Modeling an Adiabatic Quantum Computer via an Exact Map to a Gas of Particles

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 $|\dot{\lambda }{|}^{\gamma }$, where $|\dot{\lambda }|$ is the adiabaticity parameter. The scaling exponent for the escape probability is $\gamma =\frac{1}{2}$ for all levels, except the edge (bottom and top) ones, where $\gamma \lesssim \frac{1}{3}$. In principle, our method can solve arbitrarily large adiabatic quantum Hamiltonians.