Triply Efficient Shadow Tomography

We investigate the problem of efficiently estimating expectation values of large sets of observables from copies of an unknown many-body quantum state. This task, known as shadow tomography, is crucial for quantum simulation and quantum chemistry, where the relevant observables often include 𝑘-body fermionic or 𝑘-local Pauli operators. A key goal is to achieve sample efficiency, computational efficiency, and few-copy measurements. We introduce the notion of triply efficient shadow tomography to formalize these requirements. Prior work has shown that single-copy measurements suffice for 𝑘-local Pauli operators with constant 𝑘. However, we prove that for 𝑘-body fermionic observables and the full set of 𝑛-qubit Pauli operators, single-copy protocols fail to achieve sample-efficient tomography. We then present new protocols based on measuring two copies of the state at a time that overcome these lower bounds, providing a triply efficient protocol.