Title: An Introduction to the Stabilizer Formalism for Quantum Error Correction
Abstract: Quantum algorithms can provide dramatic speedups over classical information processing for certain computational tasks. However, quantum information is highly susceptible to noise, making error correction essential for building a practical quantum computer. This talk gives provides an introduction to the stabilizer formalism of quantum error correction, a central mathematical framework used to build and understand quantum error-correcting codes. The key idea is to encode the quantum information in a joint eigenspace of a suitably chosen group of commuting operators called the stabilizer group. We will see how algebraic objects such as the centralizer of the stabilizer group (in the Pauli group) are intimately connected to the error-correcting capability of the code. Further, the operators that we consider lend themselves to a convenient binary representation that provides a natural bridge to the well-studied theory of classical error correction. This leads to the Calderbank-Shor-Steane (CSS) framework which constructs quantum stabilizer codes from a pair of suitably chosen classical error-correcting codes. We will specialize the description of general stabilizer codes to CSS codes, and, time permitting, close with a discussion on the important class of quantum low-density parity-check codes. The talk will be aimed at illustrating how algebra and linear algebra combine to solve a very concrete engineering problem: protecting quantum information from noise.