Generating families of steering Hamiltonians and optimizing to achieve fast adiabatic evolution

Precise steering of quantum systems has found many modern applications, e.g. in quantum computation. One fruitful approach has been in the area of quantum transitionless driving (QTD), or more general shortcuts to adiabaticity (STA), where one drives quantum systems to a desired state that is an instantaneous eigenstate of a given time-dependent Hamiltonian H₀(t). There is some freedom in the driving Hamiltonian H₁(t), which partly depends on whether one wants strict QTD or more general approaches to STA. Unfortunately, in all but the simplest cases, direct implementation of H₁(t) is impractical (due, e.g. to its being nonlocal), and one must resort to approximation schemes. In our work, we expand upon the ideas in STA by studying the geometrical structure behind the relationship between H₀(t) and H₁(t), and we make use of this structure to generate families of the driving term H₁(t) that can be used to eliminate undesirable properties that render implementation impractical. Having families of Hamiltonians H₁(t) available also allows the possibility of optimization for a given task, e.g., driving in the presence of noise. To illustrate our ideas, we present results on driving a particle on a ring geometry with a scattering potential and threaded by a time-dependent flux.