Research Article: A new quantum approach to binary classification

Date Published: May 9, 2019

Publisher: Public Library of Science

Author(s): Giuseppe Sergioli, Roberto Giuntini, Hector Freytes, Austin Lund.


This paper proposes a new quantum-like method for the binary classification applied to classical datasets. Inspired by the quantum Helstrom measurement, this innovative approach has enabled us to define a new classifier, called Helstrom Quantum Centroid (HQC). This binary classifier (inspired by the concept of distinguishability between quantum states) acts on density matrices—called density patterns—that are the quantum encoding of classical patterns of a dataset. In this paper we compare the performance of HQC with respect to twelve standard (linear and non-linear) classifiers over fourteen different datasets. The experimental results show that HQC outperforms the other classifiers when compared to the Balanced Accuracy and other statistical measures. Finally, we show that the performance of our classifier is positively correlated to the increase in the number of “quantum copies” of a pattern and the resulting tensor product thereof.

Partial Text

In the past few decades, various methods based on quantum information theory have been used extensively to focus on a variety of problems concerning classification and clustering [1–4]. On the other hand, some classification methods developed in computer engineering have been employed to solve such problems as quantum-state discrimination [5–8], which are closely connected with certain recent developments in quantum cryptography. In view of these exchanges, quantum computation and machine learning are nowadays recognized as two closely-related connected research fields. A natural starting point for bridging these two different topics is to establish a common background. The initial idea was to represent classical patterns in terms of quantum objects, with an eye to increasing the computational efficiency of the classification algorithms. Following this intuition, in the past few years many attempts have been made to apply the quantum formalism to signal processing [9] and pattern recognition [10, 11].

The general purpose of a classification process is to classify a set of objects, i.e., to assign to each object from the set, a label that represents real classes (for instance the class of the cats, the class of the cancer cells, the class of the human faces, etc). Following a standard classification procedure of supervised learning systems (i.e., learning from a training dataset of correctly labeled objects) we initially select the d features that characterize all the objects of a given dataset. Thus, each object is represented by a d-dimensional real vector X=(x1,…,xd)∈Rd. Formally, we say that a pattern is represented by a pair (Xi, li), where Xi is the d-dimensional vector associated with the object and li labels the class of which the objects is a member. We consider a class as being merely a set of objects and we confine ourselves to the very common case where each object belongs to one class of objects only. Let L = {l1, …, lM} be the set of labels corresponding to their respective classes. The goal of the classification process is to design a classifier that will attributes (in the most accurate way possible) a label (a class) to any unlabeled object. The strategy is divided into two stages; first, training the classifier and second, performing the test proper. The dataset is therefore also divided into two parts: the first is used to train the classifier and the second is used to properly verify the accuracy of the classifier during a test [15, 20].

In this section we describe a quantum-style classification process for binary classification based on the Helstrom measurement (see [22, 23]), which was initially introduced by Helstrom in a seminal work that addressed the following question: “Suppose to deal with an unknown quantum state drawn from an ensemble of possible pure states where each state is labeled with respect to the class they came from. How well can we predict the class of this unknown quantum state?” [24]. This problem is generally known as the quantum state discrimination problem or quantum classification problem [7] when referred to machine learning. The answer clearly depends on a number of factors—e.g. crucially, on the amount of available information and on the way this quantity of information might be improved. Unlike in the classical case, in quantum information multiple copies of a quantum state provide more information about it than that is encoded in a single copy thereof. We shall exploit this property in order to improve the classification process inspired by Helstrom’s construct.

In this section we show some significant improvement to the performances of HQC and (⊗n)HQC with respect to a large set of classifiers that generally perform well for different kinds of datasets. Given the nature of HQC, we restrict the experimental setup to binary datasets, i.e., datasets with only two classes.

In this paper we have introduced an innovative technique—inspired by the formalism of quantum theory—to design a new kind of supervised classifier (HQC) and we have provided a full comparison of the performances of this new classifier with respect to many others (linear and non linear) frequently used classifiers. The HQC proves to be superior, on average, compared to all the statistical parameters we considered. As a result, we believe that the potential of the quantum formalism as an application for classification processes in the classical context is extremely promising.

Let us consider two classes of the same cardinality C1 = {ρi}i=1,⋯,n and C2 = {σi}i=1,⋯,n of 2 × 2- diagonal density matrices, where diag[ρi] = [1 − ri, ri] and diag[σi] = [1 − si, si] and where the non trivial cases 0 < ri, si < 1 are considered. Let us indicate by ρ and σ the centroids of the classes C1 and C2, respectively, and let us suppose that ρ ≠ σ. By referring to Eq 10, we show that Pguess((⊗2)ρ,(⊗2)σ)≥Pguess(ρ,σ).   Source:


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