System identification, system inversion and its multiple-signal extension to source separation, including non-blind and blind (i.e. unsupervised) configurations, are major topics in classical, i.e. nonquantum, signal processing. Their application fields especially include communications, where they are often referred to as (possibly blind and MIMO) channel identification / estimation and channel equalization. Some of their configurations have also been extended to quantum channels. In this paper, we focus on such channels and on the most challenging configurations, namely blind and multiplequbit ones, hence with main emphasis on source separation. In this framework, two classes of methods were previously proposed for two-qubit configurations and a specific type of channels. We here first define their extensions to any number of qubits and general channels. For that configuration too, we then propose a third class of methods and we analyze its relationships with the previous two. This leads us to provide new general insight into the connections between the unentanglement of a pure quantum state and the statistical independence of associated measurement components.
System identification, system inversion and its multiple-signal extension to source separation, including non-blind and blind (i.e. unsupervised) configurations, are major topics in classical, i.e. nonquantum, signal processing. Their application fields especially include communications, where they are often referred to as (possibly blind and MIMO) channel identification / estimation and channel equalization. Some of their configurations have also been extended to quantum channels. In this paper, we focus on such channels and on the most challenging configurations, namely blind and multiplequbit ones, hence with main emphasis on source separation. In this framework, two classes of methods were previously proposed for two-qubit configurations and a specific type of channels. We here first define their extensions to any number of qubits and general channels. For that configuration too, we then propose a third class of methods and we analyze its relationships with the previous two. This leads us to provide new general insight into the connections between the unentanglement of a pure quantum state and the statistical independence of associated measurement components.
System identification and inversion are two closely related problems, which lead to various configurations. In the quantum framework, system identification was only studied in the non-blind (or supervised) mode, i.e. with known input states (in addition to known output states). It is then called quantum process tomography. We here develop its blind (or unsupervised) version, operating with unknown (unentangled) input states (which are here not controlled by an automated loop). Our approach takes advantage of the blind quantum source separation (i.e. blind multi-qubit system inversion) methods based on disentanglement that we recently introduced. It is detailed for Heisenberg spin coupling and could be extended to other classes of processes. It estimates the parameter values of the Heisenberg Hamiltonian.
Many methods for processing classical or quantum data are based on estimating some parameters, e.g. those of adaptive filters, artificial neural networks (including deep learning approaches), blind source separation systems or quantum gates. For classical signals, these methods include stochastic algorithms, such as stochastic gradient descent. We here first introduce a partly related, general, type of stochastic algorithms for quantum data and we prove the asymptotic efficiency of the proposed estimator. We then show the attractiveness of this stochastic approach for the quantum version of blind multiple-input multiple-output system identification and blind source separation: the resulting model estimation methods can operate with a single copy of each considered quantum state, whereas the previous methods require many copies of the same (unknown) states to be available and are thus "less blind". Numerical tests show that good performance is obtained with 104such quantum states.
The term “machine learning” especially refers to algorithms that derive mappings, i.e., input–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 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. We consider the general single-preparation quantum information processing (SIPQIP) framework that we have recently proposed. It involves methods that can work with a single instance of each (unknown) state, whereas usual methods are quite cumbersome because they have to very accurately create many copies of each (known) state. In our previous papers, we applied the SIPQIP approach to only one task [blind quantum process tomography (BQPT)], but it opens the way to a large range of other types of methods. In this article, we, therefore, propose various new SIPQIP methods that efficiently perform tasks related to system identification (blind Hamiltonian parameter estimation (BHPE), blind quantum channel identification/estimation, and blind phase estimation), system inversion and state estimation (blind quantum source separation (BQSS), blind quantum entangled state restoration (BQSR), and blind quantum channel equalization), and classification. Numerical tests show that our SIPQIP framework, moreover, yields much more accurate estimation than the usual multiple-preparation approach. Our methods are especially useful in aquantum computer, which we propose to more briefly call a “quamputer”: BQPT and BHPE simplify the characterization of quamputer gates; BQSS and BQSR yield quantum gates that may be used to compensate for the nonidealities that alter states stored in quantum registers and open the way to very general self-adaptive quantum gates.
In the framework of Blind Quantum Source Separation, we investigate the adaptation of a separating system which receives coupled quantum bit (qubit) states and processes them with quantum means in its feedforward path, to uncouple them. We propose the first separation principle which ensures that the output qubit states of this system are disentangled and that they restore the non-entangled source qubit states up to limited indeterminacies. This separation principle exploits measurements of output spin components along two directions and has some links with the non-quantum Independent Component Analysis (ICA) principle. It opens the way to various practical separation criteria and algorithms, some of which are described here.
Blind Source Separation (BSS) is an active domain of Classical Information Processing, with well-identified methods and applications. The development of Quantum Information Processing has made possible the appearance of Blind Quantum Source Separation (BQSS), with a recent extension towards Blind Quantum Process Tomography (BQPT). This article investigates the use of several fundamental quantum concepts in the BQSS context and establishes properties already used without justification in that context. It mainly considers a pair of electron spins initially separately prepared in a pure state and then submitted to an undesired exchange coupling between these spins. Some consequences of the existence of the entanglement phenomenon, and of the probabilistic aspect of quantum measurements, upon BQSS solutions, are discussed. An unentanglement criterion is established for the state of an arbitrary qubit pair, expressed first with probability amplitudes and secondly with probabilities. The interest of using the concept of a random quantum state in the BQSS context is presented. It is stressed that the concept of statistical independence of the sources, widely used in classical BSS, should be used with care in BQSS, and possibly replaced by some disentanglement principle. It is shown that the coefficients of the development of any qubit pair pure state over the states of an orthonormal basis can be expressed with the probabilities of results in the measurements of well-chosen spin components.
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 require one to know, hence to very 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. In contrast, we propose ``single-preparation BQPT methods'' (SBQPT) , i.e., methods which can operate with only one instance of each considered input state. These two concepts are here illustrated with practical (S)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 and (ii) the considered quantum process is based on cylindrical-symmetry Heisenberg spin coupling. As a benchmark, we moreover introduce nonblind QPT methods dedicated to the considered Heisenberg process, we analyze their theoretical behavior (this requires the tools developed in this paper for random input states), and we numerically test their sensitivity to systematic and nonsystematic errors, which are most likely to occur in practice. This shows that, even for very low preparation errors (especially systematic ones), these nonblind QPT methods yield much lower performance than our SBQPT methods. Our blind and single-preparation QPT concepts may be extended, e.g., to a much wider class of processes and to SBQPT methods based on other quantum state properties, as outlined in this paper.