Quantum Supervised Learning

TL;DR

This paper presents a classical perspective on quantum machine learning (QML), emphasizing how traditional supervised learning concepts—parametric vs non-parametric methods, empirical risk minimization, and generalization—inform the interpretation of quantum approaches. It maps the QML landscape into two main strands: fault-tolerant quantum machine learning (FT-QML) leveraging algorithms like HHL for linear systems, and hybrid quantum-classical models that operate on near-term noisy devices via parametrized quantum circuits. The authors analyze key FT-QML techniques (HHL, LS-SVM, quantum splines) and two hybrid paradigms (quantum kernels and classical-inspired quantum models), while detailing current challenges in trainability, barren plateaus, and gradient estimation. They conclude that a near-term quantum advantage is not guaranteed; progress will likely come from integrating classical ML insights with quantum methods and from hardware evolution, with surrogates and problem-specific advantages playing important roles in the interim.

Abstract

Recent advancements in quantum computing have positioned it as a prospective solution for tackling intricate computational challenges, with supervised learning emerging as a promising domain for its application. Despite this potential, the field of quantum machine learning is still in its early stages, and there persists a level of skepticism regarding a possible near-term quantum advantage. This paper aims to provide a classical perspective on current quantum algorithms for supervised learning, effectively bridging traditional machine learning principles with advancements in quantum machine learning. Specifically, this study charts a research trajectory that diverges from the predominant focus of quantum machine learning literature, originating from the prerequisites of classical methodologies and elucidating the potential impact of quantum approaches. Through this exploration, our objective is to deepen the understanding of the convergence between classical and quantum methods, thereby laying the groundwork for future advancements in both domains and fostering the involvement of classical practitioners in the field of quantum machine learning.

Figures (3)

• Figure 1: Visual representation of the field of Quantum Machine Learning as the intersection between different paradigms in machine learning (parametric and non-parametric) and quantum computation capabilities (Fault-tolerant and NISQ). Fault-Tolerant Quantum Machine Learning (FT-QML) involves quantum algorithms like Quantum Support Vector Machines rebentrost2014quantum (least-square formulation ye2007svm), Quantum Splines macaluso2020quantum, and Quantum Linear Regression PhysRevA.96.012335, which require error-corrected qubits and the capability to run arbitrarily long quantum circuits. Conversely, Hybrid QML incorporates Quantum Neural Networks macaluso2020variational and Quantum Kernels mengoni2019kernel, utilizing NISQ devices characterized by noisy and shallow quantum circuits. These approaches integrate with traditional machine learning methods. In particular, parametric models, which assume linearity in the underlying function $f$ and rely on convex optimization procedures, can benefit from the HHL algorithm PhysRevLett.103.150502 in the FT-QML setting. On the other hand, non-parametric models, which do not impose such assumptions, can be enhanced by utilizing parametrized quantum circuits (PQCs) to introduce a new class of hypotheses in the Hybrid QML settings.
• Figure 2: Schema for implementing a fault-tolerant quantum machine learning approach based on the HHL algorithm. The first step involves the model specification, where a preemptive assumption is made about the linear relationship between the target variable of interest and the basis expansion of the input feature $x$, such as in the case of spline functions. The second step is to formulate a linear system of equations of the form $A\ket{x} = \ket{b}$. It is important to note that in this context, the quantum state $\ket{x}$ does not correspond to the input $x$ but instead contains information about the classical set of parameters $\theta$. Once the linear system is established, the task of finding its solution is assigned to the HHL algorithm, which primarily comprises three sub-steps: (i) state preparation, involving the quantum gate $e^{iAt}$ and quantum state $\ket{b}$; (ii) execution of the HHL quantum circuit; and (iii) post-processing of $\ket{x}$ to classically extract the relevant information for estimating $\theta$, (the swap-test in the case of quantum splines). Finally, the parameters obtained through HHL are classically used to estimate the function of interest.
• Figure 3: Scheme of a hybrid quantum-classical algorithm for supervised learning (adapted from macaluso2022variational). The quantum variational circuit is depicted in green, while the classical component is represented in blue.