🤖 AI Summary
This work proposes a novel QUBO encoding method for permutation problems based on comparison-swap networks. Requiring only $O(n \log^2 n)$ binary variables, the approach substantially reduces both the number of variables and the density of the interaction graph while ensuring a bijective mapping to the space of permutations and enabling unbiased sampling. It is the first to integrate oblivious comparison-swap networks with QUBO modeling, thereby supporting constraints such as fixed points and parity, as well as algebraic operations including permutation multiplication, inversion, and order detection. Compared to conventional permutation matrix formulations, the resulting model is significantly more compact and sparse, facilitating efficient generation of solutions with prescribed properties—such as a given order or commutativity with a target permutation—thus offering promising applications in cryptography and combinatorial design.
📝 Abstract
Quadratic Unconstrained Binary Optimization (QUBO) is a standard NP-hard optimization problem. Recently, it has gained renewed interest through quantum computing, as QUBOs directly reduce to the Ising model, on which quantum annealing devices are based. We introduce a QUBO formulation for permutations using compare-exchange networks, with only $O(n \log^2 n)$ binary variables. This is a substantial improvement over the standard permutation matrix encoding, which requires $n^2$ variables and has a much denser interaction graph. A central feature of our approach is uniformity: each permutation corresponds to a unique variable assignment, enabling unbiased sampling. Our construction also allows additional constraints, including fixed points and parity. Moreover, it provides a representation of permutations that supports the operations multiplication and inversion, and also makes it possible to check the order of a permutation. This can be used to uniformly generate permutations of a given order or, for example, permutations that commute with a specified permutation. To our knowledge, this is the first result linking oblivious compare-exchange networks with QUBO encodings. While similar functionality can be achieved using permutation matrices, our method yields QUBOs that are both smaller and sparser. We expect this method to be practically useful in areas where unbiased sampling of constrained permutations is important, including cryptography and combinatorial design.