# Quantifying Quantum Coherence Using Machine Learning Methods
## 1 Introduction  
Quantum coherence refers to the possibility of creating a superposition of a set of orthogonally distinguishable states in quantum physics and quantum information science [1]. It is an indispensable element in multi-particle interference, entanglement, and other phenomena, and can also be regarded as a quantum resource in some quantum processing tasks [2; 3; 4]. Especially in the field of thermodynamics, the theory of coherent quantum resources has brought new enlightenment and has energy value in terms of thermodynamic work [5; 6]. Recent studies have shown that quantum coherence can serve as a useful resource for improving the performance of some thermal machines [7; 8; 9] and the charging power of quantum batteries [10; 11].  
There are many types of quantum coherence measures like coherence measure based on entanglement and discord [12; 13; 14], intrinsic randomness of coherence [15], the basis independent set coherence [16], coherence distillation and cost [3; 17], measures based on Fisher information [18], and relative entropy of coherence [19; 20]. The robustness of coherence [21; 22], serving as an operational measure of quantum coherence, is not only feasible to observe experimentally but also numerically assessable through semidefinite programming. In practice, fast and accurate calculations of coherence values are crucial. However, this is a challenge because this computational process relies on convex optimization techniques, and the time required can increase significantly as the state dimension of the processing increases.  
Machine learning methods have been recently brought to the field of quantum physics and quantum information to address problems such as quantum-state tomography and estimation [23; 24], quantum error correction code [25], wave function reconstruction [26]. In particular, it is found that machine learning methods are useful in entanglement detection [27; 28; 29; 30] and quantification [31]. Meanwhile, machine learning methods are adopted inthe classification of Einstein-Podolsky-Rosen (EPR) steering [32]. Recently, [PERSON] et al. presented a semisupervised support vector machine method on EPR steering that showed a significant improvement in accuracy and labor-saving [33]. Methods of artificial neural network to quantify two-qubit steerability have also been proposed [34]. Therefore, machine learning plays a positive role in the detection and quantification of quantum information resources.  
Inspired by these recent research progresses, in this work we combine machine learning methods with the quantification of quantum coherence measured by the robustness of coherence. We first obtain the dataset of labeled quantum states by using the semidefinite programming. Then, we train artificial neural networks (ANNs) for two-qubit, qubit-qutrit and three-qubit systems. ANNs are computational models inspired by biological neural systems. In this work, we utilize a backpropagation (BP) neural network, a type of multi-layer feedforward network trained via the error backpropagation algorithm, which is among the most widely used neural network models [35]. This method can straightforwardly be extended to arbitrary dimensional quantum systems. By applying the trained model to test samples and an example of non-Markovian dynamics of coherence, we show the trained models have strong generalization ability.  
This paper is organized as follows. We briefly review the ROC from the viewpoint of quantum resource theory, the feedforward neural network and the methods for data preprocessing in Section 2. We describe our results of training the model for two-qubit, qubit-qutrit and three-qubit systems in Section 3. A conclusion is given in Section 4.