Noisy Tensor Ring approximation for computing gradients of Variational Quantum Eigensolver for Combinatorial Optimization

TL;DR

This paper introduces a tensor ring approximation method to efficiently compute gradients in Variational Quantum Eigensolver algorithms, addressing scalability issues by reducing computational complexity.

Contribution

It proposes a novel classical gradient computation technique using tensor ring approximation that preserves structure and scales linearly with qubits and gates.

Findings

Reduces classical simulation complexity from exponential to linear growth.

Enables faster gradient evaluation for VQE on classical simulators.

Maintains accuracy by truncating singular values during tensor contractions.

Abstract

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.