Code distance and thresholds
Given an error model in which, in principle, all Pauli errors are possible but low-weight 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 t for some “good” value of t.
Such a code will fail will with probability roughly equal to the total probability of any error of weight larger than t occurring.
This probability of failure is called the 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 N(\mathcal{S})\setminus\mathcal{S}, which means that
d
= \min_{P\in N(\mathcal{S})\setminus\mathcal{S}}|P|.
Now our goal is less ambitious: we are not aiming to correct all possible Pauli errors, but only those of weight at most t, where t satisfies d=2t+1.
So how can a code with distance d do this?
Firstly, note that, if we take a product of two errors E_i and E_j, each of weight at most t, then the resulting Pauli operator E_iE_j will have weight at most 2t, and by definition 2t<d.
Therefore the product of these errors can never be a logical operator, since the logical operators in N(\mathcal{S})\setminus\mathcal{S} have weight at least d.
Thus if one of these errors E_i occurs and our decoding procedure picks another error E_j that gives rise to the same syndrome (i.e. that belongs to the same error family) and applies the latter to the encoded qubits, then we know that E_iE_j\not\in N(\mathcal{S})\setminus\mathcal{S}, which means that E_iE_j\in\mathcal{S} acts as the identity on the codespace.
Needless to say, from the perspective of code distance alone, the larger the value of d the better we can correct for more errors.
For this, we need the logical errors (i.e. the logical operations on the codespace L\in N(\mathcal{S})\setminus\mathcal{S}) to have the largest possible weight — by our assumptions about our error model, these occur with low probability, and thus keep the logical error probability low.
The threshold theorem for stabiliser codes asserts that if the physical error probability p of individual qubits is below a certain threshold value p_\mathrm{th} then increasing the distance of the code will decrease the logical error probability.
This principle implies that quantum error-correction codes could theoretically suppress the logical error rate indefinitely.
However, if the physical error rate p is greater than the threshold value p_\mathrm{th}, then quantum encoding actually becomes counterproductive.
So the threshold value serves as a critical experimental benchmark for quantum computing experiments, since achieving it is essential for the feasibility of quantum error correction.
We will return to the threshold theorem in more detail in Chapter 15.
As of 2024, the upper bound for this threshold value is approximately p_\mathrm{th}=0.1.