arXiv quant-ph/04010532004-01-10quant-ph

量子化游动的谱和一个$\sqrt{δε}$规则

Spectra of Quantized Walks and a $\sqrt{δε}$ rule

Mario Szegedy

备注：: 27pages

摘要：

引入量子化二部游动，计算其谱，推广Grover \cite{g}和Ambainis \cite{amb03}的算法，将其解释为带记忆的量子游动.我们比较了基于步行的经典算法和量子算法的性能，并表明后者总体上运行得更快。设$P$是一个转移概率为$P[i,j]$，$(i ,j\in [n])$的对称马氏链.状态空间的一些元素被标记。我们保证标记元素的集合的大小为零或至少为$εn$。我们的目标是非常确定地找出上述两种情况中哪一种成立。我们的模型是一个黑盒子，可以回答某些是/否问题，并可以生成从某些分布中挑选的随机元素。更具体地说，通过请求，黑盒可以为$\wp_{0}$的成本提供均匀分布的随机元素。此外，当将元素$i$“插入”到黑盒中时，我们可以获得随机元素$j$，其中$j$根据$P[i,j]$分布。后一种操作的成本是$\wp_{1}$。最后，我们可以使用黑盒来测试元素$i$是否被标记，这将花费$\wp_{2}$。如果$δ$是$P$的特征值间隙，则存在解决上述承诺问题的具有成本$O(\wp_{0} + (\wp_{1}+\wp_{2})/δε)$的简单经典算法。(The如果$\wp_{0}$比$\wp_{1}+\wp_{2}$大得多，则算法是有效的。）相比之下，我们表明，对于“量化”版本的算法，它的成本只有$O(\wp_{0} + (\wp_{1}+\wp_{2})/\sqrt{δε})$来解决这个问题。我们将此称为$\sqrt{δε}$规则。在技术贡献中，我们给出了两个一般反射的产物的光谱公式。

英文摘要：

We introduce quantized bipartite walks, compute their spectra, generalize the algorithms of Grover \cite{g} and Ambainis \cite{amb03} and interpret them as quantum walks with memory. We compare the performance of walk based classical and quantum algorithms and show that the latter run much quicker in general. Let $P$ be a symmetric Markov chain with transition probabilities $P[i,j]$, $(i ,j\in [n])$. Some elements of the state space are marked. We are promised that the set of marked elements has size either zero or at least $εn$. The goal is to find out with great certainty which of the above two cases holds. Our model is a black box that can answer certain yes/no questions and can generate random elements picked from certain distributions. More specifically, by request the black box can give us a uniformly distributed random element for the cost of $\wp_{0}$. Also, when ``inserting'' an element $i$ into the black box we can obtain a random element $j$, where $j$ is distributed according to $P[i,j]$. The cost of the latter operation is $\wp_{1}$. Finally, we can use the black box to test if an element $i$ is marked, and this costs us $\wp_{2}$. If $δ$ is the eigenvalue gap of $P$, there is a simple classical algorithm with cost $O(\wp_{0} + (\wp_{1}+\wp_{2})/δε)$ that solves the above promise problem. (The algorithm is efficient if $\wp_{0}$ is much larger than $\wp_{1}+\wp_{2}$.) In contrast,we show that for the ``quantized'' version of the algorithm it costs only $O(\wp_{0} + (\wp_{1}+\wp_{2})/\sqrt{δε})$ to solve the problem. We refer to this as the $\sqrt{δε}$ rule. Among the technical contributions we give a formula for the spectrum of the product of two general reflections.

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摘要：

英文摘要：

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