  A major challenge for the realisation of useful universal quantum computers
is achieving high fidelity two-qubit entangling gate operations. However,
calibration errors can affect the quantum gate operations and limit their
fidelity. To reduce such errors it is desirable to have an analytical
understanding and quantitative predictions of the effects that miscalibrations
of gate parameters have on the gate performance. In this work, we study a
systematic perturbative expansion in miscalibrated parameters of the
Molmer-Sorensen entangling gate, which is widely used in trapped ion quantum
processors. Our analytical treatment particularly focuses on systematic center
line detuning miscalibrations. Via a unitary Magnus expansion, we compute the
gate evolution operator which allows us to obtain relevant key properties such
as relative phases, electronic populations, quantum state purity and
fidelities. These quantities, subsequently, are used to assess the performance
of the gate using the fidelity of entangled states as performance metric. We
verify the predictions from our model by benchmarking them against measurements
in a trapped-ion quantum processor. The method and the results presented here
can help design and calibrate high-fidelity gate operations of large-scale
quantum computers.
