Fault Tolerance Metrics

Key parameters used to measure the reliability, accuracy, and error-correction

performance.

Physical Error Rate ((p))	Probability that a physical qubit experiences an error. Smaller values indicate better hardware quality.
Logical Error Rate ((P_L))	Probability that a logical qubit fails after error correction. Fault-tolerant operation requires (P_L ll p).
Code Distance ((d))	Minimum number of physical qubit errors needed to corrupt a logical qubit. Larger code distance provides stronger protection.
QEC Threshold :math:`p_{\mathrm{th}}`	If \(p < p_{\\mathrm{th}}\), QEC improves reliability; if \(p > p_{\\mathrm{th}}\), adding more overhead no longer helps. For the surface code, \(p_{\\mathrm{th}} \\approx 1\\%\). A common scaling model is

\[ \begin{align}\begin{aligned}\begin{split}P_L(d) \\approx A\\left(\\frac{p}{p_{\\mathrm{th}}}\\right)^{(d+1)/2}\end{split}\\ which shows why increasing \(d\) helps only when \(p\) is below threshold.\end{aligned}\end{align} \]