Sampling-based Sublinear Low-rank Matrix Arithmetic Framework for Dequantizing Quantum Machine Learning
- Department of Computer Science, University of Texas at Austin, USA
- Alfréd Rényi Institute of Mathematics Hungary and Institute for Quantum Information and Matter, California Institute of Technology, USA
- Center on Frontiers of Computing Studies, School of Computer Science, Peking University China and Center for Theoretical Physics, Massachusetts Institute of Technology, USA
- University of Washington, WA, USA
- Department of Computer Science and Engineering, Pennsylvania State University, USA and Department of Computer Science, University of Texas at Austin, USA
We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all prior results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ2-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.
- Research Organization:
- US Department of Energy (USDOE), Washington, DC (United States). Office of Science, Advanced Scientific Computing Research (ASCR)
- Sponsoring Organization:
- USDOE
- OSTI ID:
- 2424704
- Journal Information:
- Journal of the Association for Computing Machinery, Journal Name: Journal of the Association for Computing Machinery Journal Issue: 5 Vol. 69; ISSN 0004-5411
- Publisher:
- Association for Computing Machinery (ACM)
- Country of Publication:
- United States
- Language:
- English
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