Efficiently estimating observables of varying locality using tensor-network-inspired classical shadows

Classical shadow tomography is a powerful randomized measurement technique for estimating $M$ state observables with $O(log M)$ measurements. While global Clifford randomized measurements are sample-optimal for some learning tasks, their implementation remains experimentally inaccessible for large systems because they require a number of gates quadratic in the system size. We show how to recover some aspects of global Clifford scaling using far fewer gates with tensor-network-inspired classical shadows. We consider two implementations: projecting the state of interest into a random matrix product state (MPS) or a random multiscale entanglement renormalization ansatz (MERA) state. We present numerical and analytic evidence that, when estimating observables across a range of scales, these tensor network structures can achieve shadow norms (a figure of merit directly related to measurement sample complexity) with more favorable scaling than the best known bounds on existing shadow schemes. For example, the best-known scheme for learning k-local Pauli observables is shallow shadows, whose variance is upper bounded by $2.28^k$. Both of our implementations outperform this bound, and we provide a framework for implementing our sample-efficient measurements in neutral atom arrays. Our work brings the advantages of randomized measurement protocols closer to experimental realization and highlights the power of making randomized measurements at multiple scales.