Statistical theory of controlled quantum dynamics

We are currently entering a new technological era where harnessing quantum mechanical effects is at the core of applications ranging from computation to cryptography and metrology. The modelling, estimation and identification of quantum open systems are key enabling tools for a broad array of quantum engineering tasks such as state preparation and tomography, quantum feedback control, error correction, and high precision metrology.

As quantum devices and measurement techniques become more sophisticated, experimenters have to interpret increasingly complex measurement datasets to obtain accurate information about the system's state and dynamics. Therefore, there is a need for expanding the mathematical foundations of quantum statistics beyond the traditional i.i.d. (independent, identically distributed) framework of state estimation, in order to tackle new inference problems involving correlated states of interacting systems, with realistic modelling of noise, and optimal experimental design.

The goal of this proposal is to build a statistical theory of quantum stochastic processes in the framework of quantum input-output (I-O) dynamics. The I-O formalism describes the system of interest (e.g. quantum device, or sensor) as a black-box interacting with the outside world via input and output channels. These can model unwanted effects such as dephasing and leakage, or can be used to monitor and control the system using quantum feedback. The central premise of the proposal is that information about the dynamics is continuously encoded into the output state, and this resource can be exploited to perform tasks such as system identification, estimation and quantum enhanced metrology. The three central objectives are: (1) to develop new mathematical theory pertaining to central limit, concentration for quantum processes, and large deviations, in close relation to the theories of finitely correlated states and hidden Markov processes; (2) to build a general statistical framework for quantum estimation with I-O systems centred around the concept of local asymptotic normality; and (3) to employ these theoretical results for the design of new quantum metrology setups which exploit features such as dynamical phase transitions, including the use of novel methods for statistical learning of time-dependent parameters.

The proposal has a distinct interdisciplinary character as we are combining Mathematics and Physics research teams and cover a broad spectrum of expertise, including quantum information, statistics, quantum control, and non-equilibrium statistical mechanics. This maximises synergies and facilitates the creation of impact by nurturing cross-disciplinary ideas and thinking. As the research is of fundamental nature, impact on society and economy will predominantly develop over a longer time span, once the newly established body of knowledge will contribute to the development of next-generation technologies. We identify the following impact strands:

Impact by knowledge: We will pursue well established routes for dissemination to the wider academic community, through seminar talks, workshop and conference presentations as well as publications in the leading journals of the contributing fields. In addition to UK conferences, we intend to present our work at major international meetings. As measurement data plays an increasingly important role in quantum technology, there is a need for a dedicated scientific forum at the interface between statistics and quantum physics. In the proposal, we have allocated resources for a three-day interdisciplinary workshop in the final year of the project.

People: The requested funding will be used to employ an early stage PDRA, and cover part of the PI's and CI's time. The proposed research programme draws substantially from the complementary skills available in the team and aims at creating a novel and interdisciplinary body of knowledge on quantum stochastic processes and open systems. This offers excellent opportunities to acquire a unique background and set of skills in several research areas, while engaging with both the mathematical physics and condensed matter groups in Nottingham.


Society and Economy: Basic research is necessary in order to stay ahead of the curve by advancing current research priorities, but furthermore, by helping to shape what the next generation of research priorities will be. This proposal aims to develop a novel mathematical framework that will underpin future applications in continuous-time quantum metrology. Examples include system identification methodology for quantum feedback control, estimation of time-dependent parameters, and learning of complex quantum dynamical systems. Our research will generate impact by delivering progress in our general understanding of the interplay between quantum statistics, input-output control and non-equilibrium statistical mechanics, which will contribute to underpinning the next revolution of "complex quantum technologies".


Outreach: The team members will actively participate in Open Days activities by giving short presentations, and engage in other outreach activities such as the series of Science Public Lectures, Nottingham's Wonder event and the Pint of Science Festival. The Sixty Symbols project (www.sixtysymbols.com) is a very successful initiative for creating public awareness of science, comprising short films aimed at explaining physics to the general public. We plan to contribute with a video about "quantum noise & signals", highlighting the work done by the Nottingham pioneers Viacheslav Belavkin and Robin Hudson.