Pedro Costa

The solution of large systems of nonlinear differential equations is needed for many applications in science and engineering. In this study, we present three main improvements to existing quantum algorithms based on the Carleman linearisation technique. First, by using a high-precision technique for the solution of the linearised differential equations, we achieve logarithmic dependence of the complexity on the error and near-linear dependence on time. Second, we demonstrate that a rescaling technique can considerably reduce the cost, which would otherwise be exponential in the Carleman order for a system of ODEs, preventing a quantum speedup for PDEs. Third, we provide improved, tighter bounds on the error of Carleman linearisation. We apply our results to a class of discretised reaction-diffusion equations using higher-order finite differences for spatial resolution. We show that providing a stability criterion independent of the discretisation can conflict with the use of the rescaling due to the difference between the max-norm and 2-norm. An efficient solution may still be provided if the number of discretisation points is limited, as is possible when using higher-order discretisations.

Abstract We compile explicit circuits and evaluate the computational cost for heuristic-based quantum algorithms for combinatorial optimization. We consider several variants of quantum-accelerated simulated annealing as well as adiabatic algorithms, quantum-enhanced population transfer, the quantum approximate optimization algorithm, and other approaches. We provide novel methods for executing the bottleneck subroutines for these heuristics, and our methods can easily be applied to other algorithms where numerical performance matters. We estimate how quickly the subroutines could be executed on a modestly sized superconducting-qubit-based quantum computer with surface code error correction. We conclude that quadratic speedups for heuristic-based quantum optimization algorithms are insufficient for early quantum computers to beat present day classical computers.