Adiabatic condition and quantum geometric potential

Necessary condition for the quantum adiabatic approximation

A gapped quantum system that is adiabatically perturbed remains approximately in its eigenstate after the evolution. We prove that, for constant gap, general quantum processes that approximately prepare the final eigenstate require a minimum time proportional to the ratio of the length of the eigenstate path to the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We also giv...

Adiabatic quantum counting by geometric phase estimation

We design an adiabatic quantum algorithm for the counting problem, i.e., approximating the proportion, α, of the marked items in a given database. As the quantum system undergoes a designed cyclic adiabatic evolution, it acquires a Berry phase 2πα. By estimating the Berry phase, we can approximate α, and solve the problem. For an error bound ǫ, the algorithm can solve the problem with cost of o...

Gauge fields, geometric phases, and quantum adiabatic pumps.

Quantum adiabatic pumping of charge and spin between two reservoirs (leads) has recently been demonstrated in nanoscale electronic devices. Pumping occurs when system parameters are varied in a cyclic manner and sufficiently slowly that the quantum system always remains in its ground state. We show that quantum pumping has a natural geometric representation in terms of gauge fields (both Abelia...

The Quantum Adiabatic Approximation and the Geometric Phase

A precise definition of an adiabaticity parameter ν of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator U(τ) = ∑ l U (l)(τ) with U (l)(τ) being at least of the order ν. In particular U (0)(τ) corresponds to the adiabatic approximation and yields Berry’s adiabatic phase. It is sh...

Non-Abelian geometric effect in quantum adiabatic transitions.