Predicting arbitrary state properties from single Hamiltonian quench dynamics

Analog quantum simulation is an essential routine for quantum computing and plays a crucial role in studying quantum many-body physics. Typically, the quantum evolution of an analog simulator is largely determined by its physical characteristics, lacking the precise control or versatility of quantum gates. This limitation poses challenges in extracting physical properties on analog quantum simulators, an essential step of quantum simulations. To address this issue, we introduce the Hamiltonian shadow protocol, which uses a single quench Hamiltonian for estimating arbitrary state properties, eliminating the need for ancillary systems and random unitaries. Additionally, we derive the sample complexity of this protocol and show that it performs comparably to the classical shadow protocol. The Hamiltonian shadow protocol does not require sophisticated control and can be applied to a wide range of analog quantum simulators. We demonstrate its utility through numerical demonstrations with Rydberg atom arrays under realistic parameter settings. The new protocol significantly broadens the application of randomized measurements for analog quantum simulators without precise control and ancillary systems.