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arXiv 0903.3465quant-ph
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量子行走的有效电路

Efficient Circuits for Quantum Walks

Chen-Fu Chiang, Daniel Nagaj, Pawel Wocjan

备注：
Modified abstract, clarified conclusion, added application section in appendix and updated references
发表信息：
Quantum Information and Computation, Vol.10, No.5&6, pp0420-0434 (2010)
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摘要：

我们提出了一个有效的通用方法来实现量子行走算子对应的任意稀疏经典随机行走。我们的方法是基于Grover和Rudolph的方法，用于准备有效可积概率分布的相干版本。这种方法旨在用于具有多项式加速比的量子行走算法，其复杂性通常是根据我们必须应用量子行走的步骤的次数来衡量的，与必要的经典马尔可夫链步骤的数量相比。我们考虑一个更精细的复杂性概念，包括以某种期望的精度实现量子行走的每一步所需的基本门的数量。各种实现方法的复杂性的差异在于，我们的方法在稀疏性参数中线性缩放，并且与所需精度的倒数多项式地缩放。以前最好的已知的一般方法要么规模平方的稀疏参数，或多项式的逆精度。我们的方法是特别相关的实现量子行走对应于经典的随机行走，如在经典算法中使用的近似永久和采样从二进制列联表。在这些算法中，稀疏参数随着问题的规模而增长，同时保持高精度的要求。

英文摘要：

We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph's method for preparing coherent versions of efficiently integrable probability distributions. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measured in terms of how many times we have to apply a step of a quantum walk, compared to the number of necessary classical Markov chain steps. We consider a finer notion of complexity including the number of elementary gates it takes to implement each step of the quantum walk with some desired accuracy. The difference in complexity for various implementation approaches is that our method scales linearly in the sparsity parameter and poly-logarithmically with the inverse of the desired precision. The best previously known general methods either scale quadratically in the sparsity parameter, or polynomially in the inverse precision. Our approach is especially relevant for implementing quantum walks corresponding to classical random walks like those used in the classical algorithms for approximating permanents and sampling from binary contingency tables. In those algorithms, the sparsity parameter grows with the problem size, while maintaining high precision is required.

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