The application of quantum computation to accelerate machine learning algorithms is one of the most promising areas of research in quantum algorithms. In this paper, we explore the power of quantum learning algorithms in solving an important class of Quantum Phase Recognition (QPR) problems, which are crucially important in understanding many-particle quantum systems. We prove that, under widely believed complexity theory assumptions, there exists a wide range of QPR problems that cannot be efficiently solved by classical learning algorithms with classical resources. Whereas using a quantum computer, we prove the efficiency and robustness of quantum kernel methods in solving QPR problems through Linear order parameter Observables. We numerically benchmark our algorithm for a variety of problems, including recognizing symmetry-protected topological phases and symmetry-broken phases. Our results highlight the capability of quantum machine learning in predicting such quantum phase transitions in many-particle systems.
The application of quantum computation to accelerate machine learning algorithms is one of the most promising areas of research in quantum algorithms. In this paper, we explore the power of quantum learning algorithms in solving an important class of Quantum Phase Recognition (QPR) problems, which are crucially important in understanding many-particle quantum systems. We prove that, under widely believed complexity theory assumptions, there exists a wide range of QPR problems that cannot be efficiently solved by classical learning algorithms with classical resources. Whereas using a quantum computer, we prove the efficiency and robustness of quantum kernel methods in solving QPR problems through Linear order parameter Observables. We numerically benchmark our algorithm for a variety of problems, including recognizing symmetry-protected topological phases and symmetry-broken phases. Our results highlight the capability of quantum machine learning in predicting such quantum phase transitions in many-particle systems.
Abstract The use of quantum computation to speed-up machine learning algorithms is among the most exciting prospective applications in the NISQ era. Here, we focus on the quantum phase learning problem, which is crucially important in understanding many-particle quantum systems. We prove that, under widely believed complexity theory assumptions, quantum phase learning problem cannot be efficiently solved by machine learning algorithms using classical resources and classical data. Whereas using quantum data, we prove the universality of quantum kernel Alphatron in efficiently predicting quantum phases, indicating clear quantum advantages in such learning problems. We numerically benchmark the algorithm for a variety of problems, including recognizing symmetry-protected topological phases and symmetry-broken phases. Our results highlight the capability of quantum machine learning in efficient prediction of quantum phases of many-particle systems.
Abstract The use of quantum computation to speed-up machine learning algorithms is among the most exciting prospective applications in the NISQ era. Here, we focus on the quantum phase learning problem, which is crucially important in understanding many-particle quantum systems. We prove that, under widely believed complexity theory assumptions, quantum phase learning problem cannot be efficiently solved by machine learning algorithms using classical resources and classical data. Whereas using quantum data, we prove the universality of quantum kernel Alphatron in efficiently predicting quantum phases, indicating clear quantum advantages in such learning problems. We numerically benchmark the algorithm for a variety of problems, including recognizing symmetry-protected topological phases and symmetry-broken phases. Our results highlight the capability of quantum machine learning in efficient prediction of quantum phases of many-particle systems.
Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground-state properties of gapped Hamiltonians after learning from other Hamiltonians in the same quantum phase of matter. By contrast, under a widely accepted conjecture, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, two-dimensional random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.
Classical machine learning has succeeded in the prediction of both classical and quantum phases of matter. Notably, kernel methods stand out for their ability to provide interpretable results, relating the learning process with the physical order parameter explicitly. Here we exploit quantum kernels instead. They are naturally related to the fidelity, and thus it is possible to interpret the learning process with the help of quantum information tools. In particular, we use a support vector machine (with a quantum kernel) to predict and characterize second-order quantum phase transitions. We explain and understand the process of learning when the fidelity per site (rather than the fidelity) is used. The general theory is tested in the Ising chain in transverse field. We show that for small-sized systems, the algorithm gives accurate results, even when trained away from criticality. Besides, for larger sizes we confirm the success of the technique by extracting the correct critical exponent $\ensuremath{\nu}$. Finally, we present two algorithms, one based on fidelity and one based on the fidelity per site, to classify the phases of matter in a quantum processor.
Machine learning techniques give impressive results in many areas. However, due to the physical limitation of integrated circuits which restricts their computational power growth, and the rapid advances in quantum computing, lots of research studies on quantum machine learning (QML) have been done recently. QML is a technique that uses quantum algorithms as parts of the implementation. Quantum algorithms use quantum mechanics and have the potential to outperform classical algorithms for a given problem. In this paper, three widely used machine learning algorithms are discussed and their quantum versions are presented, namely: quantum neural network, quantum autoencoder, and quantum kernel method. In addition, we discuss the potential capabilities of these QML algorithms and review recent work employing them. Moreover, a quantum neural network prototype is implemented using Qiskit as a proof of concept and tested on a real quantum computer. Empirical results show that quantum neural networks can be trained efficiently.
Abstract Quantum machine learning, an emerging field at the intersection of quantum computing and classical machine learning, has shown great promise in enhancing computational capabilities beyond classical bounds. A key element in this area of research is the utilization of Quantum Kernel Estimators, traditionally grounded in the symmetries of SU(2) groups associated with qubits. This study extends the conceptual framework of Quantum Kernel Estimators to incorporate a broader spectrum of symmetry groups. By harnessing the various structures of Lie groups, we develop novel quantum-inspired feature maps that offer more flexible and potentially powerful ways to encode and compress classical data into quantum states. We present a comprehensive theoretical introduction for this approach, followed by a methodology that integrates the developed feature maps into kernel-based quantum classifiers. Our results, derived from a series of computational experiments across various datasets, demonstrate the efficacy of this approach in comparison to traditional quantum and classical machine learning models. The findings not only underline the versatility of Lie-group theory in enhancing quantum machine learning algorithms but also open new avenues for exploring complex symmetries in quantum information processing. This research bridges a gap between the study of symmetries and machine learning, paving the way for more sophisticated quantum algorithms capable of tackling complex, high-dimensional data in ways previously unattainable.