Blockmodeling of a given problem represented by an $N\times N$ adjacency matrix can be found by swapping rows and columns of the matrix (i.e. multiplying matrix from left and right by a permutation matrix). Although classical matrix permutations can be efficiently done by swapping pointers for the permuted rows (or columns) of the matrix, by changing row-column order, a permutation changes the location of the matrix elements, which determines the membership of a group in the matrix based blockmodeling. Therefore, a brute force initial estimation of a fitness value for a candidate solution involving counting the memberships of the elements may require going through all the sum of the rows (or the columns). Similarly permutations can be also implemented efficiently on quantum computers, e.g. a NOT gate on a qubit. In this paper, using permutation matrices and qubit measurements, we show how to solve blockmodeling on quantum computers. In the model, the measurement outcomes of a small group of qubits are mapped to indicate the fitness value. However, if the number of qubits in the considered group is much less than $n=log(N)$, it is possible to find or update the fitness value based on the state tomography in $O(poly(log(N)))$. Therefore, when the number of iterations is less than $log(N)$ time and the size of the considered qubit group is small, we show that it may be possible to reach the solution very efficiently.

翻译：给定一个由$N\times N$邻接矩阵表示的问题，其分块建模可通过交换矩阵的行与列实现（即左乘与右乘置换矩阵）。尽管经典矩阵置换可通过交换被置换行（或列）的指针高效完成，但改变行列顺序会改变矩阵元素的位置，进而决定矩阵分块建模中元素的群组归属。因此，针对候选解初始适应的暴力估算（需统计元素归属关系）可能需要遍历所有行（或列）之和。类似地，置换操作同样可在量子计算机上高效实现，例如对量子比特施加NOT门。本文利用置换矩阵与量子比特测量，展示了如何在量子计算机上求解分块建模问题。在该模型中，通过对少量量子比特的测量结果进行映射，即可表征适应度值。然而，当所考察量子比特组的数量远小于$n=log(N)$时，可基于态层析成像在$O(poly(log(N)))$复杂度内计算或更新适应度值。因此，当迭代次数少于$log(N)$时间且考虑的量子比特组尺寸较小时，我们证明了可能以极高效率获得解。