Received 2 December 2022, accepted 14 January 2023, date of publication 18 January 2023, date of current version 24 January 2023. Digital Object Identifier 10.1109/ACCESS.2023.3237997 Shortest Path Finding in Quantum Networks With Quasi-Linear Complexity SARA SANTOS 1 , FRANCISCO A. MONTEIRO 1,2 , (Member, IEEE), BRUNO C. COUTINHO 1 , AND YASSER OMAR 3,4,5 1 Instituto de Telecomunicações, 1049-001 Lisbon, Portugal 2 ISCTE—Instituto Universitário de Lisboa, 1649-026 Lisbon, Portugal 3 Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal 4 Physics of Information and Quantum Technologies Group, Centro de Física e Engenharia de Materiais Avançados (CeFEMA), 1049-001 Lisbon, Portugal 5 PQI—Portuguese Quantum Institute, 1049-001 Lisbon, Portugal Corresponding author: Francisco A. Monteiro (francisco.monteiro@lx.it.pt) This work was supported in part by the Project Quantum Internet Alliance (QIA) of the European Union’s Horizon 2020 Research and Innovation Program under Grant 820445; in part by the John Templeton Foundation (JTF) Project NQuN under Grant 60478; and in part by the Instituto de Telecomunicações and Fundação para a Ciência e Tecnologia/Ministério da Ciência, Tecnologia e Ensino Superior (FCT/MCTES), Portugal, through National Funds co-funded European Union (EU) Funds under Project UIDB/50008/2020 and Project UIDB/04540/2020. The work of Sara Santos was supported by the Fundação Calouste Gulbenkian through the Program New Talents in Quantum Technologies. The work of Bruno C. Coutinho was supported by FCT under Project CEECINST/00117/2018/CP1495. ABSTRACT A fully-quantum network implies the creation of quantum entanglement between a given source node and some other destination node, with a number of quantum repeaters in between. This paper tackles the problem of quantum entanglement distribution by solving the routing problem over an infrastructure based on quantum repeaters and with a fnite number of pairs of entangled qubits available in each link. The network model considers that link purifcation is available such that a nested purifcation protocol can be applied at each link to generate entangled qubits with higher fdelity than the original ones. A low-complexity multi- objective routing algorithm to fnd the shortest path between any two given nodes is proposed and assessed for random networks, using a fairly general path extension mechanism that can ft a large family of particular technological requirements. Different types of quantum protocols require different levels of fdelity for the entangled qubit pairs. For that reason, the proposed algorithm identifes the shortest path between two nodes that assures an end-to-end fdelity above a specifed threshold. The minimum requirements for the end-to- end entanglement fdelity depend on the whole extension of the paths, and cannot be looked at as a local property of each link. Moreover, one needs to keep track not only of the shortest path, but also of longer paths holding more entangled qubits than the shorter paths in order to satisfy the fdelity criterion. Thus, standard single parameter shortest-path algorithms do not necessarily converge to the optimal solution. The problem of fnding the best path in a network subject to multiple criteria (known as multi-objective routing) is, in general, an NP-hard problem due to the rapid growth of the number of stored paths. This work proposes a metric that identifes and discards paths that are objectively worse than others. By doing so, the time complexity of the proposed algorithm scales near-to-linearly with respect to the number of nodes in the network, showing that the shortest-path problem in quantum networks can be solved with a complexity very close to the one of the classical counterparts. That is analytically proved for the case where all the links of a path have the same fdelity (homogeneous model). The algorithm is also adapted to a particular type of path extension, where different links along a path can be purifed to different degrees, asserting its fexibility and near-to-linearity even when heterogeneous fdelities along the sections of a path are considered. INDEX TERMS Quantum networks, quantum repeaters, path-fnding algorithm, end-to-end fdelity, multi- objective routing. The associate editor coordinating the review of this manuscript and approving it for publication was Bijoy Chand Chatterjee . I. INTRODUCTION Quantum technologies are presently undergoing a fast devel- opment, particularly in the areas of quantum computing, 7180 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ VOLUME 11, 2023