  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.
