Unitary Partitioning Approach to the Measurement Problem in the Variational Quantum Eigensolver Method

To obtain estimates of electronic energies, the Variational Quantum Eigensolver (VQE) technique performs separate measurements for multiple parts of the system Hamiltonian. Current quantum hardware is restricted to projective single-qubit measurements, and, thus, only parts of the Hamiltonian that form mutually qubit-wise commuting groups can be measured simultaneously. The number of such groups in the electronic structure Hamiltonians grows as N-4, where N is the number of qubits, thereby putting serious restrictions on the size of the systems that can be studied. Using a partitioning of the system Hamiltonian as a linear combination of unitary operators, we found a circuit formulation of the VQE algorithm that allows one to measure a group of fully anticommuting terms of the Hamiltonian in a single series of single-qubit measurements. Numerical comparison of the unitary partitioning to previously used grouping of Hamiltonian terms based on their qubit-wise commutativity is consistent with an N-fold reduction in the number of measurable groups.