  Variational Quantum algorithms, especially Quantum Approximate Optimization
and Variational Quantum Eigensolver (VQE) have established their potential to
provide computational advantage in the realm of combinatorial optimization.
However, these algorithms suffer from classically intractable gradients
limiting the scalability. This work addresses the scalability challenge for VQE
by proposing a classical gradient computation method which utilizes the
parameter shift rule but computes the expected values from the circuits using a
tensor ring approximation. The parametrized gates from the circuit transform
the tensor ring by contracting the matrix along the free edges of the tensor
ring. While the single qubit gates do not alter the ring structure, the state
transformations from the two qubit rotations are evaluated by truncating the
singular values thereby preserving the structure of the tensor ring and
reducing the computational complexity. This variation of the Matrix product
state approximation grows linearly in number of qubits and the number of two
qubit gates as opposed to the exponential growth in the classical simulations,
allowing for a faster evaluation of the gradients on classical simulators.
