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arXiv:2306.14566 · Quantum Information Processing

Estimating quantum mutual information through a quantum neural network

Myeongjin Shin, Junseo Lee, Kabgyun Jeong

Published Feb 9, 2024• 17 citations
arXivHTMLPDF

Abstract

Abstract We propose a method of quantum machine learning called quantum mutual information neural estimation (QMINE) for estimating von Neumann entropy and quantum mutual information, which are fundamental properties in quantum information theory. The QMINE proposed here basically utilizes a technique of quantum neural networks (QNNs), to minimize a loss function that determines the von Neumann entropy, and thus quantum mutual information, which is believed more powerful to process quantum datasets than conventional neural networks due to quantum superposition and entanglement. To create a precise loss function, we propose a quantum Donsker-Varadhan representation (QDVR), which is a quantum analog of the classical Donsker-Varadhan representation. By exploiting a parameter shift rule on parameterized quantum circuits, we can efficiently implement and optimize the QNN and estimate the quantum entropies using the QMINE technique. Furthermore, numerical observations support our predictions of QDVR and demonstrate the good performance of QMINE.

DOI
10.1007/s11128-023-04253-1

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Abstract We propose a method of quantum machine learning called quantum mutual information neural estimation (QMINE) for estimating von Neumann entropy and quantum mutual information, which are fundamental properties in quantum information theory. The QMINE proposed here basically utilizes a technique of quantum neural networks (QNNs), to minimize a loss function that determines the von Neumann entropy, and thus quantum mutual information, which is believed more powerful to process quantum datasets than conventional neural networks due to quantum superposition and entanglement. To create a precise loss function, we propose a quantum Donsker-Varadhan representation (QDVR), which is a quantum analog of the classical Donsker-Varadhan representation. By exploiting a parameter shift rule on parameterized quantum circuits, we can efficiently implement and optimize the QNN and estimate the quantum entropies using the QMINE technique. Furthermore, numerical observations support our predictions of QDVR and demonstrate the good performance of QMINE.
Abstract Analytically continuing the von Neumann entropy from Rényi entropies is a challenging task in quantum field theory. While the n -th Rényi entropy can be computed using the replica method in the path integral representation of quantum field theory, the analytic continuation can only be achieved for some simple systems on a case-by-case basis. In this work, we propose a general framework to tackle this problem using classical and quantum neural networks with supervised learning. We begin by studying several examples with known von Neumann entropy, where the input data is generated by representing TrρAn{\text {Tr}}\rho _A^nTrρAn​ Tr ρ A n with a generating function. We adopt KerasTuner to determine the optimal network architecture and hyperparameters with limited data. In addition, we frame a similar problem in terms of quantum machine learning models, where the expressivity of the quantum models for the entanglement entropy as a partial Fourier series is established. Our proposed methods can accurately predict the von Neumann and Rényi entropies numerically, highlighting the potential of deep learning techniques for solving problems in quantum information theory.
Quantum neural networks (QNNs) is a parameterized quantum circuit model, which can be trained by gradient-based optimizer, can be used for supervised learning, regression tasks, combinatorial optimization, etc. Although many works have demonstrated that QNNs have better learnability, generalizability, etc. compared to classical neural networks. However, as with classical neural networks, we still can't explain their working mechanism well. In this paper, we reveal the training mechanism of QNNs by mutual information. Unlike traditional mutual information in neural networks, due to quantum computing remains information conserved, the mutual information is trivial of the input and output of U operator. In our work, in order to observe the change of mutual information during training, we divide the quantum circuit (U operator) into two subsystems, discard subsystem (D) and measurement subsystem (M) respectively. We calculate two mutual information, I(Di : Mo) and I(Mi : Mo) (i and o means input or output of the corresponding subsystem), and observe their behavior during training. As the epochs increases, I(Di : Mo) gradually increases, this may means some information of discard subsystem is continuously pushed into the measurement subsystem, the information should be label-related. What's more, I(Mi : Mo) exist two-phase behavior in training process, this consistent with the information bottleneck anticipation. The first phase, I(Mi : Mo) is increasing, this means the measurement subsystem perform feature fitting. The second phase, I(Mi : Mo) is decreasing, this may means the system is generalizing, the measurement subsystem discard label-irrelevant information into the discard subsystem as many as possible. Our work discussed the working mechanism of QNNs by mutual information, further, it can be used to analyze the accuracy and generalization of QNNs.
Abstract In classical information theory, uncommon information refers to the amount of information that is not shared between two messages, and it admits an operational interpretation as the minimum communication cost required to exchange the messages. Extending this notion to the quantum setting, quantum uncommon information is defined as the amount of quantum information necessary to exchange two quantum states. While the value of uncommon information can be computed exactly in the classical case, no direct method is currently known for calculating its quantum analogue. Prior work has primarily focused on deriving upper and lower bounds for quantum uncommon information. In this work, we propose a new approach for estimating these bounds by utilizing the quantum Donsker–Varadhan representation and implementing a gradient-based optimization method. Our results suggest a pathway toward efficient approximation of quantum uncommon information using variational techniques grounded in quantum neural architectures.
Quantum Machine Learning refers to the interdisciplinary area of research, blending principles from quantum physics and machine learning that can process information on quantum computers. Quantum machine learning (QML) intentions to exploit the superposition (qubit can exist in multiple states) property and other effects of quantum mechanics for enhanced efficiency and speed in specific machine learning computations, promising exponential-power superiority in particular computations. This paper focusses on Quantum Machine Learning Models to develop the framework to learn complicated associations within the data through the adjustment of the variational parameters during training. This framework serves as a foundational structure for implementing quantum neural networks, allowing for further experimentation and growth in quantum machine learning applications. It highlights the quantum circuits to process and learn from data in a mode distinct from classical neural networks. As the retrieve of quantum computing grows, such frameworks can be adapted and expanded to explore more sophisticated quantum algorithms and architectures. In the circumstances of Quantum Neural Networks, gradient descents is used in minimizing cost or loss function by iteratively adjusting the variational parameters based onto the gradient of the loss with admire to those parameters. The output that indicates the process of installing the Quantum information science Kit (Qiskit) library in Python specifically version 3.7 along with its dependencies.
Abstract Calculating the von Neumann entanglement entropy from experimental data is challenging due to its dependence on the complete wavefunction, forcing reliance on approximations such as classical mutual information (MI). We propose a machine learning approach using a graph neural network to predict the von Neumann entropy directly from experimentally accessible bitstrings. We test this approach on a Rydberg ladder system and achieve a mean absolute error of 3.6 × 10 − 3 when evaluating within the training range on a dataset with entropy values ranging from 0 to 1.9. The model achieves a mean absolute percentage error of 1.44% and outperforms MI-based bounds. When tested beyond the training range, the model maintains reasonable accuracy. Furthermore, we demonstrate that fine-tuning the model with small datasets significantly improves performance on data outside the original training range.
Calculating the von Neumann entanglement entropy from experimental data is challenging due to its dependence on the complete wavefunction, forcing reliance on approximations like classical mutual information (MI). We propose a machine learning approach using a graph neural network for predicting the von Neumann entropy from experimentally accessible bitstrings. We tested this approach on a Rydberg ladder system and achieved a mean absolute error of 3.7×10−33.7 \times 10^{-3}3.7×10−3 when testing it inside of its training range on a dataset with entropy ranging from 0 to 1.9, we also evaluated the mean absolute percentage error on the data that have its entropy larger than 0.01 of the maximum entropy(so we can avoid the areas of our phase space with near zero entropy) and got a relative error of 1.72%, outperforming MI-based bounds. When tested beyond this range, the model still delivers reasonable results. Furthermore, we demonstrate that fine tuning the model with a small dataset significantly improves its performance on data outside its original training range.