Sign Problem in Tensor-Network Contraction

Autor(en)

Jielun Chen, Jiaqing Jiang, Dominik Hangleiter, Norbert Schuch

Abstrakt

We investigate how the computational difficulty of contracting tensor networks depends on the sign structure of the tensor entries. Using results from computational complexity, we observe that the approximate contraction of tensor networks with only positive entries has lower complexity. This raises the question how this transition in computational complexity manifests itself in the hardness of different contraction schemes. We pursue this question by studying random tensor networks with varying bias towards positive entries. First, we consider contraction via Monte Carlo sampling, and find that the transition from hard to easy occurs when the entries become predominantly positive; this can be seen as a tensor network manifestation of the Quantum Monte Carlo sign problem. Second, we analyze the commonly used contraction based on boundary tensor networks. Its performance is governed by the amount of correlations (entanglement) in the tensor network. Remarkably, we find that the transition from hard to easy (i.e., from a volume law to a boundary law scaling of entanglement) occurs already for a slight bias towards a positive mean, and the earlier the larger the bond dimension is. This is in contrast to both expectations and the behavior found in Monte Carlo contraction. We gain further insight into this early transition from the study of an effective statmech model. Finally, we investigate the computational difficulty of computing expectation values of tensor network wavefunctions, i.e., PEPS, where we find that the complexity of entanglement-based contraction always remains low. We explain this by providing a local transformation which maps PEPS expectation values to a positive-valued tensor network. This not only provides insight into the origin of the observed boundary law entanglement scaling, but also suggests new approaches towards PEPS contraction based on positive decompositions.

Organisation(en)

Institut für Mathematik, Quantenoptik, Quantennanophysik und Quanteninformation

Externe Organisation(en)

California Institute of Technology (Caltech), University of Maryland, College Park

Journal

PRX Quantum

Band

6

Anzahl der Seiten

20

ISSN

2691-3399

DOI

https://doi.org/10.48550/arXiv.2404.19023

Publikationsdatum

12-2024

Peer-reviewed

Ja

ÖFOS 2012

103036 Theoretische Physik, 103025 Quantenmechanik, 101028 Mathematische Modellierung

Schlagwörter

Link zum Portal

https://ucrisportal.univie.ac.at/de/publications/40b8a8b4-1ef6-44fc-a19a-8468a9838106