Spatial search by continuous-time quantum walks on complex networks

Spatial search by continuous-time quantum walks on complex networks is focused on using a quantum walk in continuous time in order to find a single or multiple marked vertices within the complex network. The specific formalism used here is to consider a coupling constant that shifts the state of the quantum walker from the initial state to the target state, which is the marked vertex. The thesis begins with establishing the mathematical framework of network theory, quantum walks and numerical methods that will be used in the remainder of the thesis. Then spatial search by continuous-time quantum walk is studied on regular and semi-regular graphs, where most analytical results can be found. This will get us acquainted with spatial search by quantum walk. The complex networks studied are Barabasi-Albert graphs and the Internet network on the level of autonomous systems. Different renormalized and pruned versions of the Internet network are studied. The parameters of the quantum walk that are focused on are the optimal values for the coupling constant, success probability, time and search time.Kvanttikulkujen spatiaalinen etsintä jatkuvassa ajassa kompleksisissa verkoissa keskittyy yhden tai useamman merkityn solmukohdan löytämiseen kompleksisesta verkosta käyttämällä kvanttikulkua jatkuvassa ajassa. Tässä työssä käytetty formalismi käsittelee kytkentävakiota, mikä siirtää kvanttikulkijan tilan alkutilasta tavoitetilaan, eli merkittyyn solmukohtaan. Tämä Pro Gradu alkaa matemaattisen viitekehyksen käsittelemisellä, jota tarvitaan lopputyössä. Tämä viitekehys sisältää verkkoteorian, kvanttikulut ja käytetyt numeeriset menetelmät. Tämän jälkeen kvanttikulun spatiaalista etsintää jatkuvassa ajassa tutkitaan säännöllisissä ja miltei säännöllisissä verkoissa, missä analyyttiset ratkaisut on löydettävissä. Tämän tarkoituksena on tutustua spatiaaliseen etsintään kvanttikululla. Barabasi-Albert -graafit ja Internet-verkko autonomisten järjestelmien tasolla ovat tässä työssä tutkittavat kompleksiset verkot. Tässä tutkitaan eri renormalisoituja ja karsittuja versioita Internet-verkosta. Kvanttikulun parametrit, joihin keskitytään, ovat optimaaliset arvot kytkentävakiolle, onnistumistodennäköisyydelle, ajalle ja etsintäajalle

Fermionic decay of a massive scalar in the early Universe

We derive a curved space generalization of a scalar to fermion decay rate with a Yukawa coupling in expanding Friedmann-Robertson-Walker universes. This is done using the full theory of quantum fields in curved spacetime and the added-up transition probability method. It is found that in an expanding universe the usual Minkowskian decay rates are considerably modified for early times. For conformally coupled scalars the decay rate is modified by a positive additive term proportional to the inverse of mass and related to the expansion rate of the Universe. We compare and contrast our results with previous studies on scalar to scalar decay and find that in general the decay channel into fermions is the dominant channel of decay in the very early Universe.Comment: 9 pages, 2 figures. Corrected some minor misprint

Spatial search by continuous-time quantum walks on renormalized Internet networks

Publisher Copyright: © 2022 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.We study spatial search with continuous-time quantum walks on real-world complex networks. We use smaller replicas of the Internet network obtained with a recent geometric renormalization method introduced by García-Pérez, Nat. Phys. 14, 583 (2018)10.1038/s41567-018-0072-5. This allows us to infer for the first time the behavior of a quantum spatial search algorithm on a real-world complex network. By simulating numerically the dynamics and optimizing the coupling parameter, we study the optimality of the algorithm and its scaling with the size of the network, showing that on average it is considerably better than the classical scaling O(N), but it does not reach the ideal quadratic speedup O(N) that can be achieved, e.g. in complete graphs. However, the performance of the search algorithm strongly depends on the degree of the nodes and, in fact, the scaling is found to be very close to optimal when we consider the nodes below the 99th percentile ordered according to the degree.Peer reviewe

Fermionic decay of a massive scalar in the early universe

We derive a curved space generalization of a scalar to fermion decay rate with a Yukawa coupling in expanding Friedmann-Robertson-Walker universes. This is done using the full theory of quantum fields in curved spacetime and the added-up transition probability method. It is found that in an expanding universe the usual Minkowskian decay rates are considerably modified for early times. For conformally coupled scalars the decay rate is modified by a positive additive term proportional to the inverse of mass and related to the expansion rate of the Universe. We compare and contrast our results with previous studies on scalar to scalar decay and find that in general the decay channel into fermions is the dominant channel of decay in the very early Universe

Spatial search by continuous-time quantum walks on renormalized Internet networks

Publisher Copyright: © 2022 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.We study spatial search with continuous-time quantum walks on real-world complex networks. We use smaller replicas of the Internet network obtained with a recent geometric renormalization method introduced by García-Pérez, Nat. Phys. 14, 583 (2018)10.1038/s41567-018-0072-5. This allows us to infer for the first time the behavior of a quantum spatial search algorithm on a real-world complex network. By simulating numerically the dynamics and optimizing the coupling parameter, we study the optimality of the algorithm and its scaling with the size of the network, showing that on average it is considerably better than the classical scaling O(N), but it does not reach the ideal quadratic speedup O(N) that can be achieved, e.g. in complete graphs. However, the performance of the search algorithm strongly depends on the degree of the nodes and, in fact, the scaling is found to be very close to optimal when we consider the nodes below the 99th percentile ordered according to the degree.Peer reviewe

Link Prediction with Continuous-Time Classical and Quantum Walks

Funding Information: M.G., H.S., S.M., and G.G.-P. acknowledge support from the Emmy.network foundation. S.M. and M.A.C.R. acknowledge financial support from the Academy of Finland via the Centre of Excellence program (Project No. 336810 and Project No. 336814). G.G.-P. acknowledges financial support from the Academy of Finland via the Postdoctoral Researcher program (Project No. 341985). Publisher Copyright: © 2023 by the authors.Protein–protein interaction (PPI) networks consist of the physical and/or functional interactions between the proteins of an organism, and they form the basis for the field of network medicine. Since the biophysical and high-throughput methods used to form PPI networks are expensive, time-consuming, and often contain inaccuracies, the resulting networks are usually incomplete. In order to infer missing interactions in these networks, we propose a novel class of link prediction methods based on continuous-time classical and quantum walks. In the case of quantum walks, we examine the usage of both the network adjacency and Laplacian matrices for specifying the walk dynamics. We define a score function based on the corresponding transition probabilities and perform tests on six real-world PPI datasets. Our results show that continuous-time classical random walks and quantum walks using the network adjacency matrix can successfully predict missing protein–protein interactions, with performance rivalling the state-of-the-art.Peer reviewe