Amplitude Amplification and Estimation

Marco O. Lanzagorta, Jeffrey K. Uhlmann

Debajyoti Bera

Alexander M. Dalzell, Sam McArdle, Mario Berta +10 more

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.

Yutaka Shikano

Kirill Plekhanov, Matthias Rosenkranz, Mattia Fiorentini +1 more

We propose to perform amplitude estimation with the help of constant-depth quantum circuits that variationally approximate states during amplitude amplification. In the context of Monte Carlo (MC) integration, we numerically show that shallow circuits can accurately approximate many amplitude amplification steps. We combine the variational approach with maximum likelihood amplitude estimation [Y. Suzuki et al., Quantum Inf. Process. 19, 75 (2020)] in variational quantum amplitude estimation (VQAE). VQAE typically has larger computational requirements than classical MC sampling. To reduce the variational cost, we propose adaptive VQAE and numerically show in 6 to 12 qubit simulations that it can outperform classical MC sampling.

Giacomo Nannicini

In Chs. 2 and 3 we described several quantum algorithms that are exponentially faster than classical algorithms for the same problem, under some measure of complexity. Unfortunately, the algorithms of Ch. 2 apply to rather contrived problems that do not have known direct applications. We now describe a quantum algorithm that gives only a polynomial—more specifically, quadratic—speedup with respect to classical algorithms, but it applies to a large class of problems and finds many useful applications, both directly and as an algorithmic design tool. The algorithm is known as Grover’s search [89], and its generalization is known as amplitude amplification [43]. Amplitude amplification is widely used as part of many quantum algorithms. It also serves as the basis for amplitude estimation, a way of estimating probabilities quadratically faster than with classical Monte Carlo, and for quantum minimum finding. All these topics are discussed in this chapter.