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Journal of Complexity

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

Boleslaw Z. Kacewicz

Published May 12, 2006• 70 citations
Publisher
DOI
10.1016/j.jco.2006.03.001

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We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. We define for this purpose a sequence of new algorithms, whose error/cost properties improve from step to step. This leads to new upper complexity bounds, which differ from known lower bounds only by an arbitrarily small positive parameter in the exponent, and a logarithmic factor. In both randomized and quantum settings, initial-value problems turn out to be essentially as difficult as scalar integration.
We extend the representation of quantum algorithms to the preparation of the problem setting. An initial measurement serves to set the problem, the final one as usual to read the solution. This extension highlights the possibility of ascribing the selection of a variable fraction R of the information that specifies the random outcome of the initial measurement to the final measurement. The input state of the quantum algorithm to the problem solver (Alice), of maximal ignorance of the problem setting that to her should be hidden, would consequently be projected on a state of lower entropy where she knows in advance an R-th part of the setting (i. e. of the information that specifies it). Given the appropriate value of R, the quantum algorithm turns out to be a sum over classical histories in each of which Alice knows in advance one of the possible R-th parts of the problem setting and performs the Q(R) computations (oracle queries) still needed to find the solution. Grover's search problem can be solved with any value of R from 0 to about 1/2, when the solution algorithm becomes the optimal Grover algorithm. In all the optimal quantum algorithms examined, R is close to 1/2 and Q(1/2) yields the order of magnitude of the number of queries. If this held in general, as it seems to be plausible, given any oracle problem we would know the order of magnitude of the number of queries needed for its optimal quantum solution.