Classical shadows for sample-efficient measurements of gauge-invariant observables

Abstract: Classical shadows provide a versatile framework for estimating many properties of quantum states from repeated, randomly chosen measurements without requiring full quantum state tomography. When prior information is available, such as knowledge of symmetries of states and operators, this knowledge can be exploited to significantly improve sample efficiency. In this work, we develop three classical shadow protocols tailored to systems with local (or gauge) symmetries to enable efficient prediction of gauge-invariant observables in lattice gauge theory models which are currently at the forefront of quantum simulation efforts. For such models, our approaches can offer exponential improvements in sample complexity over symmetry-agnostic methods, albeit at the cost of increased circuit complexity. We demonstrate these trade-offs using a $\mathbb{Z}_2$ lattice gauge theory, where a dual formulation enables a rigorous analysis of resource requirements, including both circuit depth and sample complexity.