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

arXiv: https://arxiv.org/abs/2409.00172 · HTML: https://mlsanity.com/paper/2409.00172/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 11 references that are in the map.
- First posted on arXiv 30 Aug 2024.

## Builds on (most cited first, 5 of 5)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [The Optimal Sample Complexity of PAC Learning](https://mlsanity.com/paper/1507.00473/) (2015)
- [Optimal Quantum Sample Complexity of Learning Algorithms](https://mlsanity.com/paper/1607.00932/) (2016)
- [A Survey of Quantum Learning Theory](https://mlsanity.com/paper/1701.06806/) (2017)

## Built on by (most cited first, 5 of 5)

- [Spectral methods: crucial for machine learning, natural for quantum computers?](https://mlsanity.com/paper/2603.24654/) (2026)
- [The role of data-induced randomness in quantum machine learning classification tasks](https://mlsanity.com/paper/2411.19281/) (2024)
- [Quantum entanglement provides a competitive advantage in adversarial games](https://mlsanity.com/paper/2603.10289/) (2026)
- [Classical Limits of Spectral Filtering in Quantum Generative Models](https://mlsanity.com/paper/2608.14169/) (2026)
- [Bernstein-Vazirani Networks: Quantum Machine Learning by Interference](https://mlsanity.com/paper/2608.19043/) (2026)

## Abstract (arXiv)

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.