## Prospective students

**Undergraduate projects:**outlines of possible projects can be found in the project book and the dissertation book on Moodle.

**PhD opportunities:**You can contact me for more information about current PhD projects and availabilities in the group. For information about the formal PhD application procedure you can consult this page. Below you can find short descriptions of two generic PhD projects. For more information please consult the research topics and publications pages.

**State estimation for large dimensional quantum systems.**This project stems from the ongoing collaboration with Theo Kypraios and Ian Dryden

(Statistics group, Nottingham). The aim is to explore and investigate new statistical methods for the estimation of quantum states of

**large dimensional**quantum systems. The efficient statistical reconstruction of such states is a crucial

**enabling tool**for current quantum engineering experiments in which multiple qubits can be controlled and prepared in exotic entangled states. However, standard maximum likelihood estimation becomes practically unfeasible for systems of merely 10 qubits, due to the exponential growth of the Hilbert space with the number of qubits.

In [1] we investigated the use of

**model selection**methods for state estimation, in particular the Akaike information criterion and the Bayesian information criterion. The general principle is to find the simplest, or most parsimonious explanation of the data, by fitting different models and choosing the estimator with the best trade-off between likelihood fit and model complexity, the latter being given by the rank of the density matrix. Another rank selection technique was considered in [2], and compressed sensing methods were investigated in [3,4].

More recently, we have looked at specific estimation methods for low rank states based on the idea of ''spectral thresholding'' [5], and at the efficiency of measurement strategies based on using a reduced number of measurement settings [6,7].

The goal of the project is to compare the efficiency of the different methods, and explore new, possibly hybrid estimators which are both accurate and computationally efficient. Possible directions to be explored include models bases on matrix product states, design of experiments, connection to inverse problems, asymptotical structure of the statistical models. The project will involve both theoretical and computational work.

[1] M. Guta, T. Kypraios and I. Dryden

Rank based model selection for multiple ions quantum tomography

New Journal of Physics, 14, 105002 (2012)

Arxiv: 1206.4032

[2] P. Alquier, C. Butucea, M. Hebiri and K. Meziani and T. Morimae

Rank penalized estimation of a quantum system

Phys. Rev. A

**88**, 032113 (2013)

arXiv:1206.1711v1

[3] D. Gross, Y. K. Liu, S. Flammia, S. Becker and J. Eisert

Physical Review Letters

**105**150401 (2010)

Arxiv:0909.3304

[4] S. Flammia, D. Gross, Y.K. Liu and J. Eisert

Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators

New Journal of Physics

**14**, 095022 (2012)

ArXiv:1205.2300

[5] Cristina Butucea, Madalin Guta, Theodore Kypraios

Spectral thresholding quantum tomography for low rank states

New Journal of Physics,

**17**113050 (2015)

Arxiv:1504.08295

[6] Anirudh Acharya, Theodore Kypraios, Madalin Guta

Statistically efficient tomography of low rank states with incomplete measurements

New Journal of Physics,

**18**043018 (2016)

Arxiv:1510.03229

[7] Anirudh Acharya, Madalin Guta

Statistical analysis of low rank tomography with compressive random measurements

arXiv:1609.03758

**System identification, metrology and control of quantum dynamical systems**. This projects aims at investigating the identification of quantum dynamical systems in the framework of input-output formalism as used in Quantum Optics [1] as well as classical Control Theory [2]. A quantum system interacts with an input "quantum noise" (e.g. atom interacting with the electromagnetic field) and output fields (e.g. emitted photons) emerging from the interaction can be measured, in order to learn about the system's dynamical parameters (e.g. its hamiltonian). The goal is to find optimal system identification strategies which may involve input state preparation, output measurement design, and quantum feedback control. An interesting related question is to understand the information-disturbance trade-off which in the context of quantum dynamical systems becomes identification-control trade-off.

The first steps in this direction were made in [3,4] which introduce the concept of asymptotic Fisher information for "non-linear" quantum Markov processes, and [5] which investigates system identification for linear quantum systems, using transfer functions techniques. In [6] we completed the study of "information geometry" of quantum Markov processes, identifying the underlying Riemannian geometry of the quantum Fisher information and the associated canonical commutation relations algebra of the output process. The "power spectrum identification" of quantum linear systems was analysed in [7].

Important future problems concern the design of efficient output measurements, a general central limit theory of the output process,

and understanding the interplay between feedback and metrology in the open dynamics setting.

[1] C. Gardiner, P. Zoller

Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics

Springer Series in Synergetics (2004)

[2] K. Zhou, J.C. Doyle and K. Glover

Robust and Optimal Control

Prentice Hall, (1995)

[3] M. Guta

Quantum information Fisher information and asymptotic normality in system identification for quantum Markov chains

Physical Review A,

**83**, 062324 (2011)

Arxiv:1007.0434

[4] C. Catana, M. van Horssen and M. Guta

Asymptotic inference in system identification for the atom maser

Philosophical Transactions of the Royal Society A

**370**, 5308-5323 (2012)

Arxiv:1112.2080

[5] M. Guta and N. Yamamoto

System identification for passive quantum linear systems

IEEE Transactions on Control,

**61**, 921 - 936 (2016)

arXiv:1303.3771v2

[6] Madalin Guta, Jukka Kiukas

Information geometry and local asymptotic normality for multi-parameter estimation of quantum Markov dynamics

arXiv:1601.04355

[7] Matthew Levitt, Madalin Guta, Hendra I. Nurdin

Power Spectrum Identification for Quantum Linear Systems

arXiv:1612.02681