  Constructing an optimal mixer for Quantum Approximate Optimization Algorithm
(QAOA) Hamiltonian is crucial for enhancing the performance of QAOA in solving
combinatorial optimization problems. We present a systematic methodology for
constructing the QAOA tailored mixer Hamiltonian, ensuring alignment with the
inherent symmetries of classical optimization problem objectives. The key to
our approach is to identify an operator that commutes with the action of the
group of symmetries on the QAOA underlying Hilbert space and meets the
essential technical criteria for effective mixer Hamiltonian functionality. We
offer a construction method specifically tailored to the symmetric group $S_d$,
prevalent in a variety of combinatorial optimization problems. By rigorously
validating the required properties, providing a concrete formula and
corresponding quantum circuit for implementation, we establish the viability of
the proposed mixer Hamiltonian. Furthermore, we demonstrate that the classical
mixer $B$ commutes only with a subgroup of $S_d$ of significantly smaller order
than the group itself, enhancing the efficiency of the proposed approach. To
evaluate the effectiveness of our methodology, we compare two QAOA variants
utilizing different mixer Hamiltonians: conventional $B=\sum X_i$ and the newly
proposed $H_M$ in edge coloring and graph partitioning problems across various
graphs. We observe statistically significant differences in mean values, with
the new variant consistently demonstrating superior performance across multiple
independent simulations. Additionally, we analyze the phenomenon of poor
performance in alternative warm-start QAOA variants, providing a conceptual
explanation supported by recent literature findings.
