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.
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.
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.
Within the context of hybrid quantum-classical optimization, gradient descent based optimizers typically require the evaluation of expectation values with respect to the outcome of parameterized quantum circuits. In this work, we explore the consequences of the prior observation that estimation of these quantities on quantum hardware results in a form of stochastic gradient descent optimization. We formalize this notion, which allows us to show that in many relevant cases, including VQE, QAOA and certain quantum classifiers, estimating expectation values with k measurement outcomes results in optimization algorithms whose convergence properties can be rigorously well understood, for any value of k. In fact, even using single measurement outcomes for the estimation of expectation values is sufficient. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values - originating, e.g., from a sum over local terms of a Hamiltonian, a parameter shift rule, or a sum over data-set instances - and we show that in these cases k-shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain "doubly stochastic" gradient descent optimizers. For all algorithms we prove convergence guarantees, providing a framework for the derivation of rigorous optimization results in the context of near-term quantum devices. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for state-of-the-art results with significantly fewer circuit executions and measurements.
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.
Drawing inspiration from gradient-descent methods developed for data processing in quantum state tomography [\href{https://iopscience.iop.org/article/10.1088/2058-9565/ae0baa}{Quantum Sci.~Technol.~\textbf{10} 045055 (2025)}] and quantum process tomography [\href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.130.150402}{Phys.~Rev.~Lett.~\textbf{130}, 150402 (2023)}], we introduce stochastic gradient descent (SGD) algorithms for fast quantum measurement tomography (QMT), applicable to both discrete- and continuous-variable quantum systems -- thus completing the tomography trio. A measurement device or detector in a quantum experiment is characterized by a set of positive operator-valued measure (POVM) elements; the goal of QMT is to estimate these operators from experimental data. To ensure physically valid (positive and complete) POVM reconstructions, we propose two distinct parameterization schemes within the SGD framework: one leveraging optimization on a Stiefel manifold and one based on Hermitian operator normalization via eigenvalue scaling. Within the SGD-QMT framework, we further investigate two loss functions: mean squared error, equivalent to L2 or Euclidean norm, and average negative log-likelihood, inspired by maximum likelihood estimation. We benchmark performance against state-of-the-art constrained convex optimization methods. Numerical simulations demonstrate that, compared to standard methods, our SGD-QMT algorithms offer significantly lower computational cost, superior reconstruction fidelity, and enhanced robustness to noise. We make a Python implementation of the SGD-QMT algorithms publicly available at \href{https://github.com/agtomo/SGD-QMT}{github.com/agtomo/SGD-QMT}.
Within the context of hybrid quantum-classical optimization, gradient descent based optimizers typically require the evaluation of expectation values with respect to the outcome of parameterized quantum circuits. In this work, we explore the consequences of the prior observation that estimation of these quantities on quantum hardware results in a form of s t o c h a s t i c gradient descent optimization. We formalize this notion, which allows us to show that in many relevant cases, including VQE, QAOA and certain quantum classifiers, estimating expectation values with k measurement outcomes results in optimization algorithms whose convergence properties can be rigorously well understood, for any value of k . In fact, even using single measurement outcomes for the estimation of expectation values is sufficient. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values -- originating, e.g., from a sum over local terms of a Hamiltonian, a parameter shift rule, or a sum over data-set instances -- and we show that in these cases k -shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain ``doubly stochastic'' gradient descent optimizers. For all algorithms we prove convergence guarantees, providing a framework for the derivation of rigorous optimization results in the context of near-term quantum devices. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for state-of-the-art results with significantly fewer circuit executions and measurements.