Improved bounds on the randomized and quantum complexity of initial-value problems

Abstract

We study the problem, initiated by Kacewicz [Randomized and quantum algorithms yield a speed-up for initial-value problems, J. Complexity 20 (2004) 821-834; see also http://arXiv.org/abs/quant-ph/ 0311148], of finding randomized and quantum complexity of initial-value problems. We showed in Kacewicz (2004) that a speed-up in both settings over the worst-case deterministic complexity is possible. In the present paper we prove, by defining new algorithms, that further improvement in upper bounds on the randomized and quantum complexity can be achieved. In the Hölder class of righthand side functions with r continuous bounded partial derivatives, with rth derivative being a Hölder function with exponent , the ε-complexity is shown to be O (1/ε) 1/(r+ +1/3) in the randomized setting, and O (1/ε) 1/(r+ +1/2) on a quantum computer (up to logarithmic factors). This is an improvement for the general problem over the results from Kacewicz (2004). The gap still remaining between upper and lower bounds on the complexity is further discussed for a special problem. We consider scalar autonomous problems, with the aim of computing the solution at the end point of the interval of integration. For this problem, we fill up the gap by establishing (essentially) matching upper and lower complexity bounds. We show that the complexity in this case is(1/ε) 1/(r+ +1/2) in the randomized setting, and (1/ε) 1/(r+ +1) in the quantum setting (again up to logarithmic factors). Hence, this problem is essentially as hard as the integration problem.

Minimal asymptotic error for one-point approximation of SDEs with time-irregular coefficients

“…The positive constant M is known and neither depend on i nor n. (Such functions are often used in proving lower bounds and their construction is well-known, see [22][23][24].) We have that h(t n j ) = 0 for all j = 1, 2, .…”

Adaptive Itô–Taylor algorithm can optimally approximate the Itô integrals of singular functions

“…By (14) and (17) the total cardinality of the information used by the algorithm is bounded from above by 2(r + 1)n. Hence, A IT n ∈ Ψ 2(r+1)n .…”

“…Construction of such functions is well know, since these functions are often used in proving lower bounds, see [14,15]. Hence e(A n , F r,…”

Almost optimal solution of initial-value problems by randomized and quantum algorithms

“…(For an arbitrary s, intervals in which l s is a polynomial will be established in the next section.) Let us remark that the algorithms A 1 and A 2 defined above are the well known Taylor algorithm and the algorithm analyzed in [6], respectively. For k 3, algorithms A k are new.…”