We show that approximating the second eigenvalue of stochastic operators is BPL-complete, thus giving a natural problem complete for this class. We also show that approximating any eigenvalue of a stochastic and Hermitian operator with constant accuracy can be done in BPL. This work together with related work on the subject reveal a picture where the various space-bounded classes (e.g., probabilistic logspace, quantum logspace and the class DET) can be characterized by algebraic problems (such as approximating the spectral gap) where, roughly speaking, the difference between the classes lies in the kind of operators they can handle (e.g., stochastic, Hermitian or arbitrary).

中文翻译：

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关于概率对数空间中随机矩阵特征值的逼近

我们表明逼近随机算子的第二个特征值是 BPL 完全的，因此给出了这个类的一个自然问题。我们还表明，可以在 BPL 中以恒定精度逼近随机和厄米算子的任何特征值。这项工作与该主题的相关工作一起揭示了一幅图片，其中各种空间有界类（例如，概率对数空间、量子对数空间和 DET 类）可以用代数问题（例如近似光谱间隙）来表征，其中，粗略地说，类之间的区别在于它们可以处理的运算符类型（例如，随机、厄米或任意）。