An important question in quantum simulation is the certication of the quantumsimulators with proper readout. We examine how a detector's correlator changes
when coupled to a quantum simulator using a diagrammatic technique. From the
correlation functions calculated from the diagrammatic technique, we can determine
whether or not reliable detection of the simulator's correlator can be achieved. When
reliable detection is not possible due to detector back-action, we examine the situations
when the back-action can be negligible. In particular, we study a cavity detector
coupled to a Transverse Field Ising Model. We use a similar diagrammatic technique
to study the interaction between a cavity and a qubit in the ultrastrong coupling
regime. This cavity-qubit system is of importance in quantum computing and is a
fundamental model in cavity QED. Ultrastrong coupling strength enables novel approaches
for quantum logic operations. Our approach provides a fresh perspective
on calculating the transmission spectra and the imacpt of the ultrastrongly coupled
cavity on the qubit behavior.