Quantum regression tasks for predicting properties of quantum states are commonly addressed using variational quantum algorithms. While variational quantum circuits are highly expressive and allow to achieve reasonable accuracy, training these circuits may demand a considerable amount of time and resources. In this work, we propose an approach of constructing problem-specific quantum regression models with encoding relevant symmetries and regularizing the variance. The proposed method is based on finding the coefficients of the linear combination of suitably chosen observables. Although it requires the knowledge of the symmetries of the problem in question, the method does not involve parameterized quantum circuits, and the training is done efficiently once the observables are measured. We demonstrate this method on two examples: Prediction of the transverse field strength in the Ising model, and quantification of entanglement in bipartite qubit systems. Our approach is accurate and less resource-intensive than conventional variational methods.
Quantum regression tasks for predicting properties of quantum states are commonly addressed using variational quantum algorithms. While variational quantum circuits are highly expressive and allow to achieve reasonable accuracy, training these circuits may demand a considerable amount of time and resources. In this work, we propose an approach of constructing problem-specific quantum regression models with encoding relevant symmetries and regularizing the variance. The proposed method is based on finding the coefficients of the linear combination of suitably chosen observables. Although it requires the knowledge of the symmetries of the problem in question, the method does not involve parameterized quantum circuits, and the training is done efficiently once the observables are measured. We demonstrate this method on two examples: Prediction of the transverse field strength in the Ising model, and quantification of entanglement in bipartite qubit systems. Our approach is accurate and less resource-intensive than conventional variational methods.
Quantum computers can be considered as a natural means for performing machine learning tasks for inherently quantum labeled data. Many quantum machine learning techniques have been developed for solving classification problems, such as distinguishing between phases of matter or quantum processes. Similarly, one can consider a more general problem of regression, when the aim is to predict continuous labels quantifying properties of quantum states, such as purity or entanglement. In this work, we propose a method for predicting such properties. The method is based on the notion of parametrized quantum circuits, and it seeks to find an observable the expectation of which gives the prediction of the property of interest with a low variance. We numerically test our approach in learning to predict (i) the parameter of a parametrized channel given its output state, (ii) entanglement of two-qubit states, and (iii) the parameter of a parametrized Hamiltonian given its ground state. The results show that the proposed method is able to find observables such that they provide highly accurate predictions of the considered properties, and in some cases even saturate the Cramer-Rao bound, which characterizes the prediction error. We also compare our method with the Bayesian approach, and find that the latter prefers to minimize the prediction variance, having therefore a larger bias.
Whether parameterized quantum circuits (PQCs) can be systematically constructed to be both trainable and expressive remains an open question. Highly expressive PQCs often exhibit barren plateaus, while several trainable alternatives admit efficient classical simulation. We address this question by deriving a finite-sample, dimension-independent concentration bound for estimating the variance of a PQC cost function, yielding explicit trainability guarantees. Across commonly used ansätze, we observe an anticorrelation between trainability and expressibility, consistent with theoretical insights. Building on this observation, we propose a property-based ansatz-search framework for identifying circuits that combine trainability and expressibility. We demonstrate its practical viability on a real quantum computer and apply it to variational quantum algorithms. We identify quantum neural network ansätze with improved effective dimension using over 6× fewer parameters, and for VQE on H2 we achieve UCCSD-like accuracy at substantially reduced circuit complexity.
Variational quantum circuits have become a widely used tool for performing quantum machine learning (QML) tasks on labeled quantum states. In some specific tasks or for specific variational ansätze, one may perform measurements on a restricted part of the overall input state. This is the case for, e.g., quantum convolutional neural networks (QCNNs), where after each layer of the circuit a subset of qubits of the processed state is measured or traced out, and at the end of the network one typically measures a local observable. In this work, we demonstrate that measuring observables with restricted support results in larger label prediction variance in regression QML tasks. We show that the reason for this is, essentially, the number of distinct eigenvalues of the observable one measures after the application of a variational circuit.
Hybrid variational quantum algorithms are promising for solving practical problems, such as combinatorial optimization, quantum chemistry simulation, quantum machine learning, and quantum error correction on noisy quantum computers. However, variational quantum algorithms (derived from randomized hardware-efficient ansatz or adaptive ansatz) become a black box, not trustworthy for model interpretation, and not to mention for application deployment in informing critical decisions. In this paper, we construct the first interpretable quantum regression algorithm, in which the quantum state exactly encodes the classical data table and the variational parameters correspond directly to the regression coefficients, which are real numbers by construction, providing a high degree of model interpretability and minimal cost to optimize due to the right expressiveness. We also exploit the encoded data structure to reduce the gate complexity of computing the regression map. To reduce circuit depth in nonlinear regression, our algorithm can be extended by directly constructing nonlinear features via classical preprocessing, such as independent encoded column vectors. By design, the model performance is determined by the cost function measurement results C synchronous to the mean squared errors (MSE) for the regression models. We derived the read-out errors induced by one-hot encoding and compact encoding; the required physical qubit resources are exponentially compressed for the compact encoding to be favorable for noisy quantum devices. We also derive the cost function dependent sample complexity $ \in \mathcal{O}\left(σ^{2}(\mathcal{C}) \ln (1/α)/ε^{2}\right)$ under the error budget ε and confidence tolerance α.
The rapid progress in quantum computing (QC) and machine learning (ML) has attracted growing attention, prompting extensive research into quantum machine learning (QML) algorithms to solve diverse and complex problems. Designing high-performance QML models demands expert-level proficiency, which remains a significant obstacle to the broader adoption of QML. A few major hurdles include crafting effective data encoding techniques and parameterized quantum circuits, both of which are crucial to the performance of QML models. Additionally, the measurement phase is frequently overlooked—most current QML models rely on pre-defined measurement protocols that often fail to account for the specific problem being addressed. We introduce a novel approach that makes the observable of the quantum system—specifically, the Hermitian matrix—learnable. Our method features an end-to-end differentiable learning framework, where the parameterized observable is trained alongside the ordinary quantum circuit parameters simultaneously. Using numerical simulations, we show that the proposed method can identify observables for variational quantum circuits that lead to improved outcomes, such as higher classification accuracy, thereby boosting the overall performance of QML models.
Gaussian processes are widely known for their ability to provide probabilistic predictions in supervised machine learning models. Their non-parametric nature and flexibility make them particularly effective for regression tasks. However, training a Gaussian process model using standard methods requires matrix inversions with a cubic time complexity, which poses significant computational challenges for inference on larger datasets. Quantum algorithms, such as the HHL algorithm, have been proposed as solutions that overcome the need for classical matrix inversions by efficiently solving linear systems of equations using quantum computers. However, to gain a computational advantage over classical algorithms, these algorithms require fault-tolerant quantum computers with a large number of qubits, which are not yet available. The Variational Quantum Linear Solver is a hybrid quantum-classical algorithm that solves linear systems of equations by optimizing the parameters of a variational quantum circuit using a classical computer. This method is especially suitable for noisy intermediate-scale quantum computers, as it does not require many qubits. It can be used to compute the posterior distribution of a Gaussian process by reformulating the matrix inversion into a set of linear systems of equations. We empirically demonstrate that using the Variational Quantum Linear Solver to perform inference for Gaussian process regression delivers regression quality comparable to that of classical methods.