Inference, interference and invariance: How the Quantum Fourier Transform can help to learn from data (2409.00172v1)

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.

• The paper introduces a learning variant of the Hidden Subgroup Problem by replacing the quantum oracle with finite labeled training data.
• The paper applies the Quantum Fourier Transform to uncover invariant subspaces and proposes the Data-Annihilator Overlap to quantify subgroup likelihood.
• The paper presents a sample complexity analysis demonstrating that logarithmically many samples suffice for effective inference within the PAC framework.

Inference, Interference and Invariance: How the Quantum Fourier Transform Can Help to Learn from Data

Overview

The paper "Inference, interference and invariance: How the Quantum Fourier Transform can help to learn from data" explores how principles from quantum algorithms can be adapted to design heuristics for machine learning tasks. Specifically, it investigates how a canonical quantum procedure, involving the quantum Fourier transform (QFT), can be repurposed to facilitate inference in classical learning scenarios, focusing on the Hidden Subgroup Problem (HSP). This foundational quantum problem, which encompasses a wide range of well-known algorithms such as Shor’s algorithm and the Deutsch-Josza algorithm, serves as the blueprint for transferring quantum techniques to classical data-driven tasks.

Key Contributions

The primary contributions of the paper can be summarized as follows:

1. Learning Variant of the HSP: The authors introduce a learning variant of the HSP, where the oracle is replaced by a finite set of labeled training data. This makes the problem more akin to real-world scenarios where only limited data is available.
2. Inference via Invariant Subspaces: Based on the strategy of using the QFT to reveal invariant subspaces in the traditional HSP algorithm, the authors propose an inference principle. It involves comparing the data to candidate invariant subspaces to infer the hidden subgroup.
3. Data-Annihilator Overlap (DAO): A concrete implementation of the inference principle is suggested via the data-annihilator overlap, effectively a cost function. This cost function measures how well a candidate subgroup's annihilator explains the data and can be efficiently computed on a quantum computer.
4. Sample Complexity Analysis: The paper presents a detailed analysis of the sample complexity of learning the HSP within the PAC framework. The results indicate that logarithmically many samples relative to the group size are sufficient for inferring the hidden subgroup.

Detailed Findings

The Hidden Subgroup Problem

The HSP is a general problem that involves finding a hidden subgroup $H \leq G$ given a function $f$ that is constant on the cosets of $H$. The standard quantum algorithm for solving the HSP uses the QFT to sample from the annihilator of the subgroup, effectively reducing the problem from an exponential to a polynomial scale.

Learning Variant of the HSP

Transforming the HSP into a learning problem involves replacing the quantum oracle with a set of training examples: $\mathcal{T} = \{(g, f(g)): g \in X\},$ where $X \subseteq G$ is a set of observed points. The task becomes one of guessing the labels for new data points, making it a classification problem. With small datasets, interference patterns break down, and the signal can be lost in noise.

Inference to the Nearest Annihilator

The critical insight is that subspaces spanned by characters (resulting from the QFT) form invariant subspaces. The proposed principle of inference suggests finding the invariant subspace closest to the data subspace. This is motivated by the fact that in traditional HSP, the entire oracle subspace lies within the invariant subspace revealed by the QFT.

Data-Annihilator Overlap (DAO)

The paper defines the DAO cost function to compare the data subspace to candidate invariant subspaces. The data subspace consists of states spanned by partial cosets, while the invariant subspace consists of states spanned by character states: $$\mathcal{H}_{\mathcal{T}} = \text{span}\{|X_r\rangle \}, \quad \mathcal{H}_{\tilde{H}^\perp} = \text{span}\{|\chi \rangle : \chi \in \tilde{H}^\perp\}.$$ The DAO is computed using the fidelity between these state representations and provides a quantifiable measure for optimizing the likelihood of candidate subgroups.

Practical Implications and Future Directions

• Quantum Machine Learning: The principles outlined in the paper could be employed for designing algorithms that leverage quantum interference to infer structures in data. This is particularly relevant for algorithms running on fault-tolerant quantum computers.
• Disentangled Representation Learning: The methods proposed can be extended to identify and leverage symmetric structures in datasets, contributing to disentangled representation learning where task-relevant and nuisance factors are separated.
• Extension to Non-Abelian Groups: Although the focus is on abelian groups, the methods could be generalized to non-abelian groups such as those involved in graph symmetries and other complex data structures.

Conclusion

The paper presents a novel approach to adopting quantum computational techniques for classical learning tasks. By transforming the well-studied Hidden Subgroup Problem into a learning problem and proposing an efficient heuristic based on the Quantum Fourier Transform, the authors open new avenues for exploiting quantum algorithms in machine learning. While further exploration is needed, particularly in applying these principles to real-world data, the groundwork laid out provides a robust foundation for future developments in quantum-enhanced learning algorithms.