Abstract
Quantum error correction must carefully balance two critical tasks: efficient decoding of errors and fault-tolerant operations on encoded qubits. While codes with sparse factor graphs are well-matched to the former task, algebraic codes are often the answers for the latter task. In particular, for performing a universal set of logical gates on the encoded qubits of a quantum stabilizer code, it is necessary to fault-tolerantly realize all Clifford gates and at least one non-Clifford logical gate such as the T gate. For many codes, there are efficient methods to perform the former whereas the latter is achieved through a process called magic state distillation and injection. This process requires codes with very specific properties that guarantee fault-tolerance. After introducing the key background and motivation, we will discuss these properties in terms of the mathematical conditions. We will supplement this with a few examples that show why algebraic codes are well-suited for this problem. In contrast, we will highlight how codes with sparse graphs are appropriate for the decoding task. Finally, we will discuss an important open problem in the field that connects sparse graphs with algebraic codes and argue that a positive answer will represent groundbreaking progress in the field.