Analysis of weighted quantum secret sharing based on matrix product states
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Analysis of weighted quantum secret sharing based on matrix product states Hong Lai1 · Josef Pieprzyk2,3 · Lei Pan4 Received: 1 July 2020 / Accepted: 29 October 2020 / Published online: 17 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, motivated by the usefulness of tensor networks to quantum information theory for the progress in study of entanglement in quantum many-body systems, we apply the hexagon tensor network algorithms in terms of Holland’s theory to study the weight allocation and dynamic problems in weighted quantum secret sharing that are not well solved by existing approaches with near-term devices and avoid the instability in the allocation of participants. To be exact, the variety of matrix product state representation of any quantum many-body state can be used to realize dynamic quantum state secret sharing. Keywords Tensor networks · Quantum many-body systems · Hexagon tensor network · Weighted quantum secret sharing · Matrix product state
1 Introduction Based on a deep understanding of quantum entanglement, a new algorithm called tensor network state (TNS) has been developed, to explore new solutions for multidimensional problems. In 2003, Vidal [1] obtained a single dimension lattice system based on the Schmidt decomposition of quantum states. In 2004, Vidal [2] developed the matrix product state (MPS) representation of one-dimensional lattice system using the Schmidt decomposition. Based on this representation, a highly efficient algorithm, namely time-evolution block decimation (TEBD), has been proposed to simulate onedimensional lattice system [2,3]. Later, Verstrateet et al. rephrased the density matrix
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Hong Lai [email protected]
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School of Computer and Information Science, Southwest University, Chongqing 400715, China
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Data61, CSIRO, Sydney, NSW 2122, Australia
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Institute of Computer Science, Polish Academy of Sciences, 01-248 Warsaw, Poland
4
School of Information Technology, Deakin University, Geelong, VIC 3220, Australia
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renormalisation group (DMRG) method from the perspective of a combination of quantum entanglement and MPS, which greatly improved the performance of the DMRG method in dealing with one-dimensional periodic systems [4]. In fact, the DMRG and MPS methods are essentially equivalent, because the form of the ground state wave function obtained from DMRG also constitutes the MPS form. Thus, MPS can be used as a variational wave function in a natural way [5]. The near-term devices can only be computed for systems of a modest size of around 6 sites, according to [6]. How we can remove the restriction of the small number of sites is the first problem that needs to be addressed. We know that the many-body Hamiltonian can be coarse-grained to a few-site Hamiltonian, which can be exactly diagonalized, while preserving its low energy subspace. In order to solve the first problem, the tensor network model can also be defined in a dual space, because the dual lattice of a
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