Geometric quantum gates are conjectured to be more resilient than dynamical gates against certain types of error, which makes them ideal for robust quantum computing. However, there are conflicting claims within the literature about the validity of that robustness conjecture. Here we use dynamical invariant theory in conjunction with filter functions in order to analytically characterize the noise sensitivity of an arbitrary quantum gate. Under certain conditions, we find that there exists a transformation of the Hamiltonian that leaves invariant the final gate and noise sensitivity (as characterized by the filter function) while changing the phase from geometric to dynamical. Our result holds for a Hilbert space of arbitrary dimensions, but we illustrate our result by examining experimentally relevant single-qubit scenarios and providing explicit constructions of such a transformation.
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