  Response theory has a successful history of connecting experimental
observations with theoretical predictions. Of particular interest is the
optical response of matter, from which spectroscopy experiments can be
modelled. However, the calculation of response properties for quantum systems
is often prohibitively expensive, especially for nonlinear spectroscopy, as it
requires access to either the time evolution of the system or to excited
states. In this work, we introduce a generalized quantum phase estimation
framework designed for multi-variate phase estimation. This allows the
treatment of general correlation functions enabling the recovery of response
properties of arbitrary orders. The generalized quantum phase estimation
circuit has an intuitive construction that is linked with a physical process of
interest, and can directly sample frequencies from the distribution that would
be obtained experimentally. In addition, we provide a single-ancilla
modification of the new framework for early fault-tolerant quantum computers.
Overall, our framework enables the efficient simulation of spectroscopy
experiments beyond the linear regime, such as Raman spectroscopy. This opens up
an exciting new field of applications for quantum computers with potential
technological impact.
