A unitary coupled cluster (UCC) form for the wavefunction in the variational quantum eigensolver has been suggested as a systematic way to go beyond the mean-field approximation and include electron correlation in solving quantum chemistry problems on a quantum computer. Although being exact in the limit of including all possible coupled cluster excitations, practically, the accuracy of this approach depends on how many and what kind of terms are included in the wavefunction parametrization. Another difficulty of UCC is a growth of the number of simultaneously entangled qubits even at the fixed fermionic excitation rank. Not all quantum computing architectures can cope with this growth. To address both problems we introduce a qubit coupled cluster (QCC) method that starts directly in the qubit space and uses energy response estimates for ranking the importance of individual entanglers for the variational energy minimization. Also, we provide an exact factorization of a unitary rotation of more than two qubits to a product of two-qubit unitary rotations. Thus, the QCC method with the factorization technique can be limited to only two-qubit entanglement gates and allows for very efficient use of quantum resources in terms of the number of coupled cluster operators. The method performance is illustrated by calculating ground-state potential energy curves of H2 and LiH molecules with chemical accuracy, ≤ 1 kcal/mol.