Quantum and machine learning applications involving matrices

Annie Naveh, Imogen Fitzgerald, Phan, Anna +2 more

Enhancing classical machine learning (ML) algorithms through quantum kernels is a rapidly growing research topic in quantum machine learning (QML). A key challenge in using kernels -- both classical and quantum -- is that ML workflows involve acquiring new observations, for which new kernel values need to be calculated. Transferring data back-and-forth between where the new observations are generated & a quantum computer incurs a time delay; this delay may exceed the timescales relevant for using the QML algorithm in the first place. In this work, we show quantum kernel matrices can be extended to incorporate new data using a classical (chordal-graph-based) matrix completion algorithm. The minimal sample complexity needed for perfect completion is dependent on matrix rank. We empirically show that (a) quantum kernel matrices can be completed using this algorithm when the minimal sample complexity is met, (b) the error of the completion degrades gracefully in the presence of finite-sampling noise, and (c) the rank of quantum kernel matrices depends weakly on the expressibility of the quantum feature map generating the kernel. Further, on a real-world, industrially-relevant data set, the completion error behaves gracefully even when the minimal sample complexity is not reached.

Priyanka Jammwal

Jincheng Zhang

This paper introduces a novel quantum machine learning framework based on self-organizing matrices. We propose a method for automated quantum machine learning model construction and optimization, leveraging the inherent properties of self-organizing matrices to achieve more efficient training. Traditional quantum machine learning often relies on manual design, while our approach offers a paradigm shift toward automated and enhanced learning capabilities. The core mechanism involves utilizing the self-organizing matrix structure to inherently guide the model's evolution, leading to improved training speed and potentially higher accuracy. We detail the framework's architecture, demonstrate its efficacy through a series of benchmark tasks, and analyze its potential impact on quantum machine learning. The research explores the practical implications of this new approach, aiming to address key challenges in the field.

Marco Marcozzi, Ernestas Filatovas, Remigijius Paulavičius

Anna Bernasconi, Alessandro Berti, Gianna M. Del Corso +1 more

We propose an efficient quantum subroutine for matrix multiplication that computes a state vector encoding the entries of the product of two matrices in superposition. The subroutine exploits efficient state preparation techniques and shows a potential speed-up with respect to classical methods. The most important benefit of our subroutine is that it encodes the entries of the matrix product directly in the state vector, which can be used for further computations within the same quantum circuit. All scenarios involving the computation of non-homomorphic functions of the product of two matrices can benefit from our technique. As a possible application, we discuss the computation of the variance of the entries of a matrix product, which can be a useful tool for some machine learning algorithms.