Protocols for the classification of topological phases

Topological quantum states are exotic states of matter with global properties that cannot be identified from local observations of the system. Such states, like fractional quantum Hall states, are particularly difficult to understand conceptually, and to char- acterize experimentally in solid states. However, interacting topological quantum states can be now created artificially with systems of atoms and ions, and with an excellent level of coherence and control. These quantum simulation experiments will allow for a crucial better understanding of interacting topological quantum matter, and therefore to answer fundamental theoretical questions in theoretical condensed matter physics. The goal of this theory project is to provide the tools to measure and characterize, in quantum simulation, interacting topological quantum states. In particular, we will present proto- cols that can classify topological quantum states, i.e, identify topological quantum phases formally in the framework of condensed matter theory. The main ingredient in our protocols are random measurements. This consists in subjecting a system realized in an experiment to a sequence of random operations and in extracting information from statistical correlations between measurements. In pre- vious works, we have used these techniques to propose and realize protocols measuring entanglement, taking advantage of local addressing techniques and high repetition rates available in current experiments. In this project, we will extend these tools to protocols characterizing topology: these protocols will be implementable in current atomic systems and will enable the first formal classification of an interacting topological phase. This project has also a high-risk-high gain component: we will develop approaches aim- ing at characterizing the true topological order of fractional quantum hall state and quantum spin liquids, which are states that have a much more complex structure. Our project consists in translating the theory of topological order into simple pro- tocols that can be realized in current experiments. We thus believe that our results will have an important impact in various fields of physics, in particular in the AMO quantum simulation, quantum information and condensed matter communities. To realize this interdisciplinary project, we will combine our expertise in quantum optics theory of measurement protocols, random matrix theories, and numerical tensor- network simulations, which we built from recent previous works. We will also make use of our current collaborations with experimentalists implementing our previous random measurements protocols, in order to propose the most relevant approaches for their sys- tems.

The TOPORAND project consisted in deriving measurement protocols for quantum sys- tems. In particular, the goal was to provide experimentalists building quantum computers and quantum simulators with methods to characterize quantum states. The main outcome of the project is the development of randomized measurements protocols for qubit systems (such as a quantum computer). We showed that we can faithfully extract from these systems key quantum properties: topological invariants, entanglement, etc. Three main tasks were accomplished: In the first task, we showed that randomized measure- ments can be used to experimentally access topological invariants. Such quantities characterize uniquely a topological quantum state, which is an exotic state of quantum matter that cannot be described with local order parameters. In the second task, we proposed strategies to reduce the number of times an experiment has to be run in order to provide a faithful estimation based on randomized measurements. Finally, we developed general theoretical results showing how to derive estimators for measuring a given physical quantity via randomized measurements, and to assess the corresponding statistical errors. The project was supported by various theory collaborations that enabled us to combine different expertise on random matrices, quantum measurements, and many-body entanglement. Some of our theoretical proposals could be then experimentally demonstrated in quantum tech- nological platforms of trapped ions and superconducting quantum circuits. Most of our results are presented in a recent review [Nature Reviews Physics volume 5, pages 9-24 (2023)], which we aimed to be pedagogical and focused on practical aspects regarding experimental implications.