Measurement of a quantum system only reveals limited, probabilistic information about the structure of the underlying state. In the early days of the development of quantum theory, Einstein’s discomfort with this probabilistic nature of how we can extract information from the vast state space of a quantum system led him to famously proclaim “God does not play dice.” Somewhat surprisingly, doubling down on the probabilistic nature of quantum mechanics and performing random measurements on copies of a quantum system allows one to efficiently learn many physically relevant properties of a quantum state without having to go through the costly process of learning the full quantum state. Here, I will discuss these so-called randomized measurement protocols and describe our work on how to “weight God’s dice,” leveraging known symmetries of the physical system in question to implement more sophisticated randomized measurement protocols. These symmetry-conscious randomized measurement schemes provide clear advantages over symmetry-blind approaches by reducing measurement costs, enabling symmetry-based error mitigation in experiments, allowing measurement of entanglement structure in (lattice) gauge theories, and, potentially, the verification of so-called topologically ordered states in existing and near-term experiments. Based on arXiv:2303.15519.
