Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing Quantum machine learning

Abstract

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, suffices to generalize all recent 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: ℓ 2-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.

Quantum algorithms for escaping from saddle points

“…As far as we know, the only work along this direction is by [75], which gives a quantum algorithm for finding the negative curvature of a point in timeÕ(poly(r, 1/ )), where r is the rank of the Hessian at that point. However, the algorithm has a few drawbacks: 1) The cost is expensive when r = Θ(n); 2) It relies on a quantum data structure [52] which can actually be dequantized to classical algorithms with comparable cost [20,66,67]; 3) It can only find the negative curvature for a fixed Hessian. In all, it is unclear whether this quantum algorithm achieves speedup for escaping from saddle points.…”

Quantum semi-supervised kernel learning

“…The fundamental difference between quantuminspired algorithms and traditional classical algorithms is that via importance sampling, their runtime is independent of the dimension of input data, and thus it builds a setting comparable with quantum machine learning algorithms aided by QRAM. Recently (Chia et al 2020) introduced an algorithmic framework for quantum-inspired classical algorithms on low-rank matrices that generalizes a series of previous work, recovers existing quantum-inspired algorithms such as quantum-inspired recommendation systems (Tang 2019), quantum-inspired principal component analysis (Tang 2018), quantum-inspired low-rank matrix inversion (Gilyén et al 2018), and quantum-inspired support vector machine (Ding et al 2019).…”

Variational Learning for Quantum Artificial Neural Networks

“…Among the most relevant results obtained in Quantum Machine Learning it is worth mentioning the use of trainable parametrized digital and continuous-variable quantum circuits as a model for quantum neural networks [12]- [21], the realization of quantum Support Vector Machines (qSVMs) [22] working in quantum-enhanced feature spaces [23], [24] and the introduction of quantum versions of artificial neuron models [25]- [32]. However, it is true that very few clear statements have been made concerning the concrete and quantitative achievement of quantum advantage in machine learning applications, and many challenges still need to be addressed [8], [33], [34].…”