## 14.8 Code distance and thresholds

Given an error model in which, in principle, all Pauli errors are possible but low-weight^{310} errors are more likely than the high-weight errors, it makes perfect sense to look for an error correcting a code which can perfectly correct errors with weight at most **logical error probability**.
The goal of quantum error correction is to use all the tricks we have discussed so far (and many more) to realise *logical* qubits with logical error rates *below* the error rate of the constituent *physical* qubits.

As in the case of classical codes, the **distance** of a quantum code is defined as the minimum weight error that can go undetected by the code.
In other words, it is the minimum weight Pauli operator than can transform one codeword state into another.
But as we’ve seen, all such operators are in *all* possible Pauli errors, but only those of weight at most

Firstly, note that, if we take a product of two errors

Needless to say, from the perspective of code distance alone, the larger the value of

The **threshold theorem** for stabiliser codes asserts that if the physical error probability *could theoretically* suppress the logical error rate indefinitely.
However, if the physical error rate *greater than* the threshold value

As of 2024^{311}, the upper bound for this threshold value is approximately

Recall that the weight

|P| of a Pauli operatorP=P_1\otimes\ldots\otimes P_n is the number of non-identityP_i . For example,\mathbf{1}\mathbf{1}\mathbf{1} has weight0 ,Z\mathbf{1}\mathbf{1} and\mathbf{1}X\mathbf{1} have weight1 , andXXX has weight3 .↩︎Giving precise numbers is precarious due to the rapid advancements in quantum error correction technology.↩︎