Efficient Tensor Network 
<i>Ansatz</i>
 for High-Dimensional Quantum Many-Body Problems

Felser, Timo; Notarnicola, Simone; Montangero, Simone

doi:10.1103/physrevlett.126.170603

Cited by 49 publications

(54 citation statements)

References 81 publications

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“…While the locality properties of the Hilbert ordering are directly mapped in the MPS chain, in the case of TTNs they are enhanced by the logarithmic scaling of distances within the TTN structure. By recalling that the standard way to use TTNs for two-dimensional systems relies directly on the construction of a binary 2D TTN [48], it is worth noting that the binary 2D TTN and the Hilbert curve mapping generate the same long range interactions within the corresponding networks. Therefore, the former benefits from the mathematical locality-preserving properties of the Hilbert curve.…”

Section: Discussionmentioning

confidence: 99%

“…In particular, this is carried out by compressing the exponentially large wavefunctions into a network of tensors interconnected through auxiliary indices with bond dimension m. The main Anzätze for the representation of quantum many-body states based on TNs include Matrix Product States (MPS) for 1D systems [30][31][32], Projected Entangled Pair States (PEPS) [33][34][35], Tree Tensor Networks (TTN) [36][37][38][39] and Multiscale Entanglement Renormalization Ansatz (MERA) [40,41] which can be defined in any dimension. However, while for one-dimensional systems MPS are the established TN ansatz, the development of efficient TN algorithms for higher-dimensional systems is still ongoing [42][43][44][45][46][47][48][49].…”

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confidence: 99%

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Hilbert curve vs Hilbert space: exploiting fractal 2D covering to increase tensor network efficiency

Abedi

Magnifico

et al. 2021

Quantum

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We present a novel mapping for studying 2D many-body quantum systems by solving an effective, one-dimensional long-range model in place of the original two-dimensional short-range one. In particular, we address the problem of choosing an efficient mapping from the 2D lattice to a 1D chain that optimally preserves the locality of interactions within the TN structure. By using Matrix Product States (MPS) and Tree Tensor Network (TTN) algorithms, we compute the ground state of the 2D quantum Ising model in transverse field with lattice size up to 64×64, comparing the results obtained from different mappings based on two space-filling curves, the snake curve and the Hilbert curve. We show that the locality-preserving properties of the Hilbert curve leads to a clear improvement of numerical precision, especially for large sizes, and turns out to provide the best performances for the simulation of 2D lattice systems via 1D TN structures.

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Section: Discussionmentioning

confidence: 99%

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confidence: 99%

Hilbert curve vs Hilbert space: exploiting fractal 2D covering to increase tensor network efficiency

Abedi

Magnifico

et al. 2021

Quantum

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“…Here is where methods based on the Hamiltonian formalism in Minkowski spacetime can provide a way out. In fact, tensor network methods have seen very rapid developments in the recent years towards the possibility of simulations in 2 + 1 and 3 + 1 space-time dimensions [1,2]. And the number of qubits available on real quantum devices is ever increasing.…”

Section: Introductionmentioning

confidence: 99%

“…The aim is then to find points on S 3 depending on some parameter m which are dense in S 3 as m approaches infinity. In this paper we investigate all the discrete subgroups of SU (2) and several representative discretisations of S 3 . We study the freezing transition as a function of the number of elements in these discretisations and show that the discretisation based on so-called Fibonacci lattices behaves optimally.…”

Section: Introductionmentioning

confidence: 99%

Digitising SU(2) Gauge Fields and the Freezing Transition

Hartung,

Jakobs,

Jansen

et al. 2022

Preprint

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Efficient discretisations of gauge groups are crucial with the long term perspective of using tensor networks or quantum computers for lattice gauge theory simulations. For any Lie group other than U(1), however, there is no class of asymptotically dense discrete subgroups. Therefore, discretisations limited to subgroups are bound to lead to a freezing of Monte Carlo simulations at weak couplings, necessitating alternative partitionings without a group structure. In this work we provide a comprehensive analysis of this freezing for all discrete subgroups of SU(2) and different classes of asymptotically dense subsets. We find that an appropriate choice of the subset allows unfrozen simulations for arbitrary couplings, though one has to be careful with varying weights of unevenly distributed points. A generalised version of the Fibonacci spiral appears to be particularly efficient and close to optimal.

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“…By quantitatively validating quantum simulators in out-of-equilibrium situations, even if only in lower dimensions, MPS and TN methods play a very important role towards establishing quantum simulators as reliable tools in quantum physics. Finally, recently the applications of TN to higher-dimensional LGTs has start becoming within reach of the available algorithms and numerical resources [51][52][53][54][55][56][57], providing an important stimulus to further develop these techniques and explore their application to LGTs.…”

Section: Introductionmentioning

confidence: 99%

Loop-free tensor networks for high-energy physics

Montangero

Rico

Silvi

2021

Phil. Trans. R. Soc. A.

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This brief review introduces the reader to tensor network methods, a powerful theoretical and numerical paradigm spawning from condensed matter physics and quantum information science and increasingly exploited in different fields of research, from artificial intelligence to quantum chemistry. Here, we specialize our presentation on the application of loop-free tensor network methods to the study of high-energy physics problems and, in particular, to the study of lattice gauge theories where tensor networks can be applied in regimes where Monte Carlo methods are hindered by the sign problem. This article is part of the theme issue ‘Quantum technologies in particle physics’.

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Efficient Tensor Network Ansatz for High-Dimensional Quantum Many-Body Problems

Cited by 49 publications

References 81 publications

Hilbert curve vs Hilbert space: exploiting fractal 2D covering to increase tensor network efficiency

Hilbert curve vs Hilbert space: exploiting fractal 2D covering to increase tensor network efficiency

Digitising SU(2) Gauge Fields and the Freezing Transition

Loop-free tensor networks for high-energy physics

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