Yun Shang

Xujun Bai, Yun Shang, Honghong Lin

The traveling salesman problem (TSP) is a significant classical NP-hard combinatorial optimization problem. In this work, we demonstrate that combining classical dynamic programming with quantum search can yield an achievable quantum advantage for TSP on the basis of excellent work by the authors of~\cite{ambainis2019quantum}. We design the quantum divide and conquer strategy to provide a parameterized spectrum for this combination. The hybrid algorithm proposed in~\cite{ambainis2019quantum} corresponds to a specific case in this spectrum, while the two extremes of the spectrum represent the purely classical Held-Karp and the purely quantum search algorithm, respectively. Within our parameterized spectrum, we prove that the optimal query complexity is $O^*(1.865666\ldots^n)$, achieved with the 4-subset scheme, while the counting in~\cite{ambainis2019quantum} overlooked half of the recursive branches. The correct query complexity of their algorithm is $O^*(2.225880\ldots^n)$ at their chosen parameter ($α\approx0.055362$), and cannot fall below $O^*(2^n)$ for any $α$ - meaning their $8$-subset scheme, correctly analyzed, never surpasses the classical Held-Karp bound. Furthermore, in previous studies on quantum advantages for NP-hard combinatorial optimization problems, researchers focused only on improvements in query complexity. Our work, however, points out that the quantum advantage stems not only from the quadratic speedup of quantum search but also from the structured quantum state preparation. We argue that structured state preparation is indispensable for realizing the oracle operator while maintaining the total time complexity of $O^*(1.865666\ldots^n)$. Therefore, we design an elegant method for preparing the set partition state, which makes our TSP solver practically executable.

Xiao Shi, Yun Shang

Tensor networks, which have been traditionally used to simulate many-body physics, have recently gained significant attention in the field of machine learning due to their powerful representation capabilities. In this work, we propose a density-based clustering algorithm inspired by tensor networks. We encode classical data into tensor network states on an extended Hilbert space and train the tensor network states to capture the features of the clusters. Here, we define density and related concepts in terms of fidelity, rather than using a classical distance measure. We evaluate the performance of our algorithm on six synthetic data sets, four real world data sets, and three commonly used computer vision data sets. The results demonstrate that our method provides state-of-the-art performance on several synthetic data sets and real world data sets, even when the number of clusters is unknown. Additionally, our algorithm performs competitively with state-of-the-art algorithms on the MNIST, USPS, and Fashion-MNIST image data sets. These findings reveal the great potential of tensor networks for machine learning applications.