4,531 research outputs found
Improved Bounds on the Randomized and Quantum Complexity of Initial-Value Problems
We deal with the problem, initiated in [8], of finding randomized and quantum
complexity of initial-value problems. We showed in [8] 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
H\"older class of right-hand side functions with r continuous bounded partial
derivatives, with r-th derivative being a H\"older function with exponent \rho,
the \epsilon-complexity is shown to be O((1/\epsilon)^{1/(r+\rho+1/3)}) in the
randomized setting, and O((1/\epsilon)^{1/(r+\rho+1/2)}) on a quantum computer
(up to logarithmic factors). This is an improvement for the general problem
over the results from [8]. 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 of order
(1/\epsilon)^{1/(r+\rho+1/2)} in the randomized setting, and
(1/\epsilon)^{1/(r+\rho+1)} in the quantum setting (again up to logarithmic
factors).Comment: 17 pages, extended version (new section added), to appear in the
Journal of Complexit
Almost Optimal Solution of Initial-Value Problems by Randomized and Quantum Algorithms
We establish essentially optimal bounds on the complexity of initial-value
problems in the randomized and quantum settings. For this purpose we define a
sequence of new algorithms whose error/cost properties improve from step to
step. These algorithms yield new upper complexity bounds, which differ from
known lower bounds by only an arbitrarily small positive parameter in the
exponent, and a logarithmic factor. In both the randomized and quantum
settings, initial-value problems turn out to be essentially as difficult as
scalar integration.Comment: 16 pages, minor presentation change
The Quantum Query Complexity of Algebraic Properties
We present quantum query complexity bounds for testing algebraic properties.
For a set S and a binary operation on S, we consider the decision problem
whether is a semigroup or has an identity element. If S is a monoid, we
want to decide whether S is a group.
We present quantum algorithms for these problems that improve the best known
classical complexity bounds. In particular, we give the first application of
the new quantum random walk technique by Magniez, Nayak, Roland, and Santha
that improves the previous bounds by Ambainis and Szegedy. We also present
several lower bounds for testing algebraic properties.Comment: 13 pages, 0 figure
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Randomized Low-Memory Singular Value Projection
Affine rank minimization algorithms typically rely on calculating the
gradient of a data error followed by a singular value decomposition at every
iteration. Because these two steps are expensive, heuristic approximations are
often used to reduce computational burden. To this end, we propose a recovery
scheme that merges the two steps with randomized approximations, and as a
result, operates on space proportional to the degrees of freedom in the
problem. We theoretically establish the estimation guarantees of the algorithm
as a function of approximation tolerance. While the theoretical approximation
requirements are overly pessimistic, we demonstrate that in practice the
algorithm performs well on the quantum tomography recovery problem.Comment: 13 pages. This version has a revised theorem and new numerical
experiment
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