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.
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.
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.
The rapid advancements in quantum computing (QC) and machine learning (ML) have sparked significant interest, driving extensive exploration of quantum machine learning (QML) algorithms to address a wide range of complex challenges. The development of high-performance QML models requires expert-level expertise, presenting a key challenge to the widespread adoption of QML. Critical obstacles include the design of effective data encoding strategies and parameterized quantum circuits, both of which are vital for the performance of QML models. Furthermore, the measurement process is often neglected-most existing QML models employ predefined measurement schemes that may not align with the specific requirements of the targeted problem. We propose an innovative framework that renders the observable of a quantum system-specifically, the Hermitian matrix-trainable. This approach employs an end-to-end differentiable learning framework, enabling simultaneous optimization of the neural network used to program the parameterized observables and the standard quantum circuit parameters. Notably, the quantum observable parameters are dynamically programmed by the neural network, allowing the observables to adapt in real time based on the input data stream. Through numerical simulations, we demonstrate that the proposed method effectively programs observables dynamically within variational quantum circuits, achieving superior results compared to existing approaches. Notably, it delivers enhanced performance metrics, such as higher classification accuracy, thereby significantly improving the overall effectiveness of QML models.
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.
Conventional Variational Quantum Circuits (VQCs) for Quantum Machine Learning typically rely on a fixed Hermitian observable, often built from Pauli operators. Inspired by the Heisenberg picture, we propose an adaptive non-local measurement framework that substantially increases the model complexity of the quantum circuits. Our introduction of dynamical Hermitian observables with evolving parameters shows that optimizing VQC rotations corresponds to tracing a trajectory in the observable space. This viewpoint reveals that standard VQCs are merely a special case of the Heisenberg representation. Furthermore, we show that properly incorporating variational rotations with non-local observables enhances qubit interaction and information mixture, admitting flexible circuit designs. Two non-local measurement schemes are introduced, and numerical simulations on classification tasks confirm that our approach outperforms conventional VQCs, yielding a more powerful and resource-efficient approach as a Quantum Neural Network.
Quantum computing has shown considerable promise for compute-intensive tasks in recent years. For instance, classification tasks based on quantum neural networks (QNN) have garnered significant interest from researchers and have been evaluated in various scenarios. However, the majority of quantum classifiers are currently limited to binary classification tasks due to either constrained quantum computing resources or the need for intensive classical post-processing. In this paper, we propose an efficient quantum multi-classifier called MORE, which stands for measurement and correlation based variational quantum multi-classifier. MORE adopts the same variational ansatz as binary classifiers while performing multi-classification by fully utilizing the quantum information of a single readout qubit. To extract the complete information from the readout qubit, we select three observables that form the basis of a two-dimensional Hilbert space. We then use the quantum state tomography technique to reconstruct the readout state from the measurement results. Afterward, we explore the correlation between classes to determine the quantum labels for classes using the variational quantum clustering approach. Next, quantum label-based supervised learning is performed to identify the mapping between the input data and their corresponding quantum labels. Finally, the predicted label is determined by its closest quantum label when using the classifier. We implement this approach using the Qiskit Python library and evaluate it through extensive experiments on both noise-free and noisy quantum systems. Our evaluation results demonstrate that MORE, despite using a simple ansatz and limited quantum resources, achieves advanced performance.
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.