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. Lately, quantum computers have been used to explore symmetries of the physical systems they evolve. Previous works have focused on devising quantum algorithms to ascertain symmetries by means of fidelity-based symmetry measures. Presented in this work are alternative symmetry-testing quantum algorithms that are efficiently implementable on quantum computers. The proposed 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. It is applied to a number of scenarios involving open quantum systems, including the amplitude damping channel and a spin chain, and symmetry tests are performed both within and outside the finite symmetry group of the Hamiltonian and Lindblad operators.