This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ech T Press Science DOI: 10.32604/jqc.2022.033712 Article New Quantum Color Codes Based on Hyperbolic Geometry Avaz Naghipour 1 , * and Duc Manh Nguyen 2 1 Department of Computer Engineering, University College of Nabi Akram, Tabriz, Iran 2 CIT Lab, University of Ulsan, Ulsan, 44610, Korea *Corresponding Author: Avaz Naghipour. Email: naghipour@ucna.ac.ir Received: 25 June 2022; Accepted: 14 April 2023; Published: 15 May 2023 Abstract: In this paper, hyperbolic geometry is used to constructing new quantum color codes. We use hyperbolic tessellations and hyperbolic polygons to obtain them by pairing the edges on compact surfaces. These codes have minimum distance of at least 4 and the encoding rate near to 1, which are not mentioned in other literature. Finally, a comparison table with quantum codes recently proposed by the authors is provided. Keywords: Color codes; compact surfaces; hyperbolic geometry; tessellations 1 Introduction Channel coding theory is one of the widely used branches of telecommunication, whose purpose is to send information from the sender to the receiver through a physical channel with disturbance. Since the foundation of this theory by Claude Shannon in [1], many efforts have been made to achieve the desired codes and famous codes such as Hamming codes, Golay codes, Reed-Muller codes, convolutional codes, BCH codes, Reed-Solomon codes, turbo codes, and finally Low Density Parity Check (LDPC) codes were proposed. While researching and examining classical codes, researchers also showed interest in quantum codes and in the last few decades, various types of quantum codes have been presented with different methods in the literature. Since the introduction of the first quantum error-correcting code by Shor in [2], Calderbank et al. [3] introduced a systematic way for constructing the QECs from classical error-correcting code. The problem of constructing toric quantum codes has motivated considerable interest in the literature. This problem was generalized within the context of surface codes [4] and color codes [5]. The most popular toric code was proposed for the first time by Kitaev’s [6]. This code defined on a square lattice of size m × m on the torus. Leslie proposed a new type of sparse CSS quantum error correcting codes based on the homology of hypermaps defined on an m × m square lattice [7]. The parameters of hypermap-homology codes are  3 2  m 2 , 2, m  . These codes are more efficient than Kitaev’s toric codes. This seemed suggests good quantum that is constructed by using hypergraphs. But there are other surface codes with better parameters than the [[2m 2 , 2, m]] toric code. There exist surface codes with parameters [[m 2 + 1, 2, m]], called homological