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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.

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Tensor network theory and quantum simulation are, respectively, the key classical and quantum computing methods in understanding quantum many-body physics. Here, we introduce the framework of hybrid tensor networks with building blocks consisting of measurable quantum states and classically contractable tensors, inheriting both their distinct features in efficient representation of many-body wave functions. With the example of hybrid tree tensor networks, we demonstrate efficient quantum simulation using a quantum computer whose size is significantly smaller than the one of the target system. We numerically benchmark our method for finding the ground state of 1D and 2D spin systems of up to 8 × 8 and 9 × 8 qubits with operations only acting on 8 þ 1 and 9 þ 1 qubits, respectively. Our approach sheds light on simulation of large practical problems with intermediate-scale quantum computers, with potential applications in chemistry, quantum many-body physics, quantum field theory, and quantum gravity thought experiments.