Support Vector Machines with Quantum State Discrimination

Abstract

We analyze possible connections between quantum-inspired classifications and support vector machines. Quantum state discrimination and optimal quantum measurement are useful tools for classification problems. In order to use these tools, feature vectors have to be encoded in quantum states represented by density operators. Classification algorithms inspired by quantum state discrimination and implemented on classic computers have been recently proposed. We focus on the implementation of a known quantum-inspired classifier based on Helstrom state discrimination showing its connection with support vector machines and how to make the classification more efficient in terms of space and time acting on quantum encoding. In some cases, traditional methods provide better results. Moreover, we discuss the quantum-inspired nearest mean classification.

An efficient geometric approach to quantum-inspired classifications

“…The low-dimensional experiments, whose results are reported in Table 1 , are performed encoding feature vectors of into quantum states on by means of ( 21 ). In this case, we observe that the performances of the Helstrom classifier are comparable to those of the linear SVM as expected 7 , except for the datasets moons and prnn_synth where the SVM turns out to be definitely more accuarate. However, for the linearly_separable dataset, Helstrom reaches a high average accuracy and for the datasets analcatdata_boxing2 and lupus it is the most accurate classifier, with a tiny margin, over the classical and the quantum-inspired ones.…”

“…We considered algorithms based on the construction of an optimal measurement for state discrimination: the Helstrom classifier based on the well-known Helstrom’s theory of quantum discrimination 2 , a classifier based on the so-called Pretty Good measurement 10 and a classifier based on the geometric construction of the minimum-error measurement 11 . Moreover we considered quantum-inspired nearest mean classifiers based on the encoding of data into density operators and the calculation of distances which quantify the distinguishability of quantum states in the spirit of other works on this subject 1 , 7 . The considered operator distances were: trace distance, Bures distance, Hellinger distance, Jensen–Shannon distance.…”

“…The corresponding Helstrom measurement is , and the Helstrom bound satisfies 6 : Enlarging the Hilbert space of the quantum encoding, one increases the Helstrom bound obtaining a more accurate classifier. Since the Helstrom classifier is similar to a support vector machine with linear kernel 7 , considering many copies of the encoding quantum states give rise to a kernel trick. The corresponding computational cost is evident; however, in the following, we observe that in the case of real input vectors, the space can be enlarged saving time and space by means of the encoding into Bloch vectors.…”

“…Let us consider a binary classification problem (the multi-class generalization is straightforward). As suggested in 7 , we can define a classification algorithm that evaluates the Bures distance between the pure state encoding a test point and the centroids that correspond to mixed states. The fidelity between density operators, defined as , reduces to when .…”

Quantum-Inspired Applications for Classification Problems

“…In particular, we perform multi-class classification directly (without using binary classifiers) based on Helstrom discrimination following an approach suggested by Blanzieri and Melgani [ 2 ], where an unlabeled data instance is classified by finding its k nearest training elements before running a support vector machine (SVM) over the k training elements. This local approach improves the accuracy in classification and motivates the integration with the quantum-inspired Helstrom classifier since the latter can be interpreted as a SVM with linear kernel [ 3 ]. It has the potential to offer comparable performance using less complexity because it uses few training points per test point.…”