arXiv 2005.042142020-05-26quant-phcs.DS

更快的经典玻色子采样

Faster classical Boson Sampling

Peter Clifford, Raphaël Clifford

备注：: 10 pages. arXiv admin note: text overlap with arXiv:1706.01260

摘要：

自从玻色子采样被引入以来，它一直是量子计算领域的热门研究课题。任务是根据与子矩阵的持久式相关的概率分布，从由较大的$m \times n$复矩阵的可能重复的行构建的所有$n \times n$子矩阵的集合中独立地进行采样。利用量子光子效应的实验系统原则上可以高速执行任务。在经典计算的框架中，Aaronson和Arkhipov（2011）证明了精确玻色子采样问题不能在多项式时间内解决，除非多项式层次崩溃到第三层。事实上，多年来，已知最快的精确经典算法在每个样本的$O({m+n-1 \choose n} n 2^n )$时间内运行，强调了量子计算的潜在速度优势。Clifford和Clifford（2018）减少了这一优势，他们给出了一个明显更快的经典解，需要$O(n 2^n + \operatorname{poly}(m,n))$时间和线性空间，当$m$在$n$中是多项式时，与计算单个矩阵的积和式的复杂性相匹配。我们继续提出一个算法玻色子采样的平均情况下的时间复杂度是快得多时，$m$是成比例的$n$。特别是，当$m = n$我们的算法运行在大约$O(n\cdot1.69^n)$时间平均。这一结果进一步增加了通过玻色子采样建立量子计算霸权所需的问题规模。

英文摘要：

Since its introduction Boson Sampling has been the subject of intense study in the world of quantum computing. The task is to sample independently from the set of all $n \times n$ submatrices built from possibly repeated rows of a larger $m \times n$ complex matrix according to a probability distribution related to the permanents of the submatrices. Experimental systems exploiting quantum photonic effects can in principle perform the task at great speed. In the framework of classical computing, Aaronson and Arkhipov (2011) showed that exact Boson Sampling problem cannot be solved in polynomial time unless the polynomial hierarchy collapses to the third level. Indeed for a number of years the fastest known exact classical algorithm ran in $O({m+n-1 \choose n} n 2^n )$ time per sample, emphasising the potential speed advantage of quantum computation. The advantage was reduced by Clifford and Clifford (2018) who gave a significantly faster classical solution taking $O(n 2^n + \operatorname{poly}(m,n))$ time and linear space, matching the complexity of computing the permanent of a single matrix when $m$ is polynomial in $n$. We continue by presenting an algorithm for Boson Sampling whose average-case time complexity is much faster when $m$ is proportional to $n$. In particular, when $m = n$ our algorithm runs in approximately $O(n\cdot1.69^n)$ time on average. This result further increases the problem size needed to establish quantum computational supremacy via Boson Sampling.

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