This chapter covers the quantum algorithmic primitives of amplitude amplification and amplitude estimation. Amplitude amplification is a generalization of Grover’s quantum algorithm for the unstructured search problem. Amplitude estimation can be understood in a similar framework, where it utilizes quantum phase estimation to estimate the value of the amplitude or probability associated with a quantum state. Both amplitude amplification and amplitude estimation provide a quadratic speedup over their classical counterparts, and feature prominently as an ingredient in many end-to-end algorithms.This chapter covers the quantum algorithmic primitives of amplitude amplification and amplitude estimation. Amplitude amplification is a generalization of Grover’s quantum algorithm for the unstructured search problem. Amplitude estimation can be understood in a similar framework, where it utilizes quantum phase estimation to estimate the value of the amplitude or probability associated with a quantum state. Both amplitude amplification and amplitude estimation provide a quadratic speedup over their classical counterparts, and feature prominently as an ingredient in many end-to-end algorithms.

Cambridge University Press eBooksApr 24, 20250 citations

Arbitrary phases in quantum amplitude amplification

Peter Høyer

We consider the use of arbitrary phases in quantum amplitude amplification, which is a generalization of quantum searching. We prove that the phase condition in amplitude amplification is given by $\mathrm{tan}(\ensuremath{\varphi}/2)=\mathrm{tan}(\ensuremath{\varphi}/2)(1\ensuremath{-}2a),$ where \ensuremath{\varphi} and \ensuremath{\varphi} are the phases used and where a is the success probability of the given algorithm. Thus the choice of phases depends nontrivially and nonlinearly on the success probability. Utilizing this condition, we give methods for constructing quantum algorithms that succeed with certainty and for implementing arbitrary rotations. We also conclude that phase errors of order up to $1/\sqrt{a}$ can be tolerated in amplitude amplification.We consider the use of arbitrary phases in quantum amplitude amplification, which is a generalization of quantum searching. We prove that the phase condition in amplitude amplification is given by $\mathrm{tan}(\ensuremath{\varphi}/2)=\mathrm{tan}(\ensuremath{\varphi}/2)(1\ensuremath{-}2a),$ where \ensuremath{\varphi} and \ensuremath{\varphi} are the phases used and where a is the success probability of the given algorithm. Thus the choice of phases depends nontrivially and nonlinearly on the success probability. Utilizing this condition, we give methods for constructing quantum algorithms that succeed with certainty and for implementing arbitrary rotations. We also conclude that phase errors of order up to $1/\sqrt{a}$ can be tolerated in amplitude amplification.

arXiv:quant-ph/0006031Physical Review AOct 11, 2000160 citations

Parametric Amplification

Thomas Koryu Ishii

Elsevier eBooksJan 1, 19900 citations

Phase sensitive amplification

Radan Slavı́k

Aug 27, 20140 citations

1/\sqrt{a}

can be tolerated in amplitude amplification.