Predicting Features of Quantum Systems from Very Few Measurements

Predicting features of complex, large-scale quantum systems is essential to the characterization and engineering of quantum architectures. We present an efficient approach for constructing an approximate classical description, called the classical shadow, of a quantum system from very few quantum measurements that can later be used to predict a large collection of features. This approach is guaranteed to accurately predict M observables with bounded Hilbert-Schmidt norm from only order of log(M) measurements. This is completely independent of the system size and saturates fundamental lower bounds from information theory. A distinct realization of the concept can also be used to predict M few-body observables using classical descriptions constructed from single-qubit measurements on log(M) copies of the system. These protocols have applications in variational quantum algorithms, estimating quantum fidelity, or inferring entanglement properties of the quantum system. We support our theoretical findings with numerical experiments over a wide range of problem sizes (2 to 162 qubits). These highlight advantages compared to existing machine learning approaches.