Scalable Numerical Methods for Adiabatic Quantum Preparation
Zusammenfassung der Projektergebnisse
The physics results can be summarized briefly as follows. From the numerical efficiency perspective, we learned that it is advisable to store a spin Hamiltonian with a flexible number of terms. This is simpler for modern programming languages that provide for flexible data storage containers. Already in the beginning of the project, it has become quite clear from the analogy between adiabatic quantum computation and quenches across a quantum-critical point that the influence of a thermal reservoir on the adiabatic computational power would be quite harmful, unless the temperature would be small im comparison also to the minimum energy gap encountered throughout the evolution. No matter how cold the reservoir is, due to the closing spectral gap one will always have excitations even when the evolution is made adiabatic. Since for large problems with a nearly closing energy gap this would require even lower temperatures, one should rather aim at finding adiabatic paths that optimize the energy gap above the ground state. Indeed, we could establish that adiabatic algorithms may profit significantly from the modification of the path connecting the simple initial Hamiltonian with the final problem Hamiltonian. This can be achieved by either choosing a different initial Hamiltonian or by replacing the straight-line interpolation by a more complex path, which aims at gradually reducing the number of solutions. Beyond these central results, there have been a number of important additional insights resulting from the project. One of these was an adiabatic algorithm to transfer between different superposition states of an N-level system. Additionally, the project triggered several studies on quantum-critical systems subject to a non-equilibrium environment. The non-equilibrium physics was either achieved by periodic driving or by connecting the system to two thermal reservoirs held at different temperatures. In the first case we learned that driven critical models may give rise to an even richer phase diagram, whereas in the second case we found that the heat current may serve as a sensitive indicator of quantum criticality, even in far-from-equilibrium configurations. The mathematical working group designed an algorithm to compute the desired ground state and the first exited states. As the problem size scales with the number of particles, care was taken that the algorithm is able to cope with an extremely large problem dimension (larger than N = 2100 is not uncommon). To overcome the curse of dimensionality we used the data sparse tensor format instead of high dimensional vectors. Thus, all matrix-vector operations can be only carried out with an approximation error. In the presence of inaccuracies most of the classical methods loose the orthogonality of the computed basis very quickly, which possibly affects the result of the spectral approximations. Addressing this issue, we designed an inexact Arnoldi method that keeps the distance to orthogonality on the level of the accuracy of the vector operations. This was proven by several bounds on the distance to orthogonality. To analyze the algorithm further, we performed a backward error analysis and derived upper and lower bounds on the backward error. The bounds are shown to be on the same level of accuracy as the vector operations themselves. To complete the analysis of the inexact Arnoldi method we provided a convergence analysis. With regard to the small energy gap between the ground state and the first exited state, care was taken to ensure that the spectral error bounds are also applicable when the gap between the desired and the remaining eigenvalues is very small. Finally, we implemented the inexact Arnoldi method using the TT-format. This allows us to trace eigenvalues through critical points, i.e., where the energy gap is small, even though the size of the eigenvalue problem is very large.
Projektbezogene Publikationen (Auswahl)
- Criticality in transport through the quantum Ising chain, Physical Review Letters 109, 240402 (2012)
M. Vogl, G. Schaller, and T. Brandes
(Siehe online unter https://doi.org/10.1103/PhysRevLett.109.240402) - Nonequilibrium Quantum Phase Transitions in the Dicke Model, Physical Review Letters 108, 043003 (2012)
V. M. Bastidas, C. Emary, B. Regler, and T. Brandes
(Siehe online unter https://doi.org/10.1103/PhysRevLett.108.043003) - Nonequilibrium Quantum Phase Transitions in the Ising Model, Physical Review A 86, 063627 (2012)
V. M. Bastidas, C. Emary, G. Schaller, and T. Brandes
(Siehe online unter https://doi.org/10.1103/PhysRevA.86.063627) - A priori convergence analysis for inexact Hermitian Krylov methods, Oberwolfach reports, 3248-3250, (2013)
U. Kandler and C. Schröder
- Backward error analysis of an inexact Arnoldi method, PAMM 13, Issue 1, 417-418, (2013)
U. Kandler and C. Schröder
- Implementation of stimulated Raman adiabatic passage in degenerate systems by dimensionality reduction, Physical Review A 88, 013404 (2013)
G. Bevilacqua, G. Schaller, T. Brandes, and F. Renzoni
(Siehe online unter https://doi.org/10.1103/PhysRevA.88.013404) - Probing nonlinear adiabatic paths with a universal integrator, Physical Review A 89, 032308 (2014)
Michael Hofmann and Gernot Schaller
(Siehe online unter https://doi.org/10.1103/PhysRevA.89.032308) - Transport as a sensitive indicator of quantum criticality, Journal of Physics: Condensed Matter 26, 265001 (2014)
Gernot Schaller, Malte Vogl, Tobias Brandes
(Siehe online unter https://doi.org/10.1088/0953-8984/26/26/265001) - Backward error analysis of the shift-and-invert Arnoldi algorithm, Numerische Mathematik, (2015)
C. Schröder and L. Taslaman
(Siehe online unter https://doi.org/10.1007/s00211-015-0759-9)
