Measuring expectation values of observables is an essential task in quantum experiments and quantum algorithms, for example, Quantum Neural Networks (QNN), the Variational Quantum Eigensolver (VQE) and the Quantum Approximate Optimization Algorithm (QAOA) . These hybrid quantum-classical algorithms are one of the leading candidates for obtaining quantum advantage in the current era. However, they may require such high precision in estimating the expectation values of observables that makes them impractical due to the many sources of noise in current quantum hardware.
Based on [2] we propose a hybrid quantum-classical algorithm that estimates the expectation value of observables acting on pure states. This algorithm, like the method presented in [2] is effective for concentrated states, i.e., when only a small number of amplitudes in a measurement basis are non-negligible, a property often satisfied for Molecular Hamiltonians. Moreover, this algorithm improves the one presented in [2], since it is also designed to be effective for Hamiltonian depending on Pauli strings that contain a small fixed number of
Our algorithm is implemented in Qiskit and detailed in this tutorial, which provides the theoretical framework and numerical simulations to evaluate its performance.
[2] Kohda, Masaya and Imai, Ryosuke and Kanno, Keita and Mitarai, Kosuke and Mizukami, Wataru and Nakagawa, Yuya O. Quantum expectation-value estimation by computational basis sampling. Phys. Rev. Res, 4:033173, Sep 2022. doi: 10.1103/PhysRevResearch.4.033173.