@article {3260, title = {On the stability of solutions to Schr{\"o}dinger{\textquoteright}s equation short of the adiabatic limit}, year = {2023}, month = {3/23/2023}, abstract = {

We prove an adiabatic theorem that applies at timescales short of the adiabatic limit. Our proof analyzes the stability of solutions to Schrodinger\&$\#$39;s equation under perturbation. We directly characterize cross-subspace effects of perturbation, which are typically significantly less than suggested by the perturbation\&$\#$39;s operator norm. This stability has numerous consequences: we can (1) find timescales where the solution of Schrodinger\&$\#$39;s equation converges to the ground state of a block, (2) lower bound the convergence to the global ground state by demonstrating convergence to some other known quantum state, (3) guarantee faster convergence than the standard adiabatic theorem when the ground state of the perturbed Hamiltonian (H) is close to that of the unperturbed H, and (4) bound tunneling effects in terms of the global spectral gap when H is {\textquoteleft}{\textquoteleft}stoquastic\&$\#$39;\&$\#$39; (a Z-matrix). Our results apply to quantum annealing protocols with faster convergence than usually guaranteed by a standard adiabatic theorem. Our upper and lower bounds demonstrate that at timescales short of the adiabatic limit, subspace dynamics can dominate over global dynamics. Thus, we see that convergence to particular target states can be understood as the result of otherwise local dynamics.

}, url = {https://arxiv.org/abs/2303.13478}, author = {Jacob Bringewatt and Michael Jarret and T. C. Mooney} } @article {1992, title = {Substochastic Monte Carlo Algorithms}, year = {2017}, month = {2017/04/28}, abstract = {

In this paper we introduce and formalize Substochastic Monte Carlo (SSMC) algorithms. These algorithms, originally intended to be a better classical foil to quantum annealing than simulated annealing, prove to be worthy optimization algorithms in their own right. In SSMC, a population of walkers is initialized according to a known distribution on an arbitrary search space and varied into the solution of some optimization problem of interest. The first argument of this paper shows how an existing classical algorithm, \"Go-With-The-Winners\" (GWW), is a limiting case of SSMC when restricted to binary search and particular driving dynamics.\ 
Although limiting to GWW, SSMC is more general. We show that (1) GWW can be efficiently simulated within the SSMC framework, (2) SSMC can be exponentially faster than GWW, (3) by naturally incorporating structural information, SSMC can exponentially outperform the quantum algorithm that first inspired it, and (4) SSMC exhibits desirable search features in general spaces. Our approach combines ideas from genetic algorithms (GWW), theoretical probability (Fleming-Viot processes), and quantum computing. Not only do we demonstrate that SSMC is often more efficient than competing algorithms, but we also hope that our results connecting these disciplines will impact each independently. An implemented version of SSMC has previously enjoyed some success as a competitive optimization algorithm for Max-k-SAT.

}, url = {https://arxiv.org/abs/1704.09014}, author = {Michael Jarret and Brad Lackey} }