Amplitudes

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.

Daniel F. Styer

Biswajit Sen, Swapan Mandal

Coherent light coupled to a third-order nonlinear medium is responsible for spontaneous and for stimulated Raman processes. The analytical solutions up to second order in coupling constants of various field modes for spontaneous Raman and for stimulated Raman processes are utilized to obtain the amplitude-squared and amplitude-cubed squeezing of the input coherent light. The amplitude-squared and amplitude-cubed squeezing are completely ruled out for all modes for spontaneous Raman processes. For stimulated Raman processes, we observe that the squeezing for amplitude-squared and amplitude-cubed are possible for pump and for vibrational phonon modes. It is found that the percentage of higher ordered amplitude squeezing decreases with the increase of the orders. The squeezing in Stokes and in anti-Stokes modes are ruled out even for stimulated Raman processes.

Stanley Gudder

Lov K. Grover

Quantum search/amplitude amplification algorithms are designed to be able to amplify the amplitude in the target state linearly with the number of operations. Since the probability is the square of the amplitude, this results in the success probability rising quadratically with the number of operations. This paper presents a new kind of quantum search algorithm in which the amplitude of the target state, itself increases quadratically with the number of operations. However, the domain of applications of this is much more limited than standard amplitude amplification.

Stanley Gudder