Predicting features of complex, large-scale quantum systems is essential to the characterization and engineering of quantum architectures. We present an efficient approach for predicting a large number of linear features using classical shadows obtained from very few quantum measurements. This approach is guaranteed to accurately predict @math linear functions with bounded Hilbert-Schmidt norm from only @math measurement repetitions. This sampling rate is completely independent of the system size and saturates fundamental lower bounds from information theory. 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.
Restricting attention to highly structured subsets of quantum states sometimes allows for overcoming the exponential bottleneck that plagues general tomography. Matrix product state (MPS) tomography @cite_41 is the most prominent example for such an approach. It only requires a polynomial number of samples, provided that the underlying quantum state is well approximated by a MPS with low bond dimension. In quantum many body physics this assumption is often justifiable @cite_54 . However, MPS representations of general states have exponentially large bond dimension. In this case, MPS tomography offers no advantage over general tomography.
@cite_41 @cite_54
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Direct quantum state tomography—deducing the state of a system from measurements—is mostly unfeasible due to the exponential scaling of measurement number with system size. The authors present two new schemes, which scale linearly in this respect, and can be applied to a wide range of quantum states. Traditionally quantum state tomography is used to characterize a quantum state, but it becomes exponentially hard with the system size. An alternative technique, matrix product state tomography, is shown to work well in practical situations.