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

Measurement Optimization in the Variational Quantum Eigensolver Using a Minimum Clique Cover