We map adiabatic quantum evolution on the classical Hamiltonian dynamics of a 1D gas (Pechukas gas) and simulate the latter numerically. This approach turns out to be both insightful and numerically efficient, as seen from our example of a CNOT gate simulation. For a general class of Hamiltonians we show that the escape probability from the initial state scales no faster than |lambda|gamma, where |lambda| is the adiabaticity parameter. The scaling exponent for the escape probability is gamma=1/2 for all levels, except the edge (bottom and top) ones, where gamma approximately < 1/3. In principle, our method can solve arbitrarily large adiabatic quantum Hamiltonians.