Given a connected graph G , a set of vertices X ⊂ V ( G ) is a weak k -resolving set of G if for each two vertices y , z ∈ V ( G ) , the sum of the values | d G ( y , x ) − d G ( z , x ) | over all x ∈ X is at least k , where d G ( u , v ) stands for the length of a shortest path between u and v . The cardinality of a smallest weak k -resolving set of G is the weak k -metric dimension of G , and is denoted by wdim k ( G ) . In this paper, wdim k ( K n □ K n ) is determined for every n ≥ 3 and every 2 ≤ k ≤ 2 n . An improvement of a known integer linear programming formulation for this problem is developed and implemented for the graphs K n □ K m . Conjectures regarding these general situations are posed.

A physical limitation in quantum circuit design is the fact that gates in a quantum system can only act on qubits that are physically adjacent in the architecture. To overcome this problem, SWAP gates need to be inserted to make the circuit physically realizable. The nearest neighbour compliance problem (NNCP) asks for an optimal embedding of qubits in a given architecture such that the total number of SWAP gates to be inserted is minimized. In this paper we study the NNCP on general quantum architectures. Building upon an existing linear programming formulation, we show how the model can be reduced by exploiting the symmetries of the graph underlying the formulation. The resulting model is equivalent to a generalized network flow problem and follows from an in-depth analysis of the automorphism group of specific Cayley graphs. As a byproduct of our approach, we show that the NNCP is polynomial time solvable for several classes of symmetric quantum architectures. Numerical tests on various architectures indicate that the reduction in the number of variables and constraints is on average at least 90%. In particular, NNCP instances on the star architecture can be solved for quantum circuits up to 100 qubits and more than 1000 quantum gates within a very short computation time. These results are far beyond the computational capacity when solving the instances without the exploitation of symmetries.