  We calculate the correlation functions of strings of spin operators for
integrable quantum circuits exactly. These observables can be used for
calibration of quantum simulation platforms. We use algebraic Bethe Ansatz, in
combination with computational algebraic geometry to obtain analytic results
for medium-size (around 10-20 qubits) quantum circuits. The results are
rational functions of the quantum circuit parameters. We obtain analytic
results for such correlation functions both in the real space and Fourier
space. In the real space, we analyze the short time and long time limit of the
correlation functions. In Fourier space, we obtain analytic results in
different parameter regimes, which exhibit qualitatively different behaviors.
Using these analytic results, one can easily generate numerical data to
arbitrary precision.
