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Quantum algorithms designed for noisy intermediate-scale quantum devices usually require repeatedly perform a large number of quantum measurements in estimating observable expectation values of a many-qubit quantum state. Exploiting the ideas of importance sampling, observable compatibility, and classical shadows of quantum states, different advanced quantum measurement schemes have been proposed to greatly reduce the large measurement cost. Yet, the underline cost reduction mechanisms seem distinct to each other, and how to systematically find the optimal scheme remains a critical theoretical challenge. Here, we address this challenge by firstly proposing a unified framework of quantum measurements, incorporating the advanced measurement methods as special cases. Our framework further allows us to introduce a general scheme -- overlapped grouping measurement, which simultaneously exploits the advantages of the existing methods. We show that an optimal measurement scheme corresponds to partitioning the observables into overlapped groups with each group consisting of compatible ones. We provide explicit grouping strategies and numerically verify its performance for different molecular Hamiltonians. Our numerical results show great improvements to the overall existing measurement schemes. Our work paves the way for efficient quantum measurement with near-term quantum devices.
The main results on quantum walk search are scattered over different, incomparable frameworks, most notably the hitting time framework, originally by Szegedy, the electric network framework by Belovs, and the MNRS framework by Magniez, Nayak, Roland
We introduce a framework for simulating quantum measurements based on classical processing of a set of accessible measurements. Well-known concepts such as joint measurability and projective simulability naturally emerge as particular cases of our fr
Experimental determination of an unknown quantum state usually requires several incompatible measurements. However, it is also possible to determine the full quantum state from a single, repeated measurement. For this purpose, the quantum system whos
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