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arXiv:2107.05224 · EPJ Quantum Technology

Fock state-enhanced expressivity of quantum machine learning models

Beng Yee Gan, Daniel Leykam, Dimitris G. Angelakis

Published Jun 20, 2022• 32 citations
arXivHTMLPDF

Abstract

Abstract The data-embedding process is one of the bottlenecks of quantum machine learning, potentially negating any quantum speedups. In light of this, more effective data-encoding strategies are necessary. We propose a photonic-based bosonic data-encoding scheme that embeds classical data points using fewer encoding layers and circumventing the need for nonlinear optical components by mapping the data points into the high-dimensional Fock space. The expressive power of the circuit can be controlled via the number of input photons. Our work sheds some light on the unique advantages offered by quantum photonics on the expressive power of quantum machine learning models. By leveraging the photon-number dependent expressive power, we propose three different noisy intermediate-scale quantum-compatible binary classification methods with different scaling of required resources suitable for different supervised classification tasks.

DOI
10.1140/epjqt/s40507-022-00135-0

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Reservoir models offer a hardware-efficient learning paradigm for noisy intermediate-scale quantum devices by exploiting untrained quantum dynamics as a fixed feature map and restricting optimization to a simple classical readout layer. We propose a quantum extreme learning machine implemented using gaussian boson sampling and an encoding strategy that achieves high classification accuracy while reducing optical resource requirements. Classical inputs are jointly encoded in the squeezing parameters and in the interferometer unitary, enabling sampling-based, highly nonlinear feature maps while leveraging large-scale GBS output statistics, which are conjectured to be classically intractable. We systematically compare multiple families of quantum features accessible in the same setup and find that photon-number sampling probabilities provide the best performance, consistent with their higher effective feature dimensionality. Finally, we benchmark against classical nonlinear baselines and analyse robustness under noisy scenarios, showing competitive performance with fewer trainable parameters and indicating practical promise for near-term photonic implementations.
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. Although 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.
Abstract Photons are promising candidates for quantum information technology due to their high robustness and long coherence time at room temperature. Inspired by recent advances in photonic computing techniques, research has increasingly turned to quantum machine learning (QML) on photonic platforms. Although photons provide a high-dimensional quantum feature space suitable for computation, a general understanding of how photon number can be harnessed for learning tasks remains limited. Here, we establish both theoretically and experimentally a learning-capacity advantage of multi-photon states over single-photon states in photonic QML. We prove that the learning capacity of linear optical circuits, quantified by the rank of the data quantum Fisher information matrix, scales polynomially with the photon number. This scaling enables multi-photon models to generalize from fewer training states than corresponding single-photon models and to achieve lower test loss under the same architecture and learning protocol. Moreover, we experimentally corroborate these findings through unitary learning and metric learning tasks, by performing online training on a fully programmable photonic integrated platform. Our work highlights the potential of photonic QML and paves the way for achieving quantum enhancement in practical machine learning applications.
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.
Encoding classical inputs 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. 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 quantum feature map is a universal approximator of continuous functions under its typical settings in many practical applications. We also study the capability of the quantum feature map in the classification of disjoint regions. Our work enables an important theoretical analysis to ensure that quantum-enhanced 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.