Scalable Numerical Methods for Adiabatic Quantum Preparation

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Many physically relevant models include time-dependent Hamilton operators. When the time-dependence is slow in comparison to the energy gaps of the spectrum, the adiabatic theorem implies that a quantum system prepared in the ground state of an initial Hamiltonian will completely follow the instantaneous ground state. This can be exploited to prepare interesting ground states from simple ones by slowly deforming control parameters in the Hamiltonian. The final ground states can - for example - be entangled, be useful for one-way quantum computation or even directly encode the solution to a difficult problem. The scheme has nice robustness features: When the reservoir temperature is significantly smaller than the time-dependent energy gap above the ground state, the decoherent interactions may drag the system towards its ground state and may therefore be even helpful in the goal to solve the problem. However, for critical parameter values the ground state itself may change in a quite abrupt manner, which is typically associated with a nearly vanishing energy gap. In the infinite size limit this corresponds to a quantum phase transition, whereas for finite-size systems one usually has a finite energy gap. This scaling behavior of the energy gap poses a severe problem both for the adiabatic implementation (diverging adiabatic runtime) and its robustness against thermal excitations. There exist only a few exactly solvable models that exhibit a quantum phase transition in the infinite size limit, such that in general one has to use numerical simulations. The present proposal suggests the development of intelligent numerical methods for the time-dependent tracking of eigenvalue problems and to study the performance of adiabatic algorithms in the case of open and closed quantum systems. More specifically, these new algorithms must have an adequate accuracy even in critical parameter regions. This shall be achieved by adaptively matching not only the discretization level, the step size of the eigenvalue tracer but also the number of calculated eigenvalues (and therefore also the size of the traced subspace) via error and condition estimators to the behavior of the system.