位置权矩阵匹配的量子算法
Quantum algorithm for position weight matrix matching
摘要：
我们提出了两个量子算法的问题，在生物信息学中，位置权重矩阵（PWM）匹配，其目的是找到段（序列基序）在生物序列，如DNA和蛋白质，具有高分数定义的PWM，从而与生物功能相关的信息的重要性。两种算法，朴素迭代法和基于蒙特卡罗的方法，输出匹配的片段，给出了对生物序列和PWM中的条目的预言访问。前者使用量子幅度放大（QAA）进行序列模体搜索，导致查询复杂度在序列长度$n$、序列模体长度$m$和PWM数目$K$上缩放为$\widetilde{O}\left(m\sqrt{Kn}\right)$，这意味着相对于现有经典算法在$n$和$K$方面的加速。后者还使用QAA，并进一步使用量子蒙特卡罗积分进行分段得分计算，而不是在朴素迭代方法中迭代地操作量子线路进行算术运算;然后在某些情况下，它提供了相对于$m$的额外加速。作为缺点，这些算法使用量子随机存取存储器，并且它们的初始化花费$O(n)$时间。然而，我们的算法保持的优势，特别是当我们搜索匹配的序列中的许多PWM并行。
英文摘要：
We propose two quantum algorithms for a problem in bioinformatics, position weight matrix (PWM) matching, which aims to find segments (sequence motifs) in a biological sequence such as DNA and protein that have high scores defined by the PWM and are thus of informational importance related to biological function. The two proposed algorithms, the naive iteration method and the Monte-Carlo-based method, output matched segments, given the oracular accesses to the entries in the biological sequence and the PWM. The former uses quantum amplitude amplification (QAA) for sequence motif search, resulting in the query complexity scaling on the sequence length $n$, the sequence motif length $m$ and the number of the PWMs $K$ as $\widetilde{O}\left(m\sqrt{Kn}\right)$, which means speedup over existing classical algorithms with respect to $n$ and $K$. The latter also uses QAA, and further, quantum Monte Carlo integration for segment score calculation, instead of iteratively operating quantum circuits for arithmetic in the naive iteration method; then it provides the additional speedup with respect to $m$ in some situation. As a drawback, these algorithms use quantum random access memories and their initialization takes $O(n)$ time. Nevertheless, our algorithms keep the advantage especially when we search matches in a sequence for many PWMs in parallel.
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