Informationally complete measurements and tensor networks for high-precision measurements

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Abstract:

Accurate estimation of quantum observables is a fundamental challenge in quantum computing. In this talk, we explore a novel approach to providing unbiased estimators of multiple observables with low statistical error by leveraging informationally (over)complete measurements and tensor networks. The technique involves an observable-specific classical optimization of measurement data using tensor networks, leading to low-variance estimations. Compared to other observable estimation protocols based on classical shadows and measurement frames, our approach offers several advantages: (i) it can be optimized to achieve lower statistical error, reducing the measurement budget required for a given estimation precision; (ii) it scales efficiently to large numbers of qubits due to the tensor network structure; (iii) it applies to any measurement protocol with operators that admit an efficient tensor network representation. We benchmark the method through various numerical examples, including spin and chemical systems in both infinite- and finite-statistics scenarios, and demonstrate how optimal estimation can be achieved even with tensor networks of low bond dimension. .

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