Adiabatic dynamics for a class of quantum Hamiltonians

Artem Pimachev

We describe a class of the Hamiltonians, for which the local adiabaticity condition is satisfied automatically. We consider the controllable adiabatic interpolations between the initial and final Hamiltonians. We show that the quantum search by the adiabatic quantum computation can be mapped into this class. Assuming that the interpolation is infinitely differentiable and the target adiabatic ground state is nondegenerate and separated by a gap from the rest of the spectrum, it is shown that one can obtain an exponentially small error in the algorithm run time T betweenthe finale adiabatic eigenstate and the actual state of the system. The optimal for quantum search scaling of T as square root of the system size is also satisfied.

Ning Wu, Tunan Ruan

Leon Bess

Aurel Bulgac, Dimitri Kusnezov

We explore the dynamics of mixed classical-quantum Hamiltonians, a class of systems that frequently appears in the study of physical systems characterized by both slow and fast modes. Traditionally such systems are studied within the adiabatic approximation and as a rule they display level crossings. We show that level crossings can have peculiar effects on the dynamics and, in the regime between the adiabatic and diabatic limits in both classical and quantum approaches, the full dynamics has a series of non-intuitive features.

Philip Grech

Alexandre M. Zagoskin, Sergey E. Savel'ev, Franco Nori

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