Noisy tensor-ring approximation for computing gradients of a variational quantum eigensolver for combinatorial optimization

Variational quantum algorithms, especially quantum approximate optimization and the variational quantum eigensolver (VQE), have demonstrated 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 the 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.

Automated summary

This research proposes an improved VQE algorithm by utilizing a classical gradient computation method that uses tensor-ring approximation. By truncating singular values and preserving the structure of the tensor ring, this method allows for faster evaluation of gradients on classical simulators, addressing the scalability challenge of VQE.