Single-preparation unsupervised quantum machine learning: concepts and applications

The term "machine learning" especially refers to algorithms that derive mappings, i.e. intput/output transforms, by using numerical data that provide information about considered transforms. These transforms appear in many problems, related to classification/clustering, regression, system identification, system inversion and input signal restoration/separation. We here first analyze the connections between all these problems, in the classical and quantum frameworks. We then focus on their most challenging versions, involving quantum data and/or quantum processing means, and unsupervised, i.e. blind, learning. Moreover, we propose the quite general concept of SIngle-Preparation Quantum Information Processing (SIPQIP). The resulting methods only require a single instance of each state, whereas usual methods have to very accurately create many copies of each fixed state. We apply our SIPQIP concept to various tasks, related to system identification (blind quantum process tomography or BQPT, blind Hamiltonian parameter estimation or BHPE, blind quantum channel identification/estimation, blind phase estimation), system inversion and state estimation (blind quantum source separation or BQSS, blind quantum entangled state restoration or BQSR, blind quantum channel equalization) and classification. Numerical tests show that our framework moreover yields much more accurate estimation than the standard multiple-preparation approach. Our methods are especially useful in a quantum computer, that we propose to more briefly call a "quamputer": BQPT and BHPE simplify the characterization of the gates of quamputers; BQSS and BQSR allow one to design quantum gates that may be used to compensate for the non-idealities that alter states stored in quantum registers, and they open the way to the much more general concept of self-adaptive quantum gates (see longer version of abstract in paper).Comment: 7 figure

Quantum process tomography with unknown single-preparation input states

Quantum Process Tomography (QPT) methods aim at identifying, i.e. estimating, a given quantum process. QPT is a major quantum information processing tool, since it especially allows one to characterize the actual behavior of quantum gates, which are the building blocks of quantum computers. However, usual QPT procedures are complicated, since they set several constraints on the quantum states used as inputs of the process to be characterized. In this paper, we extend QPT so as to avoid two such constraints. On the one hand, usual QPT methods requires one to know, hence to precisely control (i.e. prepare), the specific quantum states used as inputs of the considered quantum process, which is cumbersome. We therefore propose a Blind, or unsupervised, extension of QPT (i.e. BQPT), which means that this approach uses input quantum states whose values are unknown and arbitrary, except that they are requested to meet some general known properties (and this approach exploits the output states of the considered quantum process). On the other hand, usual QPT methods require one to be able to prepare many copies of the same (known) input state, which is constraining. On the contrary, we propose "single-preparation methods", i.e. methods which can operate with only one instance of each considered input state. These two new concepts are here illustrated with practical BQPT methods which are numerically validated, in the case when: i) random pure states are used as inputs and their required properties are especially related to the statistical independence of the random variables that define them, ii) the considered quantum process is based on cylindrical-symmetry Heisenberg spin coupling. These concepts may be extended to a much wider class of processes and to BQPT methods based on other input quantum state properties

Use and misuse of variances for quantum systems in pure or mixed states

As a consequence of the place ascribed to measurements in the postulates of quantum mechanics, if two differently prepared systems are described with the same density operator \r{ho}, they are said to be in the same quantum state. For more than fifty years, there has been a lack of consensus about this postulate. In a 2011 paper, considering variances of spin components, Fratini and Hayrapetyan tried to show that this postulate is unjustified. The aim of the present paper is to discuss major points in this 2011 article, and in their reply to a 2012 paper by Bodor and Diosi claiming that their analysis was irrelevant. Facing some ambiguities or inconsistencies in the 2011 paper and in the reply, we first try to guess their aim, then establish results useful in this context, and finally discuss the use or misuse of several concepts implied in this debate

Blind Signal Separation Methods for the Identification of Interstellar Carbonaceous Nanoparticles

The use of Blind Signal Separation methods (ICA and other approaches) for the analysis of astrophysical data remains quite unexplored. In this paper, we present a new approach for analyzing the infrared emission spectra of interstellar dust, obtained with NASA's Spitzer Space Telescope, using FastICA and Non-negative Matrix Factorization (NMF). Using these two methods, we were able to unveil the source spectra of three different types of carbonaceous nanoparticles present in interstellar space. These spectra can then constitute a basis for the interpretation of the mid-infrared emission spectra of interstellar dust in the Milky Way and nearby galaxies. We also show how to use these extracted spectra to derive the spatial distribution of these nanoparticles

Inertia-Constrained Pixel-by-Pixel Nonnegative Matrix Factorisation: a Hyperspectral Unmixing Method Dealing with Intra-class Variability

Blind source separation is a common processing tool to analyse the constitution of pixels of hyperspectral images. Such methods usually suppose that pure pixel spectra (endmembers) are the same in all the image for each class of materials. In the framework of remote sensing, such an assumption is no more valid in the presence of intra-class variabilities due to illumination conditions, weathering, slight variations of the pure materials, etc... In this paper, we first describe the results of investigations highlighting intra-class variability measured in real images. Considering these results, a new formulation of the linear mixing model is presented leading to two new methods. Unconstrained Pixel-by-pixel NMF (UP-NMF) is a new blind source separation method based on the assumption of a linear mixing model, which can deal with intra-class variability. To overcome UP-NMF limitations an extended method is proposed, named Inertia-constrained Pixel-by-pixel NMF (IP-NMF). For each sensed spectrum, these extended versions of NMF extract a corresponding set of source spectra. A constraint is set to limit the spreading of each source's estimates in IP-NMF. The methods are tested on a semi-synthetic data set built with spectra extracted from a real hyperspectral image and then numerically mixed. We thus demonstrate the interest of our methods for realistic source variabilities. Finally, IP-NMF is tested on a real data set and it is shown to yield better performance than state of the art methods

Temporal and time-frequency correlation-based blind source separation methods. Part I : Determined and underdetermined linear instantaneous mixtures

We propose two types of correlation-based blind source separation (BSS) methods, i.e. a time-domain approach and extensions which use time-frequency (TF) signal representations and thus apply to much more general conditions. Our basic TF methods only require each source to be isolated in a tiny TF area, i.e. they set very limited constraints on the source sparsity and overlap, unlike various previously reported TF-BSS methods. Our approaches consist in identifying the columns of the (scaled permuted) mixing matrix in TF areas where these methods detect that a source is isolated. Both the detection and identification stages of these approaches use local correlation parameters of the TF transforms of the observed signals. Two such Linear Instantaneous TIme-Frequency CORRelation-based BSS methods are proposed, using Centered or Non-Centered TF transforms. These methods, which are resp. called LI-TIFCORR-C and LI-TIFCORR-NC, are especially suited to non-stationary sources. We derive their performance from many tests performed with mixtures of speech signals. This demonstrates that their output SIRs have a low sensitivity to the values of their TF parameters and are quite high, i.e. typically 60 to 80 dB, while the SIRs of all tested classical methods range about from 0 to 40 dB. We also extend these approaches to achieve partial BSS for underdetermined mixtures and to operate when some sources are not isolated in any TF area