Quantum machine learning in feature Hilbert spaces (1803.07128v1)

Abstract: The basic idea of quantum computing is surprisingly similar to that of kernel methods in machine learning, namely to efficiently perform computations in an intractably large Hilbert space. In this paper we explore some theoretical foundations of this link and show how it opens up a new avenue for the design of quantum machine learning algorithms. We interpret the process of encoding inputs in a quantum state as a nonlinear feature map that maps data to quantum Hilbert space. A quantum computer can now analyse the input data in this feature space. Based on this link, we discuss two approaches for building a quantum model for classification. In the first approach, the quantum device estimates inner products of quantum states to compute a classically intractable kernel. This kernel can be fed into any classical kernel method such as a support vector machine. In the second approach, we can use a variational quantum circuit as a linear model that classifies data explicitly in Hilbert space. We illustrate these ideas with a feature map based on squeezing in a continuous-variable system, and visualise the working principle with $2$-dimensional mini-benchmark datasets.

• The paper proposes that encoding classical data into quantum states acts as a nonlinear feature map to a quantum Hilbert space, foundational for building quantum machine learning models.
• One approach discussed is using quantum devices to estimate classically intractable kernels for integration into classical methods like Support Vector Machines.
• Another method involves using variational quantum circuits as trainable linear models directly within the quantum feature space for classification tasks.

Quantum Machine Learning in Feature Hilbert Spaces: A Detailed Examination

The paper "Quantum machine learning in feature Hilbert spaces" by Maria Schuld and Nathan Killoran proposes a compelling theoretical foundation linking quantum computing and kernel methods in machine learning. This linkage hinges on their mutual objective of efficiently performing computations in large Hilbert spaces—spaces that, classically, could be computationally intractable. The authors explore the prospects of designing quantum machine learning algorithms by considering quantum state-based data encoding as a nonlinear feature map to a quantum Hilbert space. This formulation supports the development of sophisticated quantum models for data classification, leveraging two principal approaches—quantum kernel methods and variational quantum circuits.

Approaches to Quantum Models for Classification

The research presented considers two methodologies for constructing quantum classification models:

1. Quantum-enhanced kernel estimation: The first method employs the capability of quantum devices to estimate inner products of quantum states, thereby computing kernels that are classically difficult to determine. These kernels are then integrated into classical machine learning techniques like support vector machines. The quantum device's role as a subroutine in the kernel evaluation can potentially resolve computational challenges in classical high-dimensional kernel estimation.
2. Variational quantum circuits as linear models: In contrast, the second approach employs variational quantum circuits directly within the quantum feature space. These circuits function as linear models, effectively serving as supervised learning classifiers. The quantum devices are trained classically and can manage complex patterns through adjustable circuit parameters. This approach aligns with the growing interest in hybrid quantum-classical algorithms which facilitate smaller, near-term quantum devices' integration into practical machine learning workflows.

Key Theoretical Insights and Implications

The paper's central thesis that quantum state preparation can be interpreted as a nonlinear feature map opens new avenues for exploring quantum models in machine learning. This view aligns with the traditional kernel trick, where data transformations through feature maps lead to simplified linear separations. In the quantum context, switching data encoding strategies showcases analogous adaptability to changing kernel functions in classical settings, albeit executed in hardware within quantum systems.

Numerical simulations demonstrated significant findings: the squeezing feature map made linearly inseparable classical data separable in feature space due to its property of mapping inputs to distinct, linearly independent quantum states. Moreover, these findings underscore the potential capability of quantum-assisted models to perform efficient pattern recognition and classification, particularly using optical quantum computers where squeezing operations are prevalent.

Practical and Theoretical Impact on Quantum Machine Learning

Practically, these techniques herald potential shifts in how quantum and classical information systems may interact, especially concerning problems where classical computational resources are limited by the polynomial overhead of explicitly handling large Hilbert spaces or multifaceted feature transformations. The role of quantum devices as differentiable components within broader machine learning systems reflects the intermediate-term feasibility of quantum-enhanced computations, including an emphasis on scalability, modularity, and integration with established machine learning frameworks.

Theoretically, the introduction of feature Hilbert spaces expands the frontier of applying Hilbert space formalism traditionally reserved for quantum mechanics into the domain of machine learning, fostering novel algorithms and models with unique capabilities in handling vast, non-linear, and complex data spaces. Future research directions could deepen our understanding of Hilbert space properties, develop more efficient, classically difficult kernel functions, and refine training algorithms for variational quantum models.

Schuld and Killoran's paper situates the quantum enhancement of machine learning at a strategic intersection of theoretical innovation and practical application, guiding future developments of quantum machine learning frameworks that capitalize on quantum systems' unique strengths while addressing classical computational limitations.