Sie können Bookmarks mittels Listen verwalten, loggen Sie sich dafür bitte in Ihr SLUB Benutzerkonto ein.
Medientyp:
E-Artikel
Titel:
Convex optimization using quantum oracles
Beteiligte:
van Apeldoorn, Joran;
Gilyén, András;
Gribling, Sander;
de Wolf, Ronald
Erschienen:
Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften, 2020
Erschienen in:
Quantum, 4 (2020), Seite 220
Sprache:
Englisch
DOI:
10.22331/q-2020-01-13-220
ISSN:
2521-327X
Entstehung:
Anmerkungen:
Beschreibung:
We study to what extent quantum algorithms can speed up solving convex optimization problems. Following the classical literature we assume access to a convex set via various oracles, and we examine the efficiency of reductions between the different oracles. In particular, we show how a separation oracle can be implemented using O~(1) quantum queries to a membership oracle, which is an exponential quantum speed-up over the Ω(n) membership queries that are needed classically. We show that a quantum computer can very efficiently compute an approximate subgradient of a convex Lipschitz function. Combining this with a simplification of recent classical work of Lee, Sidford, and Vempala gives our efficient separation oracle. This in turn implies, via a known algorithm, that O~(n) quantum queries to a membership oracle suffice to implement an optimization oracle (the best known classical upper bound on the number of membership queries is quadratic). We also prove several lower bounds: Ω(n) quantum separation (or membership) queries are needed for optimization if the algorithm knows an interior point of the convex set, and Ω(n) quantum separation queries are needed if it does not.