We construct a tensor network that delivers an unnormalized quantum state whose coefficients are the solutions to a given instance of 3SAT, an NP-complete problem. The tensor network contraction that corresponds to the norm of the state counts the number of solutions to the instance. It follows that exact contractions of this tensor network are in the #P-complete computational complexity class, thus believed to be a hard task. Furthermore, we show that for a 3SAT instance with n bits, it is enough to perform a polynomial number of contractions of the tensor network structure associated to the computation of local observables to obtain one of the explicit solutions to the problem, if any. Physical realization of a state described by a generic tensor network is equivalent to finding the satisfying assignment of a 3SAT instance and, consequently, this experimental task is expected to be hard.
Tropical Tensor Network for Ground States of Spin Glasses
我们提出了一个统一的精确张量网络方法来计算自旋玻璃的基态能量，确定最佳构型，并计算溶液的数量。该方法是基于张量网络与热带代数定义的半环。收缩热带张量网络得到基态能量;通过张量网络收缩微分得到基态构型;混合热带代数和普通代数计算基态简并度。该方法汇集了图形模型，张量网络，微分编程和量子线路模拟的概念，并很容易利用图形处理单元（GPU）的计算能力。在应用方面，我们在单个GPU上计算了正方晶格上1024个自旋、立方晶格上216个自旋和3个规则随机图上220个自旋的Ising自旋玻璃的精确基态能量;在D-的嵌合图上，我们得到了（+/-）J Ising自旋玻璃的精确基态能量在不到100秒的时间内完成了512个量子比特的波量子退火，并研究了嵌合体图上（+/-）J自旋玻璃的剩余熵的精确值;最后，我们研究了18 x 18的正方形晶格上三态Potts玻璃的基态能量和熵。我们的方法提供了基线和基准精确算法的自旋眼镜和组合优化问题，并评估启发式算法和平均场论。
Exact NMR simulation of protein-size spin systems using tensor train formalism
University of Southampton, School of Chemistry（南安普顿大学化学学院）；University of Southampton, Centre for Biological Systems（南安普顿大学生物系统中心）；Max-Planck Institute for Mathematics in the Sciences（马普科学数学研究所）
TENSO: Software Package for Numerically Exact Open Quantum Dynamics Based on Efficient Tree Tensor Network Decomposition of the Hierarchical Equations of Motion
Prime Factorization Equation from a Tensor Network Perspective
Instituto Tecnológico de Castilla y León（卡斯蒂利亚和莱昂理工学院）；Andalusian Research Institute in Data Science and Computational Intelligence (DaSCI)（安达卢西亚数据科学与计算智能研究所（DaSCI））；University of Jaén（哈恩大学）
