Supervised quantum machine learning models are kernel methods (2101.11020v2)

Abstract: With near-term quantum devices available and the race for fault-tolerant quantum computers in full swing, researchers became interested in the question of what happens if we replace a supervised machine learning model with a quantum circuit. While such "quantum models" are sometimes called "quantum neural networks", it has been repeatedly noted that their mathematical structure is actually much more closely related to kernel methods: they analyse data in high-dimensional Hilbert spaces to which we only have access through inner products revealed by measurements. This technical manuscript summarises and extends the idea of systematically rephrasing supervised quantum models as a kernel method. With this, a lot of near-term and fault-tolerant quantum models can be replaced by a general support vector machine whose kernel computes distances between data-encoding quantum states. Kernel-based training is then guaranteed to find better or equally good quantum models than variational circuit training. Overall, the kernel perspective of quantum machine learning tells us that the way that data is encoded into quantum states is the main ingredient that can potentially set quantum models apart from classical machine learning models.

• The paper's main contribution is showing that supervised quantum machine learning models can be reformulated as kernel methods through their data encoding strategies.
• It reveals that quantum models operate as linear classifiers in high-dimensional Hilbert spaces, analogous to support vector machines via reproducing kernel Hilbert spaces.
• The findings underscore that selecting appropriate quantum feature maps is crucial for achieving expressivity, universality, and efficient training in quantum machine learning.

Overview of "Supervised quantum machine learning models are kernel methods"

The paper "Supervised quantum machine learning models are kernel methods" by Maria Schuld posits that many supervised quantum machine learning models can be understood fundamentally as kernel methods. This perspective clarifies the mathematical relationship between quantum machine learning and classical kernel methods, offering significant insights into the expressiveness and trainability of quantum models.

Core Concepts and Theoretical Implications

1. Quantum Models as Kernel Methods: The paper asserts that quantum machine learning models, often termed "quantum neural networks," align more closely with kernel methods. They operate by analyzing data in high-dimensional Hilbert spaces accessed through inner products. Importantly, the kernel perspective positions quantum models as linear classifiers in a Hilbert space defined by quantum states, contrasting with the more common neural network paradigm.
2. Data Encoding Strategy as Feature Mapping: Quantum models utilize a data encoding strategy that maps classical data into quantum states, forming a feature map. This process establishes the foundation for a quantum kernel, which measures the similarity between data points by evaluating inner products of these encoded states. The structure and properties of the quantum kernel are determined by the chosen quantum feature map, encapsulating the model’s inductive biases.
3. Quantum Kernels and Reproducing Kernel Hilbert Spaces (RKHS): The RKHS framework underpins the analytical approach to quantum models. The quantum kernel induces an RKHS where quantum models reside, implying that any function realizable by a quantum model can also be expressed in the RKHS. This equivalence facilitates the translation of classical kernel methods into the quantum domain, allowing for the leveraging of classical insights into training and regularization.
4. Expressivity and Universality: The expressiveness of quantum models hinges on the form of the quantum kernel. The paper outlines conditions under which quantum kernels can be universal, capable of approximating any function within a certain class. This universality is conditional on the feature map's properties, such as being injective, highlighting the importance of the data encoding strategy.
5. Training Quantum Models: The representer theorem from classical kernel theory is employed to delineate how optimal quantum models can be expanded in terms of training data. This draws a direct line to support vector machines, where optimal decision boundaries are expressed via kernel functions evaluated on the training set. The paper demonstrates that optimally trained quantum models possess measurements expanded in the space of data-encoded quantum states, emphasizing the finite-dimensional nature of the optimization problem inherent in quantum machine learning.
6. Challenges and Prospective Quantum Advantage: While kernel-based training offers theoretical advantages, such as global optimality and convexity in optimization, it incurs computational costs that scale quadratically with the number of data points. The paper notes that larger, fault-tolerant quantum computers could mitigate these costs by efficiently implementing complex kernels inaccessible to classical machines.

Practical Implications and Future Directions

The paper's insights have profound implications for both theoretical and practical aspects of quantum machine learning. Understanding quantum models as kernel methods sharpens the focus on the design and selection of quantum feature maps, as these inherently dictate model performance. Practically, these insights suggest a paradigm where quantum processors might primarily serve as accelerators for kernel evaluations, rather than solely emulating neural network architectures traditionally.

The future of this research lies in identifying and constructing quantum kernels that demonstrate provable advantages over classical counterparts in specific applications. Such advancements could cement the role of quantum computers in solving complex machine learning problems, potentially outpacing classical systems when handling massive datasets or complex, high-dimensional data structures.

In conclusion, the kernel-based perspective of quantum machine learning bridges classical and quantum paradigms, offering a robust framework for exploring the full capabilities of quantum models. This research sets the stage for further exploration into practical quantum machine learning strategies, underpinned by rigorous mathematical foundations.