Faster classical Boson Sampling

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.