t-distributed stochastic neighbor embedding (t-SNE) is a nonlinear dimensionality reduction algorithm that preserves local similarity by matching probability distributions between high- and low-dimensional spaces. Owing to its ability to capture complex manifold structures, t-SNE has become a widely used technique for high-dimensional data visualization. However, it still faces significant computational challenges when dealing with large-scale datasets. In this paper, we present a quantum-enhanced t-SNE algorithm that achieves a quadratic speedup with respect to the number of data samples. Specifically, three unitary operators are constructed using quantum arithmetic and comparators to compute the probability distributions. By integrating them with the Hadamard transform, the gradients of the Kullback-Leibler divergence can be efficiently computed. Moreover, since a complete quantum procedure is provided, our method surpasses the performance of existing quantum variants.
t-distributed stochastic neighbor embedding (t-SNE) is a nonlinear dimensionality reduction algorithm that preserves local similarity by matching probability distributions between high- and low-dimensional spaces. Owing to its ability to capture complex manifold structures, t-SNE has become a widely used technique for high-dimensional data visualization. However, it still faces significant computational challenges when dealing with large-scale datasets. In this paper, we present a quantum-enhanced t-SNE algorithm that achieves a quadratic speedup with respect to the number of data samples. Specifically, three unitary operators are constructed using quantum arithmetic and comparators to compute the probability distributions. By integrating them with the Hadamard transform, the gradients of the Kullback-Leibler divergence can be efficiently computed. Moreover, since a complete quantum procedure is provided, our method surpasses the performance of existing quantum variants.
t-stochastic neighbor embedding (t-SNE) is a nonparametric data visualization method in classical machine learning. It maps the data from the high-dimensional space into a low-dimensional space, especially a two-dimensional plane, while maintaining the relationship or similarities between the surrounding points. In t-SNE, the initial position of the low-dimensional data is randomly determined, and the visualization is achieved by moving the low-dimensional data to minimize a cost function. Its variant called parametric t-SNE uses neural networks for this mapping. In this paper, we propose to use quantum neural networks for parametric t-SNE to reflect the characteristics of high-dimensional quantum data on low-dimensional data. We use fidelity-based metrics instead of Euclidean distance in calculating high-dimensional data similarity. To verify our proposed method, we visualize both classical (Iris dataset) and quantum (time-depending Hamiltonian dynamics) data for classification tasks. Since this method allows us to represent a quantum dataset in a higher dimensional Hilbert space by a quantum dataset in a lower dimension while keeping their similarity, the proposed method can also be used to compress quantum data for further quantum machine learning.
Classical machine learning algorithms can provide insights on high-dimensional processes that are hardly accessible with conventional approaches. As a notable example, t -distributed Stochastic Neighbor Embedding ( t -SNE) represents the state of the art for visualization of data sets of large dimensionality. An interesting question is then if this algorithm can provide useful information also in quantum experiments with very large Hilbert spaces. Leveraging these considerations, in this work we apply t -SNE to probe the spatial distribution of n -photon events in m -dimensional Hilbert spaces, showing that its findings can be beneficial for validating genuine quantum interference in boson sampling experiments. In particular, we find that nonlinear dimensionality reduction is capable to capture distinctive features in the spatial distribution of data related to multi-photon states with different evolutions. We envisage that this approach will inspire further theoretical investigations, for instance for a reliable assessment of quantum computational advantage.
Neighborhood preserving embedding (NPE) is an important linear dimensionality reduction technique that aims at preserving the local manifold structure. NPE contains three steps, i.e., finding the nearest neighbors of each data point, constructing the weight matrix, and obtaining the transformation matrix. Liang et al. proposed a variational quantum algorithm (VQA) for NPE [Phys. Rev. A 101 032323 (2020)]. The algorithm consists of three quantum sub-algorithms, corresponding to the three steps of NPE, and was expected to have an exponential speedup on the dimensionality n. However, the algorithm has two disadvantages: (i) It is not known how to efficiently obtain the input of the third sub-algorithm from the output of the second one. (ii) Its complexity cannot be rigorously analyzed because the third sub-algorithm in it is a VQA. In this paper, we propose a complete quantum algorithm for NPE, in which we redesign the three sub-algorithms and give a rigorous complexity analysis. It is shown that our algorithm can achieve a polynomial speedup on the number of data points m and an exponential speedup on the dimensionality n under certain conditions over the classical NPE algorithm, and achieve a significant speedup compared to Liang et al.’s algorithm even without considering the complexity of the VQA.
Reducing the dimension of nonlinear data is crucial in data processing and visualization. The locally linear embedding algorithm (LLE) is specifically a representative nonlinear dimensionality reduction method with well maintaining the original manifold structure. In this paper, we present two implementations of the quantum locally linear embedding algorithm (qLLE) to perform the nonlinear dimensionality reduction on quantum devices. One implementation, the linear-algebra-based qLLE algorithm utilizes quantum linear algebra subroutines to reduce the dimension of the given data. The other implementation, the variational qLLE algorithm utilizes a variational quantum-classical hybrid procedure to acquire the low-dimensional data. The classical LLE algorithm requires polynomial time complexity of N, where N is the global number of the original high-dimensional data. In data preprocessing, we invoke the quantum k-nearest neighbors algorithm (k-NN) to find out the k nearest neighbors of the given data with quadratic speedup. For the main part of the qLLE algorithm, compared with the corresponding classical algorithm, the linear-algebra-based qLLE algorithm proposed in this paper achieves an exponential speedup in O(poly(logN)). The variational qLLE algorithm can be implemented on the near term quantum devices. In addition, the corresponding quantum circuits of the qLLE algorithm are also presented.
In this paper, we employ PCA and t-SNE analyses to gain deeper insights into the behavior of entangled and non-entangled mixing operators within the Quantum Approximate Optimization Algorithm (QAOA) at various depths. We utilize a dataset containing optimized parameters generated for max-cut problems with cyclic and complete configurations. This dataset encompasses the resulting RZ, RX, and RY parameters for QAOA models at different depths (1L, 2L, and 3L) with or without an entanglement stage within the mixing operator. Our findings reveal distinct behaviors when processing the different parameters with PCA and t-SNE. Specifically, most of the entangled QAOA models demonstrate an enhanced capacity to preserve information in the mapping, along with a greater level of correlated information detectable by PCA and t-SNE. Analyzing the overall mapping results, a clear differentiation emerges between entangled and non-entangled models. This distinction is quantified numerically through explained variance in PCA and Kullback-Leibler divergence (post-optimization) in t-SNE. These disparities are also visually evident in the mapping data produced by both methods, with certain entangled QAOA models displaying clustering effects in both visualization techniques.
Data visualization is important in understanding the characteristics of data that are difficult to see directly. It is used to visualize loss landscapes and optimization trajectories to analyze optimization performance. Popular optimization analysis is performed by visualizing a loss landscape around the reached local or global minimum using principal component analysis. However, this visualization depends on the variational parameters of a quantum circuit rather than quantum states, which makes it difficult to understand the mechanism of the optimization process through the property of quantum states. Here, we propose a quantum data visualization method using quantum kernels, which enables us to offer fast and highly accurate visualization of quantum states. In our numerical experiments, we visualize a hand-written digits data set and apply a k -nearest neighbor algorithm to the low-dimensional data to quantitatively evaluate our proposed method compared with a classical kernel method. As a result, our proposed method achieves comparable accuracy to the state-of-the-art classical kernel method, meaning that the proposed visualization method based on quantum machine learning does not degrade the separability of the input higher-dimensional data. Furthermore, we visualize the optimization trajectories of finding the ground states of the transverse field Ising model, and we successfully find the trajectory characteristics. Since quantum states are higher-dimensional objects that can only be seen via observables, our visualization method, which inherits the similarity of quantum data, would be useful in understanding the behavior of quantum circuits and algorithms. Published by the American Physical Society 2024