  Solving the electronic structure problem using the Variational Quantum
Eigensolver (VQE) technique involves measurement of the Hamiltonian expectation
value. Current hardware can perform only projective single-qubit measurements,
and thus, the Hamiltonian expectation value is obtained by measuring parts of
the Hamiltonian rather than the full Hamiltonian. This restriction makes the
measurement process inefficient because the number of terms in the Hamiltonian
grows as $O(N^4)$ with the size of the system, $N$. To optimize VQE measurement
one can try to group as many Hamiltonian terms as possible for their
simultaneous measurement. Single-qubit measurements allow one to group only the
terms that commute within corresponding single-qubit subspaces or qubit-wise
commuting. We found that qubit-wise commutativity between the Hamiltonian terms
can be expressed as a graph and the problem of the optimal grouping is
equivalent of finding a minimum clique cover (MCC) for the Hamiltonian graph.
The MCC problem is NP-hard but there exist several polynomial heuristic
algorithms to solve it approximately. Several of these heuristics were tested
in this work for a set of molecular electronic Hamiltonians. On average,
grouping qubit-wise commuting terms reduced the number of operators to measure
three times compared to the total number of terms in the considered
Hamiltonians.
