Quantum phase transition detection via quantum support vector machine

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.