From Quantum Error Detection to Error Correction

In the independent depolarizing error model with small probability $p$ of depolarization, the most likely errors are those that affect a small number of qubits. That is, if we define the ``weight'' of a product of Pauli operators to be the number of qubits affected, the dominant errors are those of small weight. Because the probability of a non-identity Pauli operator is $3p/4$ (see Eq. 25), one expects about ${3p\over 4} n$ of $n$ qubits to be changed. As a result, good error-correcting codes are considered to be those for which all errors of weight $\leq e \simeq {3p\over 4}n$ can be corrected. It is desirable that $e$ have a high ``rate'', which means that it is a large fraction of the total number of qubits, $n$ (the ``length'' of the code). Combinatorially, good codes are characterized by a high minimum distance, a concept that arises naturally in the context of error-detection.