A Simple Quantum Blockmodeling with Qubits and Permutations

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). In general, through performing this task, row and column permutations affect the fitness value in optimization: For an $N\times N$ matrix, it requires $O(N)$ computations to find (or update) the fitness value of a candidate solution. On quantum computers, permutations can be applied in parallel and efficiently, and their implementations can be as simple as a single qubit operation (a NOT gate on a qubit) which takes an $O(1)$ time algorithmic step. In this paper, using permutation matrices, we describe a quantum blockmodeling for data analysis tasks. In the model, the measurement outcome of a small group of qubits are mapped to indicate the fitness value. Therefore, we show that it is possible to find or update the fitness value in $O(log(N))$ time. This lead us to show that when the number of iterations are less than $log(N)$ time, it may be possible to reach the same solution exponentially faster on quantum computers in comparison to classical computers. In addition, since on quantum circuits the different sequence of permutations can be applied in parallel (superpositon), the machine learning task in this model can be implemented more efficiently on quantum computers.