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.
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]. An efficient "projected least squares" estimator has been analysed in [8] and shown to be rate optimal for low rank states. A comparative study of different estimation method together with an online simulation tool can be found in [9].
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, and will benefit from our established external collaborations.
[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 compressive low rank tomography with random measurements
Journal of Physics A: Mathematical and Theoretical, 50 195301 (2017)
arXiv:1609.03758
[8] Madalin Guta, Jonas Kahn, Richard Kueng, Joel A. Tropp
Fast state tomography with optimal error bounds
arXiv:0809.11162
[9] Anirudh Acharya, Madalin Guta, Theodore Kypraios
A comparative study of estimation methods in quantum tomography
arXiv:0901.xxxx
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
Journal of Mathematical Physics 58, 052201 (2017)
arXiv:1601.04355
[7] Matthew Levitt, Madalin Guta, Hendra I. Nurdin
Power Spectrum Identification for Quantum Linear Systems
Automatica 90 255-262 (2018)
arXiv:1612.02681