[Paper Review] Quantum-inspired classical algorithms for principal component analysis and supervised clustering.

Ewin Tang|arXiv (Cornell University)|Oct 31, 2018

Quantum Computing Algorithms and Architecture62 citations

TL;DR

This paper presents classical algorithms for principal component analysis and supervised clustering that mirror the efficiency of Lloyd et al.'s quantum algorithms. By assuming efficient $β^2$-norm sampling of data, the classical algorithms achieve polylogarithmic runtime in input size, matching the quantum speedup's asymptotic complexity with only polynomial slowdown, implying no exponential quantum advantage exists for these problems.

ABSTRACT

We describe classical analogues to Lloyd et al.'s quantum algorithms for principal component analysis and nearest-centroid clustering. We introduce a classical algorithm model that assumes we can efficiently perform $\ell^2$-norm samples of input data, a natural analogue to quantum algorithms assuming efficient state preparation. In this model, our classical algorithms run in time polylogarithmic in input size, matching the runtime of the quantum algorithms with only polynomial slowdown. These algorithms indicate that their corresponding problems do not yield exponential quantum speedups.

Motivation & Objective

• To develop classical counterparts to quantum algorithms for principal component analysis and nearest-centroid clustering.
• To identify whether the quantum speedups for these problems are exponential or merely polynomial.
• To formalize a classical computational model based on efficient $β^2$-norm sampling, analogous to quantum state preparation.
• To demonstrate that the runtime of classical algorithms matches the polylogarithmic scaling of the quantum versions, up to polynomial slowdown.

Proposed method

• The algorithm model assumes access to efficient $β^2$-norm sampling of input data vectors, analogous to quantum state preparation in quantum algorithms.
• The classical algorithms use iterative sampling and estimation techniques to approximate singular vectors and centroids without full data access.
• The approach leverages random projections and variance reduction to estimate principal components and cluster centroids accurately.
• Runtime analysis shows that the algorithms scale as polylogarithmic in input size, matching the quantum algorithm's asymptotic efficiency.
• The framework avoids explicit matrix inversion or full SVD by relying on sampling-based estimation of key components.
• The model ensures that the classical algorithms achieve the same asymptotic complexity as the quantum counterparts, up to polynomial factors.

Experimental results

Research questions

• RQ1 Can classical algorithms achieve runtime complexity comparable to quantum algorithms for principal component analysis?
• RQ2 Is the quantum speedup for PCA and clustering exponential or merely polynomial in nature?
• RQ3 What classical computational model enables efficient simulation of quantum algorithm primitives like state preparation?
• RQ4 Do the problems solved by these quantum algorithms admit exponential speedups over classical methods?
• RQ5 What is the role of $β^2$-norm sampling in enabling efficient classical computation of linear algebraic operations?

Key findings

• The classical algorithms achieve runtime that is polylogarithmic in input size, matching the asymptotic efficiency of the quantum algorithms.
• The classical approach matches the quantum algorithm's complexity up to a polynomial slowdown, indicating no exponential quantum advantage.
• The use of $β^2$-norm sampling as a primitive enables efficient classical simulation of quantum algorithmic steps.
• The results imply that the problems of PCA and nearest-centroid clustering do not yield exponential quantum speedups.
• The framework demonstrates that quantum-inspired classical algorithms can replicate the efficiency of quantum algorithms under realistic sampling assumptions.