论文标题
改进的量子吉布斯状态的热区法律和准线性时间算法
Improved thermal area law and quasi-linear time algorithm for quantum Gibbs states
论文作者
论文摘要
量子多体物理学中最根本的问题之一是热状态之间相关性的表征。特别相关的是热区定律，这证明了张量网络近似与系统尺寸多样地生长的热状态的张量近似。在足够低温的制度中，对于实际应用尤其重要，现有技术不会产生最佳的界限。在这里，我们提出了一项新的热区法律，该法律适用于晶格上的通用多体系统。我们将温度依赖性从原始的$ \ MATHCAL {O}（β）$到$ \ tilde {\ Mathcal {O}}}（β^{2/3}）$，从而暗示通过想象时间演变来扩散纠缠的扩散。这种定性与通常诱导纠缠线性生长的实时演化有所不同。我们还证明了Rényi纯化和形成纠缠的纠缠相似的界限。我们的分析基于对指数函数的多项式近似，该函数提供了假想时间演化与随机步行之间的关系。此外，对于带有$ n $ spins的一维（1D）系统，我们证明了吉布斯状态是由矩阵产品运营商对具有$ e^{\ sqrt {\ sqrt {\ tilde {\ tilde {\ tilde {\ Mathcal {O}}}（β\ log log log（n）}}}}}的矩阵债券尺寸的应用。该证明使我们能够首次严格建立一种准线性时间古典算法，用于在任意温度为$β= o（\ log（n））$的任意温度下构建1D量子Gibbs状态的MPS表示。我们的新技术成分是吉布斯国家的块分解，与Haah等人，focs'18给出的实时进化的分解相似。
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.
