Measurement optimization in the variational quantum eigensolver using a minimum clique cover

Solving the electronic structure problem using the Variational Quantum Eigensolver (VQE) technique involves the measurement of the Hamiltonian expectation value. The 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(N⁴) with the size of the system, N. To optimize the 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 commuting within the corresponding single-qubit subspaces or qubit-wise commuting. We found that the qubit-wise commutativity between the Hamiltonian terms can be expressed as a graph and the problem of the optimal grouping is equivalent to 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 less compared to the total number of terms in the considered Hamiltonians.