The Poisson equation has wide applications in many areas of science and engineering. Although there are some quantum algorithms that can efficiently solve the Poisson equation, they generally require a fault-tolerant quantum computer which is beyond the current technology. In this paper, we propose a Variational Quantum Algorithm (VQA) to solve the Poisson equation, which can be executed on Noise Intermediate-Scale Quantum (NISQ) devices. In detail, we first adopt the finite difference method to transform the Poisson equation into a linear system. Then, according to the special structure of the linear system, we find an explicit tensor product decomposition, with only 2 log n + 1 items, of its coefficient matrix under a specific set of simple operators, where n is the dimension of the coefficient matrix. This implies that the proposed VQA only needs O(log n) measurements, which dramatically reduce quantum resources. Additionally, we perform quantum Bell measurements to efficiently evaluate the expectation values of simple operators. Numerical experiments demonstrate that our algorithm can effectively solve the Poisson equation.
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: (1) It is not known how to efficiently obtain the input of the third sub-algorithm from the output of the second one. (2) 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 significant speedup compared to Liang et al.’s algorithm even without considering the complexity of the VQA.
The application of quantum computation to accelerate machine learning algorithms is one of the most promising areas of research in quantum algorithms. In this paper, we explore the power of quantum learning algorithms in solving an important class of Quantum Phase Recognition (QPR) problems, which are crucially important in understanding many-particle quantum systems. We prove that, under widely believed complexity theory assumptions, there exists a wide range of QPR problems that cannot be efficiently solved by classical learning algorithms with classical resources. Whereas using a quantum computer, we prove the efficiency and robustness of quantum kernel methods in solving QPR problems through Linear order parameter Observables. We numerically benchmark our algorithm for a variety of problems, including recognizing symmetry-protected topological phases and symmetry-broken phases. Our results highlight the capability of quantum machine learning in predicting such quantum phase transitions in many-particle systems.
In recent years, container technologies have attracted intensive attention due to the features of lightweight and easy-portability. The performance isolation between containers is becoming a significant challenge, especially in terms of network throughput and disk I/O. In traditional VM environments, the performance isolation is often calculated based on performance loss ratio. In container environments, the performance loss of well-behaved containers may be incurred not only by misbehaving containers but also by container orchestration and management. Therefore, the measurement models that only take performance loss into consideration will be not accurate enough. In this paper, we proposed a novel performance isolation measurement model that combines the performance loss and resource shrinkage of containers. Experimental results validate the effectiveness of our proposed model. Our results highlight the performance isolation between containers is different with the issue in VM environments.
By introducing the BHT algorithm into the slide attack on 1K-AES and the related-key attack on PRINCE, we present the corresponding quantum attacks in this paper. In the proposed quantum attacks, we generalize the BHT algorithm to the situation where the number of marked items is unknown ahead of time. Moreover, we give an implementation scheme of classifier oracle based on Quantum Phase Estimation algorithm in presented quantum attacks. The complexity analysis shows that the query complexity, time complexity and memory complexity of the presented quantum attacks are all $\mathcal{O}(2^{n/3})$ when the success probability is about $63\%$, where $n$ is the block size. Compared with the corresponding classical attacks, the proposed quantum attacks can achieve subquadratic speed-up under the same success probability no matter on query complexity, time complexity or memory complexity. Furthermore, the query complexity of the proposed quantum slide attack on 1K-AES is less than Grover search on 1K-AES by a factor of $2^{n/6}.$ When compared with the Grover search on PRINCE, the query complexity of the presented quantum attack on PRINCE is reduced from $\mathcal{O}(2^{n})$ to $\mathcal{O}(2^{n/2}).$ When compared with the combination of Grover and Simon’s algorithms on PRINCE, the query complexity of our quantum attack on PRINCE is reduced from $\mathcal{O}(n\cdot 2^{n/2})$ to $\mathcal{O}(2^{n/2}).$ Besides, the proposed quantum slide attack on 1K-AES indicates that the quantum slide attack could also be applied on Substitution-Permutation Network construction, apart from the iterated Even-Mansour cipher and Feistel constructions.
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