一种近似配分函数的次线性时间量子算法
A Sublinear-Time Quantum Algorithm for Approximating Partition Functions
摘要：
我们提出了一种新的量子算法估计吉布斯配分函数在次线性时间相对于对数的状态空间的大小。这是第一次在Štefankovi，Vempala和Vigoda [JACM，2009]的开创性近线性时间算法上获得这种类型的加速。我们的结果也保留了二次加速精度和频谱间隙在以前的工作中取得的利用量子马尔可夫链的性质。作为一个应用程序，我们获得了新的多项式改进的最知名的算法计算的伊辛模型的配分函数，计数的$k$ -着色，匹配或独立集的一个图形，并估计凸体的体积。我们的方法依赖于开发量子相位和幅度估计算法的新变体，这些算法返回具有低方差的几乎无偏的估计，并且不破坏它们的初始量子态。我们将这些子程序扩展到一个几乎无偏的量子平均估计，减少方差平方比经典的经验平均更快。在我们的工作之前，没有这样的估计器。这些属性，这是普遍感兴趣的，导致更好的收敛保证内的范例模拟退火计算配分函数。
英文摘要：
We present a novel quantum algorithm for estimating Gibbs partition functions in sublinear time with respect to the logarithm of the size of the state space. This is the first speed-up of this type to be obtained over the seminal nearly-linear time algorithm of Štefankovič, Vempala and Vigoda [JACM, 2009]. Our result also preserves the quadratic speed-up in precision and spectral gap achieved in previous work by exploiting the properties of quantum Markov chains. As an application, we obtain new polynomial improvements over the best-known algorithms for computing the partition function of the Ising model, counting the number of $k$-colorings, matchings or independent sets of a graph, and estimating the volume of a convex body. Our approach relies on developing new variants of the quantum phase and amplitude estimation algorithms that return nearly unbiased estimates with low variance and without destroying their initial quantum state. We extend these subroutines into a nearly unbiased quantum mean estimator that reduces the variance quadratically faster than the classical empirical mean. No such estimator was known to exist prior to our work. These properties, which are of general interest, lead to better convergence guarantees within the paradigm of simulated annealing for computing partition functions.
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