Quantum Amplitude Estimation

The primitive behind quadratic Monte Carlo speedups: reading an amplitude to precision eps in O(1/eps) queries instead of O(1/eps^2).

Estimating a probability amplitude

Amplitude estimation (QAE) is the algorithmic core behind the neutronics speedup and several quantum finance-style estimators. Given an operator A that prepares a state where a 'good' subspace has amplitude sqrt(a), QAE returns an estimate of a. It generalizes Grover search from finding a marked item to measuring how much amplitude sits on the marked subspace.

A|0> = sqrt(1-a) |bad,0> + sqrt(a) |good,1>
# Grover operator:  Q = A S_0 A^dag S_chi   (reflections)
# eigenphases of Q are  +/- 2*theta  with  a = sin^2(theta)
# run phase estimation on Q with m ancilla:
#   estimate theta to ~2^{-m}  =>  estimate a to error O(1/M), M=2^m

The quadratic advantage, made precise

target error eps in the amplitude a:
 classical sampling : M ~ O(1/eps^2)  measurements
 amplitude estimation: M ~ O(1/eps)   applications of Q
# a genuine but only quadratic improvement

Modern low-depth variants

• Textbook QAE uses phase estimation and deep circuits requiring error correction.
• Maximum-likelihood and iterative QAE trade some coherence for shallower circuits, edging toward NISQ feasibility.
• Even these need far cleaner hardware than today's for any real neutronics or chemistry estimator.

Where Kronos would use it

Any expectation value or integrated response computed by Monte Carlo is a candidate: breeding-ratio integrals, activation inventories, dose estimates. Because the speedup is quadratic it only pays off once fault-tolerant depth is cheap. Kronos documents QAE as the shared engine behind neutronics Monte Carlo and behind eigenvalue readout in phase estimation, and validates it on analytically known toy integrals before trusting any nontrivial case.