Analysis of graph-based quantum error-correcting codes

With the advent of quantum computers, there has been a growing interest in the practicality of this device. Due to the delicate conditions that surround physical qubits, one could wonder whether any useful computation could be implemented on such devices. As we describe in this work, it is possible to exploit concepts from classical information theory and employ quantum error-correcting techniques. Thanks to the Threshold Theorem, if the error probability of physical qubits is below a given threshold, then the logical error probability corresponding to the encoded data qubit can be arbitrarily low. To this end, we describe decoherence which is the phenomenon that quantum bits are subject to and is the main source of errors in quantum memories. From the cause of error of a single qubit, we then introduce the error models that can be used to analyze quantum error-correcting codes as a whole. The main type of code that we studied comes from the family of topological codes and is called surface code. Of these codes, we consider both the toric and planar structures. We then introduce a variation of the standard planar surface code which better captures the symmetries of the code architecture. Once the main properties of surface codes have been discussed, we give an overview of the working principles of the algorithm used to decode this type of topological code: the minimum weight perfect matching. Finally, we show the performance of the surface codes that we introduced, comparing them based on their architecture and properties. These simulations have been performed with different error channel models to give a more thorough description of their performance in several situations showing relevant results.