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arXiv:2407.08847 · Physical Review Research

Predicting properties of quantum systems by regression on a quantum computer

Andrey Kardashin, Yerassyl Balkybek, Vladimir V. Palyulin, K. V. Antipin

Published Feb 25, 2025• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevresearch.7.013201

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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.
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.
Machine learning has emerged recently as a powerful tool for predicting properties of quantum many-body systems. For many ground states of gapped Hamiltonians, generative models can learn from measurements of a single quantum state to reconstruct the state accurately enough to predict local observables. Alternatively, classification and regression models can predict local observables by learning from measurements on different but related states. In this work, we combine the benefits of both approaches and propose the use of conditional generative models to simultaneously represent a family of states, learning shared structures of different quantum states from measurements. The trained model enables us to predict arbitrary local properties of ground states, even for states not included in the training data, without necessitating further training for new observables. We first numerically validate our approach on 2D random Heisenberg models using simulations of up to 45 qubits. Furthermore, we conduct quantum simulations on a neutral-atom quantum computer and demonstrate that our method can accurately predict the quantum phases of square lattices of 13$\times$13 Rydberg atoms.
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.
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This study investigates the application of machine learning techniques for predicting gate depth in quantum computational systems, focusing on the relationship between quantum circuit parameters and computational performance. Using a dataset of 50 observations encompassing qubit count, gate depth, entanglement rate, and logical error rate, we developed and compared Linear Regression (LR) and Random Forest Regression (RFR) models for gate depth prediction. The research addresses critical challenges in quantum fault tolerance and circuit optimization, which are essential for achieving scalable quantum computation. Our analysis revealed strong negative correlation (-0.65) between qubit count and logical error rate, while gate depth showed moderate positive correlation (0.62) with error rates. The Random Forest model demonstrated superior performance with R² values of 0.96068 on training data and 0.87262 on testing data, significantly outperforming Linear Regression (R²: 0.87552 training, 0.60257 testing). Error metrics further confirmed RFR's superiority, achieving RMSE of 2.33 (training) and 2.97 (testing) compared to LR's 4.15 and 5.25 respectively. The study highlights the importance of quantum error correction mechanisms and the inherent trade-offs between circuit complexity and computational reliability. Both models showed expected performance degradation from training to testing, with RFR maintaining better generalization capabilities. These findings contribute to the development of typed models for quantum mechanics in computational applications, offering insights into quantum circuit optimization and fault-tolerant design. The research demonstrates that ensemble methods like Random Forest provide more robust predictions for quantum system parameters, supporting the advancement of practical quantum computing implementations and algorithm development. KeyWords: Quantum Computation, Gate Depth Prediction, Random Forest Regression, Quantum Error Correction, Machine Learning, Fault Tolerance.
In the last two decades machine learning and quantum computing have seen massive growth in terms of application and implementation. Quantum computers can provide the solution to certain classes of problem that classical computers fail to solve. Hence a greater number of optimized quantum algorithms are being developed. Today quantum computers are used in various fields to solve difficult problems like optimization and critical prediction problems in finance, artificial intelligence, and medicine. Quantum computing is based on the laws of quantum mechanics and uses superposition and entanglement properties to solve computation complexities in analyzing large data. Quantum machine learning is a new discipline for machine learning. In this survey, we will explore algorithms for quantum machine learning tasks. We will study parametrized quantum circuits for multi-class classification. This chapter also shows that quantum machine learning and near-term quantum devices open new doors for fast learning and enhanced classification.