  We propose a qubit efficient scheme to study ground state properties of
quantum many-body systems on near-term noisy intermediate scale quantum
computers. One can obtain a tensor network representation of the ground state
using a number of qubits smaller than the physical degrees of freedom. By
increasing the qubits number, one can exponentially increase the bond dimension
of the tensor network variational ansatz on a quantum computer. Moreover, we
construct circuits blocks which respect U(1) and SU(2) symmetries of the
physical system and show that they can significantly speed up the training
process and alleviate the gradient vanishing problem. To demonstrate the
feasibility of the qubit efficient variational quantum eigensolver in a
practical setting, we perform first principle classical simulation of
differentiable programming of the circuits. Using only $6$ qubits one can
obtain the ground state of a $4\times 4$ square lattice frustrated Heisenberg
model with fidelity over 97%. Arbitrarily long ranged correlations can also be
measured on the same circuit after variational optimization.
