[Paper Review] Quantum embeddings for machine learning

Seth Lloyd, Maria Schuld+3 more|arXiv (Cornell University)|Jan 10, 2020

Quantum Computing Algorithms and ArchitectureComputer Science2 references69 citations

TL;DR

The paper proposes metric learning for quantum embeddings: train the embedding to maximize separation of data clusters in Hilbert space, then use optimal quantum measurements (Helstrøm or fidelity) for classification. This yields a framework where the embedding drives performance and measurements are known given the chosen metric.

ABSTRACT

Quantum classifiers are trainable quantum circuits used as machine learning models. The first part of the circuit implements a quantum feature map that encodes classical inputs into quantum states, embedding the data in a high-dimensional Hilbert space; the second part of the circuit executes a quantum measurement interpreted as the output of the model. Usually, the measurement is trained to distinguish quantum-embedded data. We propose to instead train the first part of the circuit -- the embedding -- with the objective of maximally separating data classes in Hilbert space, a strategy we call quantum metric learning. As a result, the measurement minimizing a linear classification loss is already known and depends on the metric used: for embeddings separating data using the l1 or trace distance, this is the Helstrom measurement, while for the l2 or Hilbert-Schmidt distance, it is a simple overlap measurement. This approach provides a powerful analytic framework for quantum machine learning and eliminates a major component in current models, freeing up more precious resources to best leverage the capabilities of near-term quantum information processors.

Motivation & Objective

• Embed classical data as quantum states using a trainable quantum feature map to create maximally separated clusters in Hilbert space.
• Show that the choice of Hilbert-space metric determines the optimal measurement (Helstrøm for trace distance; fidelity/overlap for Hilbert-Schmidt distance).
• Provide an analytic framework that reduces the need for resource-intensive adaptive measurements in quantum classifiers.
• Demonstrate training of quantum embeddings using PennyLane and assess performance with small-scale examples.
• Discuss experimental feasibility and potential quantum advantages for near-term quantum devices.

Proposed method

• Define quantum embedding via a circuit Phi(x, theta) producing |x> = Phi(x, theta)|0...0>.
• Characterize two data ensembles ρ and σ from two classes A and B and relate empirical risk to metric-dependent measurements.
• Use linear loss with empirical risk -tr((ρ−σ)M) and connect to Hilbert-space metrics: for fidelity, risk corresponds to −D_hs(ρ,σ); for Helstrøm, risk corresponds to −2 D_tr(ρ,σ).
• Show that optimal measurements are the fidelity (swap test) or Helstrøm projection onto ± subspaces of ρ−σ.
• Focus practical training on Hilbert-Schmidt distance due to circuit efficiency; discuss relations to trace distance and low-rank embeddings.
• Provide circuits and analysis for implementing both measurements and discuss embedding training via PennyLane.

Experimental results

Research questions

• RQ1 Can quantum embeddings trained to maximize Hilbert-space separation lead to optimal or near-optimal classifications under known quantum measurements?
• RQ2 What is the relationship between embedding metrics (trace distance vs Hilbert-Schmidt distance) and the corresponding optimal measurements (Helstrøm vs fidelity) in quantum classifiers?
• RQ3 How feasible is training and deploying quantum embeddings on near-term quantum hardware using shallow circuits and standard optimization?
• RQ4 Does metric quantum learning reduce resource needs by front-loading separation into the embedding rather than the measurement?
• RQ5 How do low-rank embeddings influence generalization and sample requirements for quantum classifiers?

Key findings

• Training embeddings to maximize cluster separation can yield decision boundaries that correspond to optimal measurements in Hilbert space.
• Fidelity-based (Hilbert-Schmidt) optimization leads to a feasible, short quantum circuit implementation via overlap measurements such as SWAP tests.
• Helstrøm-based (trace-distance) decision making corresponds to projecting onto positive/negative subspaces of ρ−σ with an efficient circuit.
• Optimal measurement choice depends on the chosen Hilbert-space metric, providing a clear analytic framework for quantum machine learning.
• Numerical demonstrations with small datasets (e.g., 2D moons) illustrate similar, though not identical, decision boundaries between the two approaches.
• Discussion indicates embeddings can be high-dimensional and potentially inaccessible to classical computation, offering a potential quantum advantage.