Inference, interference and invariance: How the Quantum Fourier Transform can help to learn from data

TL;DR

This paper explores how quantum Fourier Transform-based interference strategies can be adapted for machine learning inference, leveraging symmetries and invariance principles to improve generalization from finite data samples.

Contribution

It introduces a novel inference framework inspired by quantum algorithms, transforming the Hidden Subgroup Problem into a classical learning task using quantum state overlaps.

Findings

Proposes a new inference principle based on invariant subspaces.

Suggests a quantum-inspired heuristic leveraging symmetries.

Provides a conceptual link between quantum algorithms and machine learning generalization.

Abstract

How can we take inspiration from a typical quantum algorithm to design heuristics for machine learning? A common blueprint, used from Deutsch-Josza to Shor's algorithm, is to place labeled information in superposition via an oracle, interfere in Fourier space, and measure. In this paper, we want to understand how this interference strategy can be used for inference, i.e. to generalize from finite data samples to a ground truth. Our investigative framework is built around the Hidden Subgroup Problem (HSP), which we transform into a learning task by replacing the oracle with classical training data. The standard quantum algorithm for solving the HSP uses the Quantum Fourier Transform to expose an invariant subspace, i.e., a subset of Hilbert space in which the hidden symmetry is manifest. Based on this insight, we propose an inference principle that "compares" the data to this invariant subspace, and suggest a concrete implementation via overlaps of quantum states. We hope that this leads to well-motivated quantum heuristics that can leverage symmetries for machine learning applications.