Applications of Symmetry Testing in Quantum Algorithms*

In quantum computing, it is often useful to know the symmetries obeyed by a given system or state. For example, Hamiltonian symmetries may limit allowed state transitions or simplify learning parameters in a machine learning application, and quantum states possessing certain types of asymmetry are also known to be potentially resourceful in various applications. Symmetry testing algorithms provide a means to identify and quantify these properties with respect to the representation of a group. Additionally, by making particular choices of groups and representations, these algorithms can be employed to gleam useful information about a state or system. In this work, we review notions of symmetry for states and Hamiltonians, and give algorithms for testing these on a quantum computer. We then demonstrate how these symmetry testing algorithms can be employed for applications such as testing for Werner states, measuring Hamming distance, and quantum machine learning.