  Symmetry is an important and unifying notion in many areas of physics. In
quantum mechanics, it is possible to eliminate degrees of freedom from a system
by leveraging symmetry to identify the possible physical transitions. This
allows us to simplify calculations and characterize potentially complicated
dynamics of the system with relative ease. Previous works have focused on
devising quantum algorithms to ascertain symmetries by means of fidelity-based
symmetry measures. In our present work, we develop alternative symmetry testing
quantum algorithms that are efficiently implementable on quantum computers. Our
approach estimates asymmetry measures based on the Hilbert--Schmidt distance,
which is significantly easier, in a computational sense, than using fidelity as
a metric. The method is derived to measure symmetries of states, channels,
Lindbladians, and measurements. We apply this method to a number of scenarios
involving open quantum systems, including the amplitude damping channel and a
spin chain, and we test for symmetries within and outside the finite symmetry
group of the Hamiltonian and Lindblad operators.
