Noise Impact on Variational Quantum Algorithms: A Challenge for Quantum Computing

Variational quantum algorithms (VQAs) are a class of algorithms with the potential to solve complex scientific problems on noisy intermediate-scale quantum (NISQ) hardware. They have a higher error tolerance than other quantum algorithms due to their hybrid nature, which allows optimization tasks to be offloaded to a classical computer.

However, implementing VQAs beyond the classical limit is challenging due to the complexity of the non-convex optimization. A recent study investigated the effect of noise on the optimal solution of a VQE algorithm, finding that noise can trigger a transition in the optimal solution, potentially deteriorating the solution’s quality unpredictably.

What are Variational Quantum Algorithms and How Do They Work?

Variational quantum algorithms (VQAs) are a promising class of algorithms that have the potential to demonstrate practical quantum advantage in problems of scientific interest on noisy intermediate-scale quantum (NISQ) hardware. The immense potential of VQAs originates from the shallow variational circuits enabled by their hybrid nature. In these algorithms, the optimization task is offloaded to a classical computer, and the optimization can mitigate certain coherent noises by shifting the cost function landscape according to the noises. This makes VQAs have a much higher error tolerance than other quantum algorithms.

In recent years, there have been many successful demonstrations of VQAs, for example, in quantum machine learning and with problems in quantum chemistry and condensed matter physics, using NISQ hardware without error correction. However, implementing VQAs on problems beyond the classical limit remains challenging due to the complexity of the non-convex optimization performed on the classical side associated with the algorithms. One of the challenges is the barren plateau where the cost-function gradient exponentially diminishes with an increase in the system size.

Local minima appear in the high-dimensional cost-function landscape, making it difficult for local optimization to obtain a good solution. Furthermore, analyzing the optimization complexity is made even more complicated by the noisy aspect resulting from the non-negligible error rates of NISQ hardware.

How Does Noise Affect the Optimal Solution of a VQE Algorithm?

In a recent study, researchers investigated the effect of noise on the optimal solution of a VQE algorithm calculating the ground state of a one-dimensional spin chain. They discovered that the noise can trigger a transition in the optimal solution. This transition can be qualitatively explained by the first-order perturbative correction to the cost-function landscape, which causes the global minimum to switch from one local minimum to another.

The researchers also demonstrated this transition on an IBM quantum processor unit (QPU) for a spin-dimer. The noise-induced transition, which may occur for some applications of VQAs, can result in the deterioration of the solution’s quality in an unpredictable manner. This study provides insights into the origin of this issue and paves the way for further studies in mitigating it to ensure the proper functioning of VQAs on NISQ hardware.

What is the Heisenberg XXX Model and How is it Used in VQAs?

The researchers studied the VQE optimal solution of a one-dimensional Heisenberg XXX model with antiferromagnetic coupling. The Hamiltonian of the spin chain with open boundary conditions is given by a specific equation. The XXX antiferromagnetic spin chain has been widely studied with rich entanglement features. Nonetheless, the researchers merely used this model as a test bed to investigate the response of the VQE solutions to noise.

VQE algorithms determine the ground state of the model by minimizing the cost function, the energy expectation value, associated with a parameterized ansatz, where the parameters are to be optimized. The optimal solution gives the approximated ground state and ground-state energy. The quality of the solution representing the true ground state and ground-state energy depends on the parameters.

How Does Noise Influence the Cost Function Response?

The researchers visualized the cost function energy landscape corresponding to the two-parameter ansatz for the spin dimer. Two minima, labeled minimum 1 and minimum 2, can be found. In the error-free case, with the bit flip error rate at zero, minimum 1 is the global minimum. With increasing error rate, the energy associated with minimum 1 increases faster than that with minimum 2. Beyond a threshold value, the global minimum switches from minimum 1 to minimum 2, and this results in a transition in the optimal solution.

This noise-induced transition, which may occur for some applications of VQAs, can result in the deterioration of the solution’s quality in an unpredictable manner. The study provides insights into the origin of this issue and paves the way for further studies in mitigating it to ensure the proper functioning of VQAs on NISQ hardware.

What are the Implications of this Study for the Future of VQAs?

The findings of this study suggest that careful analysis is crucial to avoid misinterpreting the noise-induced features as genuine algorithm results. The noise-induced transition, which may occur for some applications of VQAs, can result in the deterioration of the solution’s quality in an unpredictable manner. This study provides insights into the origin of this issue and paves the way for further studies in mitigating it to ensure the proper functioning of VQAs on NISQ hardware.

The researchers’ work on the effect of noise on optimization by studying a VQE algorithm calculating the ground state of a spin chain model is a significant contribution to the field of quantum computing. Their findings will be instrumental in the development of more robust and reliable VQAs, which are expected to play a crucial role in the future of quantum computing.