Randomized measurement for lattice gauge theory

Randomized measurement, including techniques like classical shadows and entanglement tomography, form a robust and widely applicable toolset for extracting information from quantum states. In these settings, the choice of random measurement basis determines the cost of extracting certain quantities of interest. With a focus on lattice gauge theory, we show how known symmetries can be leveraged to design more tailored randomized measurement protocols. We analyze both deep and shallow randomizing circuits, and demonstrate how, in both cases, there is a trade-off, when compared to symmetry-ignorant randomization, between the complexity of the randomization operations and the efficiency with which quantities of interest can be extracted. As our primary example, we use Z2 LGT where we can use shallow circuits to measure local observables and deep circuits to detect a phase transition from the trivial to topologically ordered phase.