Computare Errare Est: Quantum Error Correction

Abstract

We show how to diagnose and rehabilitate bit-flip errors in a classical error correcting model. We compare logical vs. physical bits, define codewords and introduce a classical error correcting model. Quantum codes must take into account the no-cloning theorem, the collapse hypothesis, and the possibility of continuous errors. We present encoding, syndrome measurement, and recovery circuits for single qubit bit-flip and phase shift errors. We review the Shor code and the role that stabilizers play in its implementation. We illustrate the use of the stabilizer formalism in the analysis of quantum error-correcting codes (QECC). We discuss the threshold theorem and its role in allowing for fault-tolerant quantum computing.