The paper presents an efficient method to construct a classical description, called a *classical shadow*, of a quantum state using very few measurements. This classical shadow can be used to predict various properties of the quantum system, such as quantum fidelities, entanglement entropies, two-point correlation functions, and expectation values of local observables. The method is based on applying random unitary transformations and measuring the resulting states in the computational basis. The number of measurements required is independent of the system size and scales logarithmically with the number of target functions to be predicted. The authors provide rigorous performance guarantees and show that the method is optimal up to information-theoretic lower bounds. Numerical experiments demonstrate the effectiveness of classical shadows compared to other methods, such as machine learning-based approaches, in predicting properties of quantum systems. The paper also discusses the application of classical shadows to quantum simulation and variational quantum algorithms.The paper presents an efficient method to construct a classical description, called a *classical shadow*, of a quantum state using very few measurements. This classical shadow can be used to predict various properties of the quantum system, such as quantum fidelities, entanglement entropies, two-point correlation functions, and expectation values of local observables. The method is based on applying random unitary transformations and measuring the resulting states in the computational basis. The number of measurements required is independent of the system size and scales logarithmically with the number of target functions to be predicted. The authors provide rigorous performance guarantees and show that the method is optimal up to information-theoretic lower bounds. Numerical experiments demonstrate the effectiveness of classical shadows compared to other methods, such as machine learning-based approaches, in predicting properties of quantum systems. The paper also discusses the application of classical shadows to quantum simulation and variational quantum algorithms.