[Paper Review] Experimental kernel-based quantum machine learning in finite feature space

Karol Bartkiewicz, Clemens Gneiting+4 more|arXiv (Cornell University)|Jun 10, 2019

Quantum Computing Algorithms and Architecture40 references78 citations

TL;DR

This paper presents the first experimental realization of kernel-based quantum machine learning (KQML) using an all-optical setup with two-photon states in an eight-dimensional finite feature Hilbert space. By optimizing feature maps to maximize kernel resolution under fixed Hilbert space dimension, the scheme achieves exponentially better qubit scaling than prior approaches and demonstrates viable nonlinear decision boundaries for classification tasks using multiphoton quantum optical circuits.

ABSTRACT

We implement an all-optical setup demonstrating kernel-based quantum machine learning for two-dimensional classification problems. In this hybrid approach, kernel evaluations are outsourced to projective measurements on suitably designed quantum states encoding the training data, while the model training is processed on a classical computer. Our two-photon proposal encodes data points in a discrete, eight-dimensional feature Hilbert space. In order to maximize the application range of the deployable kernels, we optimize feature maps towards the resulting kernels' ability to separate points, i.e., their resolution, under the constraint of finite, fixed Hilbert space dimension. Implementing these kernels, our setup delivers viable decision boundaries for standard nonlinear supervised classification tasks in feature space. We demonstrate such kernel-based quantum machine learning using specialized multiphoton quantum optical circuits. The deployed kernel exhibits exponentially better scaling in the required number of qubits than a direct generalization of kernels described in the literature.

Motivation & Objective

• To demonstrate a practical, hybrid quantum-classical approach to supervised machine learning using optical quantum systems.
• To address the challenge of limited Hilbert space dimension in finite-feature quantum machine learning by optimizing kernel resolution.
• To achieve exponential improvement in qubit scaling compared to direct generalizations of existing quantum kernels.
• To implement and validate a kernel-based quantum machine learning protocol using photonic qubits and projective measurements.
• To show that finite-dimensional quantum systems can effectively solve nonlinear classification problems via optimized feature maps.

Proposed method

• The method uses a two-photon optical setup encoding data in a discrete, eight-dimensional Hilbert space via polarization and path degrees of freedom.
• Feature maps are optimized to maximize kernel resolution—defined as the ability to distinguish data points—under fixed Hilbert space dimension.
• Kernel evaluations are performed via projective measurements on entangled photonic states, bypassing classical computation of inner products.
• The kernel function is implemented as |⟨ϕ(x′)|ϕ(x)⟩|², with the kernel value extracted from photon coincidence rates.
• A classical computer trains the model using the measured kernel values, solving a support vector machine (SVM)-like optimization problem.
• The setup employs beam splitters, wave plates, and single-photon detectors to implement the quantum feature map and measurement circuits.

Experimental results

Research questions

• RQ1 Can kernel-based quantum machine learning be experimentally realized in a finite-dimensional Hilbert space using linear optics?
• RQ2 How can kernel resolution be maximized under fixed Hilbert space dimension to improve classification performance?
• RQ3 What is the scaling advantage of the proposed kernel in terms of required qubits compared to direct generalizations of existing quantum kernels?
• RQ4 Can a hybrid quantum-classical approach using photonic systems achieve viable decision boundaries for nonlinear classification tasks?
• RQ5 How does the performance of resolution-optimized kernels compare to standard kernels like MSI or TSQ in finite-dimensional feature spaces?

Key findings

• The experiment successfully demonstrated kernel-based quantum machine learning for nonlinear classification using an all-optical setup with two-photon states.
• The resolution-optimized kernel achieved significantly improved resolution progression with increasing Hilbert space dimension N compared to MSI and TSQ kernels.
• The proposed kernel exhibited exponentially better scaling in the number of required qubits than direct generalizations of existing kernels, as shown in Figure 1b.
• The experimental kernel was extracted from photon coincidence rates, with the ratio of coincidences to total photons used to compute the kernel value.
• The method achieved viable decision boundaries for standard nonlinear classification tasks, confirming the feasibility of the approach.
• The results validate that finite-dimensional feature Hilbert spaces can be effectively utilized for quantum machine learning when feature maps are optimized for kernel resolution.