Vol.:(0123456789) Optical and Quantum Electronics ( 2020) 52:40 https://doi.org/10.1007/s11082-019-2157-5 1 3 Construction of quantum codes from new embeddings of graphs on compact surfaces Avaz Naghipour 1 Received: 24 July 2019 / Accepted: 10 December 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract In this paper we introduce a new class of sparse CSS quantum error-correcting codes based on new orientable self-dual embeddings of graphs. Each code in this class is associated with the embedding of the surface so that the qubits correspond to the edges of the embed- ding. The parameters of the new codes are [[ m(m+1) 2 , m(m-3) 2 ,3]] , where m = 4s, s ≥ 1 . We also present a table of quantum codes whose parameters had not been shown before. Keywords Quantum codes · Embedding · Orientable · self-dual 1 Introduction Quantum error-correcting codes (QECs) has played an important role in quantum informa- tion during the last decades and the importance of this area of study is illustrated by the researchers in diverse areas. Over the years, there has been an increasing interest in construction of QECs as per both experimental and theoretical studies in Engineering (Bacon et al. 2017) and Physics (Nigg et al. 2014). Since the work of Shor (Shor 1995), considerable efort has been dedicated to construction of QECs. Examples are the construction of QECs from classical error-correct- ing code (Calderbank et al. 1998), Kitaev toric codes (Kitaev 2003), color codes (Bombin and Martin-Delgado 2006), surface codes (Bombin and Martin-Delgado 2007), toric quan- tum codes (de Albuquerque et al. 2010), stabilizer codes of distance 3 (Yu et al. 2013), hypermap-homology codes (Leslie 2014) and others (de Albuquerque et al. 2010, 2014). Quantum codes based on the homology of embeddings of graphs have been discussed widely in the literature and our class of quantum codes generalize these. We use embedded graphs which are a generalization of graphs that can have edges connected to more that two vertices. In this work, the most straightforward generalization is self-dual embedding of K 5 in the torus which gives us a class of quantum codes with parameters [[ m(m+1) 2 , m(m-3) 2 ,3]] , where m = 4s, s ≥ 1 . For these quantum codes, the encoding rate is such that k n tends 1 as m goes to infnity. We also present a table of quantum codes which are diferent parameters in * Avaz Naghipour naghipour@ucna.ac.ir 1 Department of Computer Engineering, University College of Nabi Akram, No. 1283, Rah Ahan Street, Tabriz, Iran