Amplitude Estimation

Amplitude estimation determines the amplitude (and therefore probability) associated with a specific outcome of a quantum circuit using quadratically fewer circuit evaluations than classical Monte Carlo sampling would need for the same statistical precision, the same quadratic speedup pattern that Grover's algorithm exploits for search, and in fact the original algorithm builds directly on Grover's amplitude amplification technique combined with quantum phase estimation. Where Monte Carlo methods reduce estimation error proportionally to 1/√N for N samples, amplitude estimation reduces error proportionally to 1/N for N calls to the underlying circuit, which is the source of the proposed advantage. This is the theoretical basis for most proposed quantum finance applications, particularly derivative pricing and portfolio risk estimation, both of which reduce to Monte Carlo estimation classically. The original algorithm requires the same deep, phase-estimation-based circuits that make near-term hardware struggle, which is why most current implementations use iterative or amplitude-amplification-only variants that trade some of the theoretical speedup for circuits shallow enough to run on NISQ devices.