Randomized Benchmarking: Gate Fidelity Decay

Randomized benchmarking is the workhorse protocol behind quantum-hardware certification and every serious quantum-computing training curriculum: apply a random sequence of Clifford gates to a qubit, follow it with the exact inverse rotation, and measure how often the qubit still lands back on |0⟩. In a perfect device it always would; real noise shrinks the Bloch vector a little on every gate, so the survival probability decays exponentially with sequence length. This simulator runs the real protocol on a 3D Bloch sphere — animating single sequences, averaging many trials into a final-state point cloud, and sweeping sequence length to fit the exponential decay curve P(m) = A·p^m + B, exactly the calculation that turns a decay constant into the single "average gate fidelity" number quoted on every quantum processor's spec sheet.