  The success of quantum physics in description of various physical interaction
phenomena relies primarily on the accuracy of analytical methods used. In
quantum mechanics, many of such interactions such as those found in quantum
optomechanics and quantum computing have a highly nonlinear nature, which makes
their analysis extraordinarily difficult using classical schemes. Typically,
modern quantum systems of interest nowadays come with four basic properties:
(i) quantumness, (ii) openness, (iii) randomness, and (iv) nonlinearity. The
newly introduced method of higher-order operators targets analytical solutions
to such systems, and while providing at least mathematically approximate
expressions with improved accuracy over the fully linearized schemes, some
cases admit exact solutions. Many different applications of this method in
quantum and classically nonlinear systems are demonstrated throughout. This
review is purposed to provide the reader with ease of access to this recent and
well-established operator algebra, while going over a moderate amount of
literature review. The reader with basic knowledge of quantum mechanics and
quantum noise theory should be able to start using this scheme to his or her
own problem of interest.
