2006/12/13 by Marek Szczęsny, Marek Szczesny, Szczesny, Marek
Computer Science · Physics and Astronomy · #Chaos-based Image/Signal Encryption #FOS: Physical sciences #Numerical Methods and Algorithms #Polynomial and algebraic computation #Quantum Physics (quant-ph) #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0612085
openalex publication_date 2006/12/13 · arxiv created 2006/12/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study possible advantages of randomized and quantum computing over deterministic computing for scalar initial-value problems for ordinary differential equations of order k. For systems of equations of the first order this question has been settled modulo some details in \citeKacewicz05. A speed-up over deterministic computing shown in \citeKacewicz05 is related to the increased regularity of the solution with respect to that of the right-hand side function. For a scalar equation of order k (which can be transformed into a special system of the first order), the regularity of the solution is increased by k orders of magnitude. This leads to improved complexity bounds depending on k for linear information in the deterministic setting, see \citeSzczesny05. This may suggest that in the randomized and quantum settings a speed-up can also be achieved depending on k. We show in this paper that a speed-up dependent on k is not possible in the randomized and quantum settings. We establish lower complexity bounds, showing that the randomized and quantum complexities remain at the some level as for systems of the first order, no matter how large k is. Thus, the algorithms from \citeKacewicz05 remain (almost) optimal, even if we restrict ourselves to a subclass of systems arising from scalar equations of order k.