The classical-shadow formalism is a randomized measurement protocol in quantum information science for producing a succinct classical representation of an unknown quantum many-body state from few measurements, such that a large number of the state's properties can be estimated with rigorous error and success-probability guarantees. Its central result establishes that estimating M observables to constant error requires only order log M copies of the state, with a sample-complexity prefactor governed by the observable's shadow norm; for suitable ensembles (random Clifford or random single-qubit Pauli measurements) this norm is bounded, making sample complexity independent of system size for observables such as fidelity or low-weight local operators. The subject sits within quantum estimation and learning theory, generalizing and improving on full-state tomography and shadow tomography, and it interfaces with classical machine learning to yield provably efficient algorithms for predicting ground-state properties and classifying quantum phases of matter.
The classical-shadow formalism is a randomized measurement protocol in quantum information science for producing a succinct classical representation of an unknown quantum many-body state from few mea…