Tipo di tesi 
Tesi di laurea magistrale 
Autore 
CAVAZZONI, SIMONE

URN 
etd09242021151115 
Titolo 
Transport Efficiency of continuoustime quantum walks on graphs in the presence of environmental noise 
Titolo in inglese 
Transport Efficiency of continuoustime quantum walks on graphs in the presence of environmental noise 
Struttura 
Dipartimento di Scienze Fisiche, Informatiche e Matematiche 
Corso di studi 
PHYSICS – FISICA (D.M.270/04) 
Commissione 
Nome Commissario 
Qualifica 
BORDONE PAOLO 
Primo relatore 
FERRARIO MAURO 
Controrelatore 
RAZZOLI LUCA 
Correlatore 

Parole chiave 
 Environmental Noise
 Graphs
 Quantum Walks
 Transfer Time
 Transport Efficiency

Data inizio appello 
20211028 
Disponibilità 
Embargo di 12 mesi 
Data di rilascio  20221028 
Riassunto analitico
This master thesis project is devoted to the study of the transport properties of continuoustime quantum walks (CTQWs) on different graphs. The focus is on analyzing the effects of different types of environmental noise which are commonly present in real systems and are therefore required to make the theoretical model closer to real physical implementations.
Quantum walks (QWs) are the quantum counterpart of classical random walks. As well as the latter, QWs are investigated in a variety of research areas, e.g., quantum computation, and modeling of transport phenomena, and have different physical implementations. QWs are widely used to develop quantum algorithms, like search algorithms and graph isomorphism.
The Hamiltonian of the system investigated in the present work is nonHermitian, as it includes an imaginary potential which phenomenologically models the absorption of the walker in a given vertex of the graph, thus the transport of the walker from its initial state to such vertex. The resulting time evolution is therefore nonunitary, and the probability of finding the walker within the graph is not conserved, it diminishes in time. This effect is related to the transport efficiency of the system, which is defined as the complement to 1 of the walker’s probability of surviving within the graph. The transport efficiency, together with the transfer time, are figures of merit to assess the transport properties of the system. Remarkably, the imaginary potential reduces in time the probability of finding the walker in the graph, without affecting some periodicity in the probability distribution of the walker.
The thesis offers an overview on the theoretical background about CTQWs and complex potentials, as well as on the FORTRAN and bash codes originally developed to numerically implement and analyze the physical system. The main aim of this thesis project is the development of a FORTRAN code to analyze the time evolution of the system and its transport properties implementing (i) different graphs (complete graph, complete bipartite graph, joined complete graph, and the simplex of complete graph), (ii) different initial states, and (iii) different types of noise (random telegraph noise, pink and brown noise). Studying the effects of the noise requires averaging over several different realizations of the time evolution of the system. Such a computation requires the code to run in a cluster of computers.
To test the code, the first step is to reproduce the analytical results already known in literature for the noiseless case, and the present numerical results correctly reproduce them. The second step, i.e, the study in the presence of a timedependent noise, provides the original results of this work. The purpose of introducing the noise is to study a more realistic system including some of the effects that should appear in real physical implementations. Depending on its strength and type, the noise modifies the probability distribution of the walker and the interference pattern over the graph vertices. If the unperturbed system shows a periodicity or a symmetry in the probability distribution, the noise completely breaks it.
Results suggest that noise improves the transport efficiency with respect to the noiseless case, and, on the other hand, that it leads to transport efficiencies which are less dependent on the initial state with respect to the noiseless case.

Abstract
This master thesis project is devoted to the study of the transport properties of continuoustime quantum walks (CTQWs) on different graphs. The focus is on analyzing the effects of different types of environmental noise which are commonly present in real systems and are therefore required to make the theoretical model closer to real physical implementations.
Quantum walks (QWs) are the quantum counterpart of classical random walks. As well as the latter, QWs are investigated in a variety of research areas, e.g., quantum computation, and modeling of transport phenomena, and have different physical implementations. QWs are widely used to develop quantum algorithms, like search algorithms and graph isomorphism.
The Hamiltonian of the system investigated in the present work is nonHermitian, as it includes an imaginary potential which phenomenologically models the absorption of the walker in a given vertex of the graph, thus the transport of the walker from its initial state to such vertex. The resulting time evolution is therefore nonunitary, and the probability of finding the walker within the graph is not conserved, it diminishes in time. This effect is related to the transport efficiency of the system, which is defined as the complement to 1 of the walker’s probability of surviving within the graph. The transport efficiency, together with the transfer time, are figures of merit to assess the transport properties of the system. Remarkably, the imaginary potential reduces in time the probability of finding the walker in the graph, without affecting some periodicity in the probability distribution of the walker.
The thesis offers an overview on the theoretical background about CTQWs and complex potentials, as well as on the FORTRAN and bash codes originally developed to numerically implement and analyze the physical system. The main aim of this thesis project is the development of a FORTRAN code to analyze the time evolution of the system and its transport properties implementing (i) different graphs (complete graph, complete bipartite graph, joined complete graph, and the simplex of complete graph), (ii) different initial states, and (iii) different types of noise (random telegraph noise, pink and brown noise). Studying the effects of the noise requires averaging over several different realizations of the time evolution of the system. Such a computation requires the code to run in a cluster of computers.
To test the code, the first step is to reproduce the analytical results already known in literature for the noiseless case, and the present numerical results correctly reproduce them. The second step, i.e, the study in the presence of a timedependent noise, provides the original results of this work. The purpose of introducing the noise is to study a more realistic system including some of the effects that should appear in real physical implementations. Depending on its strength and type, the noise modifies the probability distribution of the walker and the interference pattern over the graph vertices. If the unperturbed system shows a periodicity or a symmetry in the probability distribution, the noise completely breaks it.
Results suggest that noise improves the transport efficiency with respect to the noiseless case, and, on the other hand, that it leads to transport efficiencies which are less dependent on the initial state with respect to the noiseless case.

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