A randomised classifier flags a rare signal correctly with probability on each…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Information & Technology · The quantum circuit model and quantum algorithms

Quantum Information and Technology, Ch Grover search, amplitude amplification and counting

Problem

A randomised classifier flags a rare signal correctly with probability

a=0.02

on each independent run; it can be built into a state-preparation procedure

A

for amplitude amplification, whose good subspace has amplitude

\sqrt{a}

. ((a)) Find the number of amplitude-amplification steps

k

needed for the success probability to first reach

0.999

or more, using

\sin(\theta/2)=\sqrt{a}

and

P(\text{success after } k) = \sin^2\!\left(\frac{(2k+1)\theta}{2}\right)

. ((b)) Find the number of plain classical repetitions of the procedure needed for the success probability

1-(1-a)^{k_c}

to reach

0.99

, and compare with

k

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