Efficient Postprocessing Procedure for Evaluating Hamiltonian Expectation Values in Variational Quantum Eigensolver

We proposed a simple strategy to improve the postprocessing overhead of evaluating Hamiltonian expectation values in Variational quantum eigensolvers (VQEs). Observing the fact that for a mutually commuting observable group G in a given Hamiltonian, <b|G|b> is fixed for a measurement outcome bit string $b$ in the corresponding basis, we create a measurement memory (MM) dictionary for every commuting operator group G in a Hamiltonian. Once a measurement outcome bit string $b$ appears, we store $b$ and <b|G|b> as key and value, and the next time the same bit string appears, we can find <b|G|b> from the memory, rather than evaluate it once again. We further analyze the complexity of MM and compare it with commonly employed post-processing procedure, finding that MM is always more efficient in terms of time complexity. We implement this procedure on the task of minimizing a fully connected Ising Hamiltonians up to 20 qubits, and $H_2$, $H_4$, $LiH$, and $H_2O$ molecular Hamiltonians with different grouping methods. For Ising Hamiltonian, where all $O(N^2)$ terms commute, our method offers an $O(N^2)$ speedup in terms of the percentage of time saved. In the case of molecular Hamiltonians, we achieved over $O(N)$ percentage time saved, depending on the grouping method.