论文标题
在热力学极限中的一维量子系统的自由能的亚物质时间算法
A subpolynomial-time algorithm for the free energy of one-dimensional quantum systems in the thermodynamic limit
论文作者
论文摘要
我们引入了一种经典算法，以近似无限链尺寸的热力学极限，近似局部，翻译不变的一维量子系统的自由能。虽然这些系统的基态问题（即，对于量子计算机而言，这些系统的自由能在计算上也很难计算，但我们的算法在任何固定温度$ t> 0 $ subolynomial时间内运行$ \ varepsilon $是添加近似错误。以前，最著名的算法具有$ \ frac {1} {\ varepsilon} $的多项式运行时。我们的算法也特别简单，因为它减少了线性图的光谱半径的计算。该线性图的解释为非交通转移矩阵，并先前已经研究了以证明自由能的分析性和相关性衰减的结果。我们还表明，该地图的相应特征向量给出了吉布斯状态边缘的近似值，从而可以计算量子系统的各种热力学特性。
We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature $T = 0$) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed temperature $T > 0$ in subpolynomial time, i.e., in time $O((\frac{1}{\varepsilon})^{c})$ for any constant $c > 0$ where $\varepsilon$ is the additive approximation error. Previously, the best known algorithm had a runtime that is polynomial in $\frac{1}{\varepsilon}$. Our algorithm is also particularly simple as it reduces to the computation of the spectral radius of a linear map. This linear map has an interpretation as a noncommutative transfer matrix and has been studied previously to prove results on the analyticity of the free energy and the decay of correlations. We also show that the corresponding eigenvector of this map gives an approximation of the marginal of the Gibbs state and thereby allows for the computation of various thermodynamic properties of the quantum system.
