Low variance estimations of many observables with tensor networks and informationally-complete measurements

Accurately estimating the properties of quantum systems is a central challenge in quantum computing and quantum information. We propose a method to obtain unbiased estimators of multiple observables with low statistical error by post-processing informationally complete measurements using tensor networks. Compared to other observable estimation protocols based on classical shadows and measurement frames, our approach offers several advantages: (i) it can be optimised to provide lower statistical error, resulting in a reduced measurement budget to achieve a specified estimation precision; (ii) it scales to a large number of qubits due to the tensor network structure; (iii) it can be applied to any measurement protocol with measurement operators that have an efficient tensor-network representation. We benchmark the method through various numerical examples, including spin and chemical systems, and show that our method can provide statistical error that are orders of magnitude lower than the ones given by classical shadows.