Fault Tolerance Metrics
=======================

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

.. list-table::
   :widths: 30 70
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   * - **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 :math:`p < p_{\\mathrm{th}}`, QEC improves reliability; if 
                    :math:`p > p_{\\mathrm{th}}`, adding more overhead no longer helps. For the
                    surface code, :math:`p_{\\mathrm{th}} \\approx 1\\%`.

 
                A common scaling model is 
                

.. math::

   P_L(d) \\approx A\\left(\\frac{p}{p_{\\mathrm{th}}}\\right)^{(d+1)/2}

 
                which shows why increasing \(d\) helps only when \(p\) is below threshold.