Given a parameterized quantum circuit such that a certain setting of these real-valued parameters corresponds to Grover's celebrated search algorithm, can a variational algorithm recover these settings and hence learn Grover's algorithm? We studied several constrained variations of this problem and answered this question in the affirmative, with some caveats. Grover's quantum search algorithm is optimal up to a constant. The success probability of Grover's algorithm goes from unity for two-qubits, decreases for three-and four-qubits and returns near unity for five-qubits then oscillates ever-so-close to unity, reaching unity in the infinite qubit limit. The variationally approach employed here found an experimentally discernible improvement of 5.77% and 3.95% for three-and four-qubits respectively. Our findings are interesting as an extreme example of variational search, and illustrate the promise of using hybrid quantum classical approaches to improve quantum algorithms. This paper further demonstrates that to find optimal parameters one doesn't need to vary over a family of quantum circuits to find an optimal solution. This result looks promising and points out that there is a set of variational quantum problems with parameters that can be efficiently found on a classical computer for arbitrary number of qubits.Grover's algorithm [2] is one of the most celebrated quantum algorithms, enabling quantum computers to quadratically outperform classical computers at database search provided database access is restricted to a 'black box' -called the oracle model. In addition to the wide application scope of database search, Grover's algorithm has further applications as a subroutine used in a variety of other quantum algorithms.Variational hybrid quantum/classical algorithms have recently become an area of significant interest [3][4][5][6][7][8][9][10]. These algorithms have shown several advantages such as robustness to quantum errors and low coherence time requirements [11], which make them ideal for implementations in current quantum computer architectures. Here we take inspiration from algorithms such as the variational quantum eigensolver (VQE) [3] and the quantum approximate optimization algorithm (QAOA) [4]. The general procedure of these variational hybrid quantum/classical algorithms is the following:
Complex network theory has shown success in understanding the emergent and collective behavior of complex systems Newman 2010 Networks: An Introduction (Oxford: Oxford University Press). Many real-world complex systems were recently discovered to be more accurately modeled as multiplex networks Bianconi 2018 Multilayer Networks: Structure and Function (Oxford: Oxford University Press); Boccaletti et al 2014 Phys. Rep. 544 1–122; Lee et al 2015 Eur. Phys. J. B 88 48; Kivelä et al 2014 J. Complex Netw. 2 203–71; De Domenico et al 2013 Phys. Rev. X 3 041022—in which each interaction type is mapped to its own network layer; e.g. multi-layer transportation networks, coupled social networks, metabolic and regulatory networks, etc. A salient physical phenomena emerging from multiplexity is super-diffusion: exhibited by an accelerated diffusion admitted by the multi-layer structure as compared to any single layer. Theoretically super-diffusion was only known to be predicted using the spectral gap of the full Laplacian of a multiplex network and its interacting layers. Here we turn to machine learning (ML) which has developed techniques to recognize, classify, and characterize complex sets of data. We show that modern ML architectures, such as fully connected and convolutional neural networks (CNN), can classify and predict the presence of super-diffusion in multiplex networks with 94.12% accuracy. Such predictions can be done in situ, without the need to determine spectral properties of a network.
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