This study applies quantum algorithms to determine if a specific structure embeds into the Bruhat order of the Coxeter group H4, a problem equivalent to verifying the existence of a defined Lehmer code. Since the associated Hasse diagram represents a large directed graph with 14400 nodes, the analysis is computationally too demanding for traditional classical methods. Consequently, we want to adapt an existing quantum algorithm that scales logarithmically O(log N) by extending it to handle the asymmetric nature of directed graphs. Thanks to this efficient encoding, the problem can theoretically be solved using fewer than 100 qubits, offering a drastic reduction compared to the thousands required by standard approaches. The implementation will utilize frameworks such as PennyLane or Qiskit to leverage current utility-scale IBM Quantum processors.