New Single-Preparation Methods for Unsupervised Quantum Machine Learning Problems

Abstract

The term "machine learning" especially refers to algorithms that derive mappings, i.e., inputoutput 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 a quantum 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.INDEX TERMS Blind Hamiltonian parameter estimation (BHPE), blind quantum process tomography (BQPT), blind quantum source separation (BQSS), blind quantum state restoration (BQSR), blind/unsupervised quantum machine learning, Heisenberg exchange coupling, quantum classification, quantum computer (quamputer), single-preparation quantum information processing (SIPQIP), spin-based qubits. Engineering uantum Transactions on IEEEDeville and Deville: NEW SINGLE-PREPARATION METHODS FOR UNSUPERVISED (QM) LEARNING PROBLEMS

Statistical intrusion detection and eavesdropping in quantum channels with coupling: Multiple-preparation and single-preparation methods

“…A major property is then that all these expectations may in practice be estimated by using only one instance of each of the considered states (1). This property was theoretically justified in [16,18] and confirmed by numerical tests in [16,18,17]. Its relevance may be outlined as follows.…”

“…Their simplest version uses a single, deterministic, value of the initial states |ψ j (t 0 ) with j ∈ {1, 2}, of the final state |ψ(t) of the two-qubit system at time t and of the associated probability P (b 1 = +). This method exploits the fact that, "in general", this probability does not take the same value depending whether Case 0 or Case 1 is considered, as shown by (17) and (19). By "in general", we mean that the values in ( 17) and ( 19) are different except for very specific values of the quantities that they involve, that are related to the initial quantum state (parameters r 1 , r 2 , ∆ I ) or to the channel (parameters v and hence J xy and (t − t 0 )).…”

“…If sticking to that multi-preparation framework that is usual in QIP, there might seem to be no solution to this problem at first glance, because estimating a probability value requires a large number of trials for the considered single experiment. However, beyond that usual QIP framework, we recently developed an original concept (see [16,18,17]), called SIngle-Preparation QIP (or SIPQIP) for its general version, and especially applied to BQSS and related tasks so far, which solves the above problem, as will now be shown. Briefly, instead of estimating a single deterministic probability from many copies of a single deterministic quantum state, SIPQIP estimates the expectation of a random probability associated with a random quantum pure state, i.e.…”

“…We then especially provided a detailed report of the principles of this SIPQIP framework and of its application to BQPT in [18]. Finally, we very recently applied that framework to a variety of QIP tasks: See [17]. As shown further in this paper, this SIPQIP framework is particularly attractive for the communication applications considered here.…”

“…We then developed it especially in [18]. We also very recently [17] introduced methods for a closely related problem, namely Blind Hamiltonian Parameter Estimation (BHPE). All these QIP problems involve a quantum state transform, to be identified or inverted.…”