Robust Dequantization of the Quantum Singular value Transformation and Quantum Machine Learning Algorithms
This work addresses the robustness of classical dequantization methods for quantum algorithms, which is a problem for researchers in quantum computing and machine learning, though it is incremental as it builds on prior dequantization frameworks.
The paper tackles the robustness of dequantization results for quantum linear algebra and machine learning algorithms by introducing approximate length-squared sampling, showing that existing dequantization frameworks can be adapted to work under this weaker assumption, and applying this to robustly dequantize quantum algorithms for recommendation systems, supervised clustering, and low-rank matrix inversion.
Several quantum algorithms for linear algebra problems, and in particular quantum machine learning problems, have been "dequantized" in the past few years. These dequantization results typically hold when classical algorithms can access the data via length-squared sampling. In this work we investigate how robust these dequantization results are. We introduce the notion of approximate length-squared sampling, where classical algorithms are only able to sample from a distribution close to the ideal distribution in total variation distance. While quantum algorithms are natively robust against small perturbations, current techniques in dequantization are not. Our main technical contribution is showing how many techniques from randomized linear algebra can be adapted to work under this weaker assumption as well. We then use these techniques to show that the recent low-rank dequantization framework by Chia, Gilyén, Li, Lin, Tang and Wang (JACM 2022) and the dequantization framework for sparse matrices by Gharibian and Le Gall (STOC 2022), which are both based on the Quantum Singular Value Transformation, can be generalized to the case of approximate length-squared sampling access to the input. We also apply these results to obtain a robust dequantization of many quantum machine learning algorithms, including quantum algorithms for recommendation systems, supervised clustering and low-rank matrix inversion.