develop original analytical techniques to systematically analyze the structure of this highly complex quantum state space. The study will leverage advanced mathematical tools to provide novel insights, with potential applications in quantum computing and quantum information science.

Approach and Methodology This high-risk, high-gain project will rely on advanced tools, including:

- The use of smooth decompositions of parameter‑dependent linear transformations, an approach recently developed by our group [PRL 105, 103601 (2020)]. - Multilinear algebra techniques for the characterization of multimode non-Gaussian states. - Operator theory and quantum phase space formulations.

Candidate Profile We seek excellent and highly motivated students eager to tackle a frontier scientific challenge and develop independent research skills. Ideal candidates should possess:

- A background in theoretical physics or mathematics, with a strong inclination toward formal rigor and abstraction.

- Familiarity with linear algebra.

- The ability to engage constructively and openly with the research team, fostering collaboration and exchanging ideas in a positive manner.

- Curiosity and enthusiasm for exploring new concepts and developing original approaches.

Collaborations and Research Environment During the research project, the candidate will have the opportunity to collaborate with mathematicians and theoretical computer scientists interested in the many-body aspects of quantum information. The work will be conducted in an international research environment, enabling high-level scientific interactions and access to advanced tools for studying nonlinear canonical transformations. The PhD student will be co-supervised by Dr. Giuseppe Patera from University of Lille – PhLAM and Prof. Remy Boyer from University of Lille – CRIStAL Lab, ensuring a strong interdisciplinary framework and broadening the scope of expertise available for the project.

Objectives and Expected Impact This project aims to achieve significant advancements in understanding the structure of non-Gaussian quantum states and to develop new mathematical methodologies for their characterization. By leveraging nonlinear canonical transformations, we seek to explore new avenues for deterministic non-Gaussian state generation, which could lead to breakthrough applications in continuous-variable quantum computing and quantum technologies.

Funding To date, we have secured half of the funding from the Cross-Disciplinary Project C2EMPI (https://c2empi.univ-lille.fr/), and we have submitted requests for the remaining half to two different sources: the Hauts-de-France region and the PEPR 'OQuLus' (https://pepr-oqulus.cnrs.fr/presentation/).

Additionally, we are considering a third option: applying for a University of Lille fellowship. The selection will be based on the quality of the candidate's CV; therefore, to apply, applications should be submitted before May 2025.

Application and Timeline We are looking for a PhD candidate to start at the beginning of the 2025-2026 academic year. Therefore, applications should preferably be submitted before May 2025.

Contact Information

- Giuseppe Patera, University of Lille – PhLAM, giuseppe.patera@univ-lille.fr

- Remy Boyer, University of Lille – CRIStAL, remy.boyer@univ-lille.fr

We invite brilliant and motivated students to join this cutting-edge scientific adventure.