One of the most fundamental problems in quantum many-body physics is the characterization of correlations among thermal states. Of particular relevance is the thermal area law, which justifies the tensor network approximations to thermal states with a bond dimension growing polynomially with the system size. In the regime of sufficiently low temperatures, which is particularly important for practical applications, the existing techniques do not yield optimal bounds. Here, we propose a new thermal area law that holds for generic many-body systems on lattices. We improve the temperature dependence from the original $\mathcal{O}(β)$ to $\tilde{\mathcal{O}}(β^{2/3})$, thereby suggesting diffusive propagation of entanglement by imaginary time evolution. This qualitatively differs from the real-time evolution which usually induces linear growth of entanglement. We also prove analogous bounds for the Rényi entanglement of purification and the entanglement of formation. Our analysis is based on a polynomial approximation to the exponential function which provides a relationship between the imaginary-time evolution and random walks. Moreover, for one-dimensional (1D) systems with $n$ spins, we prove that the Gibbs state is well-approximated by a matrix product operator with a sublinear bond dimension of $e^{\sqrt{\tilde{\mathcal{O}}(β\log(n))}}$. This proof allows us to rigorously establish, for the first time, a quasi-linear time classical algorithm for constructing an MPS representation of 1D quantum Gibbs states at arbitrary temperatures of $β= o(\log(n))$. Our new technical ingredient is a block decomposition of the Gibbs state, that bears resemblance to the decomposition of real-time evolution given by Haah et al., FOCS'18.
Quasi-optimal sampling from Gibbs states via non-commutative optimal transport metrics
Fachbereich Mathematik, Universität Tübingen（蒂宾根大学数学系）；Dep. of App. Mathematics and Theo. Physics, University of Cambridge（剑桥大学应用数学与理论物理系）；Inria, Télécom Paris - LTCI, Institut Polytechnique de Paris（巴黎理工学院电信学院LTCI实验室法国国家信息与自动化研究所）
Optimal quantum algorithm for Gibbs state preparation
Inria, Télécom Paris - LTCI, Institut Polytechnique de Paris（法国国家信息与自动化研究所，巴黎电信学院-信息与通信实验室，巴黎理工学院）；Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP（里昂大学，里昂高等师范学院，里昂第一大学，法国国家科学研究中心，法国国家信息与自动化研究所，信息处理实验室）；Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学科学系）；Instituto de Física Téorica UAM/CSIC（马德里自治大学/西班牙高等科学研究理事会理论物理研究所）
Classical Estimation of the Free Energy and Quantum Gibbs Sampling from the Markov Entropy Decomposition
University of Cambridge（剑桥大学）；Universität Tübingen（蒂宾根大学）；IBM Quantum, IBM T. J. Watson Research Center（IBM量子，IBM T.J.沃森研究中心）；Inria, ENS Lyon, UCBL, LIP（法国国家信息与自动化研究所，里昂高等师范学院，里昂第一大学，计算与并行实验室）；UC Davis（加州大学戴维斯分校）；University of Sydney（悉尼大学）
Accelerating quantum Gibbs sampling without quantum walks
Virginia Tech（弗吉尼亚理工大学）；Simons Institute for the Theory of Computing, University of California, Berkeley（加州大学伯克利分校西蒙斯理论计算研究所）；Department of Mathematics, University of California, Berkeley（加州大学伯克利分校数学系）；Department of Computing and Mathematical Sciences, California Institute of Technology（加州理工学院计算与数学科学系）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
Polynomial-time thermalization and Gibbs sampling from system-bath couplings
Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP（里昂大学，巴黎高等师范学院里昂分校，里昂第一大学，法国国家科学研究中心，法国信息与自动化研究所，里昂理工学院）；Instituto de Física Téorica UAM/CSIC（马德里自治大学/西班牙国家研究委员会理论物理研究所）；Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学科学系）；Inria, Télécom Paris - LTCI, Institut Polytechnique de Paris（法国信息与自动化研究所，巴黎电信学院通信与图像处理实验室，巴黎理工学院）
An (almost) efficient classical algorithm for sampling from typical Gibbs states
Institute for Quantum Information and Matter, Caltech（加州理工学院量子信息与物质研究所）；Department of Physics, National University of Singapore（新加坡国立大学物理系）；Department of Computer Science, National University of Singapore（新加坡国立大学计算机科学系）；Walter Burke Institute for Theoretical Physics, Caltech（加州理工学院沃尔特·伯克理论物理研究所）
Efficient thermalization and universal quantum computing with quantum Gibbs samplers
Inria, Télécom Paris - LTCI, Institut Polytechnique de Paris（法国国家信息与自动化研究所，巴黎电信学院-信号与图像实验室，巴黎理工学院）；Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP（里昂大学，里昂高等师范学校，里昂第一大学，法国国家科学研究中心，法国国家信息与自动化研究所，里昂信息学实验室）；Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学科学系）；Instituto de Física Téorica UAM/CSIC（马德里自治大学/西班牙国家研究委员会理论物理研究所）
Overcoming the Quasi-Static Bottleneck: A Finite-Time Quantum Otto Information Engine Achieving Near-Unity Efficiency
College of Physics, Jilin University（吉林大学物理学院）；Department of Chemistry and Department of Physics and Astronomy, State University of New York at Stony Brook（纽约州立大学石溪分校化学系和物理与天文学系）
RWTH Aachen University（亚琛工业大学）；University of the Basque Country UPV/EHU（巴斯克大学）；Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心）；Imperial College London（帝国理工学院）
Efficient Quantum Gibbs Sampling with Local Circuits
University of Oxford（牛津大学）；IBM Quantum, IBM Thomas J. Watson Research Center（IBM量子，IBM托马斯·J·沃森研究中心）；IBM Quantum, IBM Research – Almaden（IBM量子，IBM阿尔马登研究中心）；African Institute for Mathematical Sciences (AIMS)（非洲数学科学研究所）；Department of Mathematical Sciences, Stellenbosch University（斯泰伦博斯大学数学科学系）；National Institute for Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）
