At the intersection of machine learning and quantum computing, quantum machine learning has the potential of accelerating data analysis, especially for quantum data, with applications for quantum materials, biochemistry and high-energy physics. Nevertheless, challenges remain regarding the trainability of quantum machine learning models. Here we review current methods and applications for quantum machine learning. We highlight differences between quantum and classical machine learning, with a focus on quantum neural networks and quantum deep learning. Finally, we discuss opportunities for quantum advantage with quantum machine learning. Quantum machine learning has become an essential tool to process and analyze the increased amount of quantum data. Despite recent progress, there are still many challenges to be addressed and myriad future avenues of research. 

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Challenges and opportunities in quantum machine learning

Precipitation has been of utmost importance in shaping the evolution of landscapes and human settlements in the Mediterranean. However, information on seasonal precipitation patterns through the Holocene is scarce. This study attempts to quantify the evolution of seasonal precipitation in the East Iberian Peninsula (5000 BC to AD 600) based on the carbon isotope composition (δ13C) of archaeobotanical remains. Data on Holm oak, Aleppo pine and small-grain cereals were combined, and precipitation was inferred from models relating present-day records to the δ13C of modern samples. Subsequently, charred grains were used as a proxy for ancient moisture during April–May, whereas oak and pine charcoals provided complementary rainfall estimates for September–December and January–August, respectively. The results reveal aridity changes throughout the Holocene in the western Mediterranean. Past spring–summer precipitation was consistently higher than at present. In contrast, autumn and early winter precipitation showed stronger fluctuations, particularly during the first millennium BC, and often exhibited values below those of the present. The high contribution of autumn precipitation to the annual water budget, typical of the present Mediterranean climate, was definitively established at the beginning of the current era. This study shows how a combination of species holding complementary environmental signals can contribute to a wider knowledge of local precipitation dynamics.

Holocene changes in precipitation seasonality in the western Mediterranean Basin: a multi-species approach using δ13C of archaeobotanical remains

Quantum Machine Learning (QML) models are aimed at learning from data encoded in quantum states. Recently, it has been shown that models with little to no inductive biases (i.e., with no assumptions about the problem embedded in the model) are likely to have trainability and generalization issues, especially for large problem sizes. As such, it is fundamental to develop schemes that encode as much information as available about the problem at hand. In this work we present a simple, yet powerful, framework where the underlying invariances in the data are used to build QML models that, by construction, respect those symmetries. These so-called group-invariant models produce outputs that remain invariant under the action of any element of the symmetry group G associated to the dataset. We present theoretical results underpinning the design of G -invariant models, and exemplify their application through several paradigmatic QML classiﬁcation tasks including cases when G is a continuous Lie group and also when it is a discrete symmetry group. Notably, our framework allows us to recover, in an elegant way, several well known algorithms for the literature, as well as to discover new ones. Taken together, we expect that our results will help pave the way towards a more geometric and group-theoretic approach to QML model design.

http://link.aps.org/pdf/10.1103/PRXQuantum.3.030341

Group-Invariant Quantum Machine Learning

Quantum error mitigation has been proposed as a means to combat unwanted and unavoidable errors in near-term quantum computing using no or few additional quantum resources, in contrast to fault-tolerant schemes that come along with heavy overheads. Error mitigation has been successfully applied to reduce noise in near-term applications. In this work, however, we identify strong limitations to the degree to which quantum noise can be effectively `undone' for larger system sizes. We set up a framework that rigorously captures large classes of error mitigation schemes in use today. The core of our argument combines fundamental limits of statistical inference with a construction of families of random circuits that are highly sensitive to noise. We show that even at poly log log n depth, a super-polynomial number of samples is needed in the worst case to estimate the expectation values of noiseless observables, the principal task of error mitigation. Notably, our construction implies that scrambling due to noise can kick in at exponentially smaller depths than previously thought. They also impact other near-term applications, constraining kernel estimation in quantum machine learning, causing an earlier emergence of noise-induced barren plateaus in variational quantum algorithms and ruling out exponential quantum speed-ups in estimating expectation values in the presence of noise or preparing the ground state of a Hamiltonian.

Exponentially tighter bounds on limitations of quantum error mitigation

Retomaremos neste texto a questao da cronologia da atribuicao dos estatutos juridico-administrativos a Pax Iulia, Felicitas Iulia Olisipo e Liberalitas Iulia Ebora, cidades cujas titulaturas, num evidente prenuncio do culto imperial, associavam as Virtudes quer ao divino Cesar, quer a Octaviano, uma vez que, a despeito de esta ultima associacao nao ter sido explicitamente assumida, ambos pertenciam a mesma gens.

https://dialnet.unirioja.es/descarga/articulo/750390.pdf

Pax Iulia, Felicitas Iulia, Liberalitas Iulia

La administración territorial de la Bética romana

Recent advances in classical machine learning have shown that creating models with inductive biases encoding the symmetries of a problem can greatly improve performance. Importation of these ideas, combined with an existing rich body of work at the nexus of quantum theory and symmetry, has given rise to the field of Geometric Quantum Machine Learning (GQML). Following the success of its classical counterpart, it is reasonable to expect that GQML will play a crucial role in developing problem-specific and quantum-aware models capable of achieving a computational advantage. Despite the simplicity of the main idea of GQML -- create architectures respecting the symmetries of the data -- its practical implementation requires a significant amount of knowledge of group representation theory. We present an introduction to representation theory tools from the optics of quantum learning, driven by key examples involving discrete and continuous groups. These examples are sewn together by an exposition outlining the formal capture of GQML symmetries via"label invariance under the action of a group representation", a brief (but rigorous) tour through finite and compact Lie group representation theory, a reexamination of ubiquitous tools like Haar integration and twirling, and an overview of some successful strategies for detecting symmetries.

Representation Theory for Geometric Quantum Machine Learning

Most currently used quantum neural network architectures have little-to-no inductive biases, leading to trainability and generalization issues. Inspired by a similar problem, recent breakthroughs in classical machine learning address this crux by creating models encoding the symmetries of the learning task. This is materialized through the usage of equivariant neural networks whose action commutes with that of the symmetry. In this work, we import these ideas to the quantum realm by presenting a general theoretical framework to understand, classify, design and implement equivariant quantum neural networks. As a special implementation, we show how standard quantum convolutional neural networks (QCNN) can be generalized to group-equivariant QCNNs where both the convolutional and pooling layers are equivariant under the relevant symmetry group. Our framework can be readily applied to virtually all areas of quantum machine learning, and provides hope to alleviate central challenges such as barren plateaus, poor local minima, and sample complexity.

María Luisa Cortijo Cerezo

Papers

Challenges and opportunities in quantum machine learning

Group-Invariant Quantum Machine Learning

La administración territorial de la Bética romana

Representation Theory for Geometric Quantum Machine Learning

Theory for Equivariant Quantum Neural Networks