Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces (2009.00298v3)

Abstract: Encoding classical data into quantum states is considered a quantum feature map to map classical data into a quantum Hilbert space. This feature map provides opportunities to incorporate quantum advantages into machine learning algorithms to be performed on near-term intermediate-scale quantum computers. The crucial idea is using the quantum Hilbert space as a quantum-enhanced feature space in machine learning models. While the quantum feature map has demonstrated its capability when combined with linear classification models in some specific applications, its expressive power from the theoretical perspective remains unknown. We prove that the machine learning models induced from the quantum-enhanced feature space are universal approximators of continuous functions under typical quantum feature maps. We also study the capability of quantum feature maps in the classification of disjoint regions. Our work enables an important theoretical analysis to ensure that machine learning algorithms based on quantum feature maps can handle a broad class of machine learning tasks. In light of this, one can design a quantum machine learning model with more powerful expressivity.

• The paper shows that quantum machine learning models achieve universal approximation of continuous functions through both parallel and sequential feature map setups.
• It rigorously analyzes two scenarios—the tensor product of multiple circuits and repeated single-qubit models—to demonstrate how nonlinearity is effectively harnessed.
• The findings underscore the potential of quantum-enhanced feature spaces to drive improvements in ML performance on near-term quantum devices.

Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces

The paper elucidates the universal approximation capabilities of quantum machine learning models when operating in quantum-enhanced feature spaces. Specifically, it focuses on the theoretical foundations that ensure that these models can approximate continuous functions using quantum feature maps. As classical data are encoded into quantum states via quantum feature maps, such representations provide a mechanism to exploit quantum phenomena in solving machine learning tasks on near-term quantum computers.

Key Contributions

1. Universal Approximation Property (UAP): The research rigorously demonstrates that quantum machine learning models can achieve the UAP of continuous functions across typical quantum feature maps. The paper articulates two principal scenarios for quantum feature map setups: parallel and sequential. In both scenarios, the models' capacity to approximate functions is affirmed, albeit under different conditions and assumptions.
2. Parallel Scenario: Here, the paper models quantum feature maps as a tensor product of multiple circuits acting on subsystems, with a flexible number of qubits. The paper posits that these maps can approximate any continuous function by leveraging polynomials generated in quantum feature spaces under Stone–Weierstrass theorem assumptions. The use of fixed Pauli rotations combined with simple observables provides a robust system foundation for achieving UAP.
3. Sequential Scenario: The paper considers constraints of having a fixed number of qubits, specifically exploring a single-qubit model where a simple circuit is repeated. It establishes that UAP can be achieved on finite sets of inputs, given that the rotation angles used in the circuits introduce the necessary nonlinear transformations.
4. Implications of Nonlinearity: The theoretical framework also emphasizes the critical role of nonlinearity, which can be introduced either through quantum observables or via classical pre-processing steps involving activations functions in the data encodings.
5. Approximation Rate: An analysis on approximation rates indicates how the error in approximating target functions decays as the complexity of the model increases, providing insights into the efficiency of quantum models relative to classical machine learning frameworks.

Implications for Quantum Machine Learning

The findings of this paper hold significant implications for both theoretical and practical advancements in quantum machine learning (QML). The universality of quantum models suggests they can match or potentially exceed the expressive power of classical models in complex learning tasks. Additionally, it reaffirms that quantum-enhanced feature spaces offer a potent avenue for handling a wide array of machine learning tasks, paving the way for developing more refined and effective quantum algorithms.

This research sets a precedent for how models can be structured on near-term quantum devices to maximize their approximating power, thus influencing the design and implementation strategies of future quantum models. It encourages further exploration into efficient designs, potential computational advantages, and execution strategies on noisy intermediate-scale quantum (NISQ) devices.

Future Directions

While the paper provides a solid theoretical understanding, practical implementation and empirical validation on emerging quantum hardware will be paramount. The challenge remains in optimizing these models for realistic constraints, such as decoherence and limited qubit availability. There is also room for extending the model to explore unsupervised learning paradigms and hybrid quantum-classical frameworks, where classical computations augment quantum advantages.

In conclusion, the universal approximation properties discussed provide a foundational pillar for advancing quantum machine learning, resonating with the universal expressivity and flexibility necessary for future quantum applications.