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arXiv:2310.04300 · Quantum Science and Technology

Quantum kernels for classifying dynamical singularities in a multiqubit system

Diego Tancara, José Fredes, Ariel Norambuena

Published May 30, 2024• 2 citations
arXivHTMLPDF

Abstract

Abstract Dynamical quantum phase transition is a critical phenomenon involving out-of-equilibrium states and broken symmetries without classical analogy. However, when finite-sized systems are analyzed, dynamical singularities of the rate function can appear, leading to a challenging physical characterization when parameters are changed. Here, we report a quantum support vector machine algorithm that uses quantum Kernels to classify dynamical singularities of the rate function for a multiqubit system. We illustrate our approach using N long-range interacting qubits subjected to an arbitrary magnetic field, which induces a quench dynamics. Inspired by physical arguments, we introduce two different quantum Kernels, one inspired by the ground state manifold and the other based on a single state tomography. Our accuracy and adaptability results show that this quantum dynamical critical problem can be efficiently solved using physically inspiring quantum Kernels. Moreover, we extend our results for the case of time-dependent fields, quantum master equation, and when we increase the number of qubits.

DOI
10.1088/2058-9565/ad5228

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Abstract Dynamical quantum phase transition is a critical phenomenon involving out-of-equilibrium states and broken symmetries without classical analogy. However, when finite-sized systems are analyzed, dynamical singularities of the rate function can appear, leading to a challenging physical characterization when parameters are changed. Here, we report a quantum support vector machine algorithm that uses quantum Kernels to classify dynamical singularities of the rate function for a multiqubit system. We illustrate our approach using N long-range interacting qubits subjected to an arbitrary magnetic field, which induces a quench dynamics. Inspired by physical arguments, we introduce two different quantum Kernels, one inspired by the ground state manifold and the other based on a single state tomography. Our accuracy and adaptability results show that this quantum dynamical critical problem can be efficiently solved using physically inspiring quantum Kernels. Moreover, we extend our results for the case of time-dependent fields, quantum master equation, and when we increase the number of qubits.
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.
Abstract Unveiling quantum phase transitions (QPTs) is important for characterising physical systems at low temperatures. However, the detection of these transitions is encumbered by significant challenges, especially in the face of the exponential growth in ground state complexity with system scale. The emergence of quantum machine learning has lately gained traction as a promising method for elucidating the properties of many-body systems, providing a different avenue to study QPT. In this paper, we propose a novel and efficient quantum algorithm for identifying QPT synthesising quantum feature with quantum machine learning. Our approach is anchored in the utilisation of quantum computers to directly encode the kernel matrix into Hilbert spaces, realised by the parallel implementation of the quantum feature map. Specifically, we generate a quantum state encoding the information of ground states of the given quantum systems by employing the parallel quantum feature map. The resultant state preparation circuit is then used to implement a block-encoding of the kernel matrix. Equipped with the associated labels and this encoding, we devise a new quantum support vector machine (QSVM) algorithm, forming the main ingredient of the classifier. The presented method refines the efficiency of the prevailing QSVM algorithm for processing quantum and classical data. We demonstrate the effectiveness of our quantum classifier in predicting QPT within the transverse-field Ising model. The findings affirm the efficacy of quantum machine learning in recognising QPT in many-body systems and offer insights into the design of quantum machine learning algorithms.
Quantum kernels offer a valid procedure for learning quantum phase transitions on quantum processing devices, yet issues on the scalability of the learning strategy in connection with the symmetry of the critical model have not been clarified. We derive a link between model symmetry and fidelity-kernel resource scaling. We quantify the measurement resources required to estimate fidelity-based quantum kernels for many-body ground states while preserving the structure of the resulting Gram matrix under finite-shot sampling. Crucially, we show that increasing symmetry in the underlying spin model systematically amplifies these shot requirements. Moving from the Z2\mathbb{Z}_2Z2​-symmetric Ising/XY regimes to the U(1)U(1)U(1)-symmetric XX (and XXZ) regimes leads to stronger kernel concentration and therefore substantially larger shot costs under the same bounds. We consider a tunable one-dimensional spin-12\tfrac{1}{2}21​ Hamiltonian spanning the transverse-field Ising, XY, XX, and XXZ limits, and define the kernel as the ground-state fidelity. Kernel entries are estimated using a SWAP-test estimator with SSS shots, and we adapt the ensemble spread and concentration-avoidance shot bounds to obtain practical shot requirements in terms of the interquartile range of kernel values and a representative kernel magnitude. For the free-fermion XY/XX family, we use the closed-form Bogoliubov-angle fidelity, while for the interacting XXZ chain we compute fidelities by exact diagonalization and benchmark shot-noise effects. Our symmetry-aware bounds provide a pragmatic procedure for physics-informed quantum machine learning.
This paper proposes a novel adaptive quantum kernel learning framework designed to overcome the limitations of classical kernel methods when applied to quantum state analysis. The core challenge lies in effectively mapping the high-dimensional space of quantum states into a manageable, low-dimensional feature space for traditional kernel-based learning algorithms. Existing methods often fail to capture the intricate correlations inherent in quantum states, leading to suboptimal results. Our approach introduces a self-adaptive kernel learning algorithm that dynamically adjusts the kernel function based on the observed quantum state correlations. This adaptation is achieved through a quantum-inspired optimization process utilizing variational quantum circuits. The circuits evolve the kernel parameters, guided by quantum state tomography data, effectively learning a kernel function tailored to the specific quantum system being analyzed. The key innovation is the integration of quantum mechanics directly into the kernel learning process, offering a potentially significant improvement in feature extraction and classification accuracy for quantum data. This work addresses a critical bottleneck in quantum machine learning, paving the way for more powerful and accurate quantum algorithms.
Quantum support vector machines are classification algorithms that rely on quantum-generated kernels. The fidelity quantum kernel commonly used in quantum support vector machines suffers from exponential concentration as system size increases, preventing an efficient scaling beyond fewqubit systems. We introduce the Hamming quantum kernel, a classical post-processing method that is based on the same measurement outcomes as the fidelity quantum kernel. However, it avoids the exponential concentration problem by using the full measurement statistics rather than a single fidelity value. We evaluate the approach on both classical data (MNIST) and synthetic data generated from quantum circuits, using systems ranging from 2 to 27 qubits. Throughout the simulations, the Hamming quantum kernel outperforms the fidelity quantum kernel whenever 15 or more qubits are used. Furthermore, for synthetic quantum data, our method consistently outperforms the classical Gaussian kernel. This demonstrates that the Hamming quantum kernel improves the expressivity and robustness at larger qubit scales without requiring any additional quantum ressources.
Quantum computing is an emerging field that can effectively solve several machine learning problems. The immensely growing data size has begun to create barriers for classical machine learning algorithms. Quantum computers proficiently handle and process big data composed of metrics and vectors. This paper presents the quantum machine learning model based on quantum support vector machines (QSVM) to classify brain tumours into malignant and benign. The dataset used in this experiment is Brats 2015, available on Kaggle. The QSVM-based model was run on real-time quantum machines and simulators to extract the performance metrics in terms of accuracy and execution time. The QSVM classification model is compared with its classical SVM equivalent. The kernel-based QSVM model implementation on a 32-qubit quantum simulator was 188 times faster with 95% accuracy which is 1.60% better than the classical equivalent. Similarly, the model takes 24.19% less time than the classical SVM model when realized on a 5-qubit real-time superconducting processor with the same accuracy. Hence, the results reveal that quantum machine learning models implemented on quantum computers outperformed their classical equivalents.