Smooth input preparation for quantum and quantum-inspired machine learning

TL;DR

This paper demonstrates that robust data analysis algorithms can enable efficient quantum and quantum-inspired machine learning by ensuring state preparation costs are manageable, even with high-dimensional data.

Contribution

It proves that under small perturbations, state preparation can be done with constant queries, facilitating practical quantum and quantum-inspired machine learning.

Findings

State preparation costs can be constant under robustness assumptions.

Polylogarithmic processing time is achievable for low-rank data.

Results apply to both quantum and quantum-inspired algorithms.

Abstract

Machine learning has recently emerged as a fruitful area for finding potential quantum computational advantage. Many of the quantum enhanced machine learning algorithms critically hinge upon the ability to efficiently produce states proportional to high-dimensional data points stored in a quantum accessible memory. Even given query access to exponentially many entries stored in a database, the construction of which is considered a one-off overhead, it has been argued that the cost of preparing such amplitude-encoded states may offset any exponential quantum advantage. Here we prove using smoothed analysis, that if the data-analysis algorithm is robust against small entry-wise input perturbation, state preparation can always be achieved with constant queries. This criterion is typically satisfied in realistic machine learning applications, where input data is subjective to moderate noise. Our results are equally applicable to the recent seminal progress in quantum-inspired algorithms, where specially constructed databases suffice for polylogarithmic classical algorithm in low-rank cases. The consequence of our finding is that for the purpose of practical machine learning, polylogarithmic processing time is possible under a general and flexible input model with quantum algorithms or quantum-inspired classical algorithms in the low-rank cases.