Website TUEindhoven **TU Eindhoven, Dept. of Mathematics & Computer Science**

Eindhoven University of Technology (TU/e) is a research university specializing in engineering science & technology.

## Description

For the development of a quantum computer, an understanding of the propagation of errors in a quantum

system on which error-prone operations are performed is needed. This project deals with the case that we

have a quantum state and we apply repeatedly a quantum operation (also called a Positive Operator Valued

Measure or POVM). In [3] was proven that, under certain conditions, in the limit of many repetitions the

state then “purifies” that is, becomes a state expressible as a rank one matrix. In [1, 2] this result was

generalized in the case that we perform a quantum operation which follows a certain distribution and a

geometric convergence rate was proven. In real applications, however, the quantum operations are errorprone

and so we ask if the theoretical results can be extended under existence of noise. The problem in the

case of quantum states turns out to be related to the problem of products of random matrices [4] and if a

Markov chain driving a stochastic process has a unique invariant distribution as well as if it is ergodic.

## Main Objectives

(a) Understand and learn the formalism of quantum states, operations, martingales, Markov chains, invariant

distributions, and ergodicity of Markov Chains.

(b) Understand the papers [3] and [2] as well as the error models for quantum operations which can be used.

(c) Analyse numerically or with other means the stability and behaviour of the quantum state when we

repeatedly perform the same operation with some error model, and if there is a threshold for stability.

(d) Generalise, if possible, the results of [3] and [2] in the case that we use the same quantum operation but

with a certain error model and giving sufficient conditions for stability.

Note: The reach and content dedicated to each objective can be discussed with the student depending on

her / his motivation and willingness. For example, a student more interested in a bibliographical part of the

topic could focus on objectives (a) and (b). Similarly, a student focused on a research oriented (and thus

riskier) path could focus more on objectives (b), (c) and maybe (d).

## Supervision / Prerequisites

You will work with Albert Senen-Cerda, a.senen.cerda@tue.nl.

Ideally, you already have some knowledge of quantum formalism (Hilbert spaces, quantum states, measurements in quantum mechanics, etc), martingales and convergence of stochastic processes.

## References

[1] M. Bauer, T. Benoist, and D. Bernard. Repeated quantum non-demolition measurements: convergence

and continuous time limit. In Annales Henri Poincare, volume 14, pages 639-679. Springer, 2013.

[2] T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini. Invariant measure for quantum trajectories. Proba-

bility Theory and Related Fields, 174(1-2):307-334, 2019.

[3] H. Maassen, B. Kummerer, et al. Purification of quantum trajectories. In Dynamics & stochastics, pages

252{261. Institute of Mathematical Statistics, 2006.

[4] G. Yves. On contraction properties for products of Markov driven random matrices. Journal of Mathe-

matical Physics, Analysis, Geometry, 4(4):457-489, 2008.

To apply for this job **email your details to** jaron.sanders@tue.nl