New Hybrid-HHL algorithm vs VQLS

Question

A team of researchers has realized hybrid quantum algorithm for solving a linear system of equations with exponential speedup that utilizes quantum phase estimation, the algorithm demonstrates quantum supremacy and holds high promise to meet practically relevant challenges.

https://scirate.com/arxiv/2003.12770

https://www.swissquantumhub.com/quantum-supremacy-quantum-hybrid-hhl-algorithm-for-solving-a-system-of-linear-equation/

There is also a variational hybrid quantum-classical algorithm for solving linear systems, with the aim of reducing the circuit depth and doing much of the computation classically, called VQLS.

https://arxiv.org/abs/1909.05820

https://pennylane.ai/qml/demos/tutorial_vqls.html

How can we compare both algorithms?

In the part of the near-term application of H-HHL paper, they talk about Bayesian deep learning application: "One of the promising applications related to deep neural network training was discussed in [1]: since the extension of the Bayesian approach to deep architectures is a serious challenge, one can exploit the hybrid quantum HHL algorithm developed for Gaussian processes in order to calculate a model’s predictor" [21].

Which algorithm should be better in the next-gen state-of-art 53-Qubits quantum computer for the Quantum Bayesian deep learning algorithm?

Answer 1

By a large margin, I would recommend VQLS rather than H-HHL. VQLS is significantly more well-tested, is a more significant leap from the previous state-of-the-art for hybrid quantum/classical linear solvers, and is far more in the "mainstream" realm of quantum computing, meaning that your study will be more relevant in today's quantum computing research climate if you implement VQLS instead of H-HHL.

You could also try both and compare the two!