  The Variational Quantum Eigensolver approach to the electronic structure
problem on a quantum computer involves measurement of the Hamiltonian
expectation value. Formally, quantum mechanics allows one to measure all
mutually commuting or compatible operators simultaneously. Unfortunately, the
current hardware permits measuring only a much more limited subset of operators
that share a common tensor product eigen-basis. We introduce unitary
transformations that transform any fully commuting group of operators to a
group that can be measured on current hardware. These unitary operations can be
encoded as a sequence of Clifford gates and let us not only measure much larger
groups of terms but also to obtain these groups efficiently on a classical
computer. The problem of finding the minimum number of fully commuting groups
of terms covering the whole Hamiltonian is found to be equivalent to the
minimum clique cover problem for a graph representing Hamiltonian terms as
vertices and commutativity between them as edges. Tested on few molecular
electronic Hamiltonians with up to 50 thousand terms, the introduced technique
allows for the reduction of the number of separately measurable operator groups
down to few hundreds, thus achieving up to 2 orders of magnitude reduction. It
was also shown that this gain scales at least linearly with the number of
qubits.
