Quantum computing can empower machine learning models by enabling kernel machines to leverage quantum kernels for representing similarity measures between data. Quantum kernels are able to capture relationships in the data that are not efficiently computable on classical devices. However, there is no straightforward method to engineer the optimal quantum kernel for each specific use case. While recent literature has focused on exploiting the potential offered by the presence of symmetries in the data to guide the construction of quantum kernels, we adopt here a different approach, which employs optimization techniques, similar to those used in neural architecture search and AutoML, to automatically find an optimal kernel in a heuristic manner. The algorithm we present constructs a quantum circuit implementing the similarity measure as a combinatorial object, which is evaluated based on a cost function and is then iteratively modified using a meta-heuristic optimization technique. The cost function can encode many criteria ensuring favorable statistical properties of the candidate solution, such as the rank of the Dynamical Lie Algebra. Importantly, our approach is independent of the optimization technique employed. The results obtained by testing our approach on a high-energy physics problem demonstrate that, in the best-case scenario, we can either match or improve testing accuracy with respect to the manual design approach, showing the potential of our technique to deliver superior results with reduced effort.

Exploiting the properties of quantum information to the benefit of machine learning models is perhaps the most active field of research in quantum computation. This interest has supported the development of a multitude of software frameworks (e.g. Qiskit, Pennylane, Braket) to implement, simulate, and execute quantum algorithms. Most of them allow us to define quantum circuits, run basic quantum algorithms, and access low-level primitives depending on the hardware such software is supposed to run. For most experiments, these frameworks have to be manually integrated within a larger machine learning software pipeline. The researcher is in charge of knowing different software packages, integrating them through the development of long code scripts, analyzing the results, and generating the plots. Long code often leads to erroneous applications, due to the average number of bugs growing proportional with respect to the program length. Moreover, other researchers will struggle to understand and reproduce the experiment, due to the need to be familiar with all the different software frameworks involved in the code script. We propose QuASK, an open-source quantum machine learning framework written in Python that aids the researcher in performing their experiments, with particular attention to quantum kernel techniques. QuASK can be used as a command-line tool to download datasets, pre-process them, quantum machine learning routines, analyze and visualize the results. QuASK implements most state-of-the-art algorithms to analyze the data through quantum kernels, with the possibility to use projected kernels, (gradient-descent) trainable quantum kernels, and structure-optimized quantum kernels. Our framework can also be used as a library and integrated into pre-existing software, maximizing code reuse.

Topological data analysis (TDA) has emerged as a powerful tool for extracting meaningful insights from complex data. TDA enhances the analysis of objects by embedding them into a simplicial complex and extracting useful global properties such as the Betti numbers, i.e. the number of multidimensional holes, which can be used to define kernel methods that are easily integrated with existing machine-learning algorithms. These kernel methods have found broad applications, as they rely on powerful mathematical frameworks which provide theoretical guarantees on their performance. However, the computation of higher-dimensional Betti numbers can be prohibitively expensive on classical hardware, while quantum algorithms can approximate them in polynomial time in the instance size. In this work, we propose a quantum approach to defining topological kernels, which is based on constructing Betti curves, i.e. topological fingerprint of filtrations with increasing order. We exhibit a working prototype of our approach implemented on a noiseless simulator and show its robustness by means of some empirical results suggesting that topological approaches may offer an advantage in quantum machine learning.

Building a quantum analog of classical deep neural networks represents a fundamental challenge in quantum computing. A key issue is how to address the inherent non-linearity of classical deep learning, a problem in the quantum domain due to the fact that the composition of an arbitrary number of quantum gates, consisting of a series of sequential unitary transformations, is intrinsically linear. This problem has been variously approached in the literature, principally via the introduction of measurements between layers of unitary transformations. In this paper, we introduce the Quantum Path Kernel, a formulation of quantum machine learning capable of replicating those aspects of deep machine learning typically associated with superior generalization performance in the classical domain, specifically, hierarchical feature learning. Our approach generalizes the notion of Quantum Neural Tangent Kernel, which has been used to study the dynamics of classical and quantum machine learning models. The Quantum Path Kernel exploits the parameter trajectory, i.e. the curve delineated by model parameters as they evolve during training, enabling the representation of differential layer-wise convergence behaviors, or the formation of hierarchical parametric dependencies, in terms of their manifestation in the gradient space of the predictor function. We evaluate our approach with respect to variants of the classification of Gaussian XOR mixtures - an artificial but emblematic problem that intrinsically requires multilevel learning in order to achieve optimal class separation.