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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

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?

I am particularly interested in the comparison of the H-HHL algo with that of the VQLS. I myself am interested in solving large linear systems on NISQ hardware and would like to know which of these I should experiment with. – thespaceman – 2021-01-19T22:40:43.597