A Simple Quantum Blockmodeling with Qubits and Permutations

TL;DR

The paper reframes blockmodeling as a quantum optimization problem in which a sequence of row/column permutations $P_{ab}$ acts on a data-encoded quantum state to maximize overlap with a cluster state. It formalizes qubit-encoding of the data matrix, defines a fitness measure from measurements of a small qubit group, and analyzes both classical and quantum implementations of the permutation-based search, including a superposition of permutations. A concrete Barbell-graph example demonstrates the approach and shows how restricted, targeted permutations can accelerate convergence, while highlighting trade-offs between search space and performance. The work proposes measurement-guided optimization, potential speedups via state tomography and amplitude amplification, and outlines avenues for extending to classification, other problems, and applications in image-related tasks. Overall, it provides a foundational framework for quantum-assisted blockmodeling with explicit complexity considerations and a concrete numerical demonstration.

Abstract

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.

Figures (8)

• Figure 1: Qubit states in $16\times 16$ matrices: While blue represents $\left|0\right\rangle$, white represents $\left|1\right\rangle$ states for the considered qubit.
• Figure 2: Group of two qubit-states in $4\times 4$ matrices.
• Figure 3: Group of two qubit-states in $8\times 8$ matrices.
• Figure 4: Generic quantum row and column swap operations applied to a quantum state at the $i$th iteration $\left|\psi_i\right\rangle$.
• Figure 5: Implementation of swap $P_{ab}$ with the help of an ancillary register. Note that it can be implemented also without ancilla by a sequence of multi controlled swap operations.

Theorems & Definitions (1)

• Definition 1