稀疏图上经典随机游动混合性质的BQP-完全问题
BQP-complete Problems Concerning Mixing Properties of Classical Random Walks on Sparse Graphs
摘要：
我们描述了两个关于具有一定对称性的稀疏图的性质的BQP-完全问题。该图是由有效的可计算的功能，输出相邻的顶点为每个顶点指定。设i和j是两个给定的顶点。第一个问题在于估计从j到j的长度为m的路径的数目与从i到j的长度为m的路径的数目之间的差，其中m是顶点数目的多对数。估计精度的尺度由这些差异随着m的增加而增长的一些先验已知的上界指定。对于度为4的正则图，这个问题仍然是BQP困难的。第二个问题与连续时间经典随机游动有关。遍历从某个顶点j开始。承诺是，对于某些μ>0，分别在j和i的概率之差以O（exp（-μt））衰减。问题是决定在某个时刻T之后，这个差是大于a exp（-μT）还是小于b exp（-μT），其中T是多对数的，并且差a-b是顶点数的反多对数。由于概率的差异仅为指数级的微小数值，因此如果要通过步行来回答这个问题，就需要进行指数级的试验。这个问题的一个修改，询问是否存在一对节点，其概率差至少是exp（-μT），是QCMA完全的。
英文摘要：
We describe two BQP-complete problems concerning properties of sparse graphs having a certain symmetry. The graphs are specified by efficiently computable functions which output the adjacent vertices for each vertex. Let i and j be two given vertices. The first problem consists in estimating the difference between the number of paths of length m from j to j and those which from i to j, where m is polylogarithmic in the number of vertices. The scale of the estimation accuracy is specified by some a priori known upper bound on the growth of these differences with increasing m. The problem remains BQP-hard for regular graphs with degree 4. The second problem is related to continuous-time classical random walks. The walk starts at some vertex j. The promise is that the difference of the probabilities of being at j and at i, respectively, decays with O(exp(-μt)) for some μ>0. The problem is to decide whether this difference is greater than a exp(-μT) or smaller than b exp(-μT) after some time instant T, where T is polylogarithmic and the difference a-b is inverse polylogarithmic in the number of vertices. Since the probabilities differ only by an exponentially small amount, an exponential number of trials would be necessary if one tried to answer this question by running the walk itself. A modification of this problem, asking whether there exists a pair of nodes for which the probability difference is at least a exp(-μT), is QCMA-complete.
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