Convex optimization using quantum oracles

• Main Authors: Joran van Apeldoorn, András Gilyén, Sander Gribling, Ronald de Wolf
• Format: Article
• Language: English
• Published: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften 2020-01-01
• Series: Quantum
• Online Access: https://quantum-journal.org/papers/q-2020-01-13-220/pdf/

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 $\tilde{O}(1)$ quantum queries to a membership oracle, which is an exponential quantum speed-up over the $\Omega(n)$ 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 $\tilde{O}(n)$ 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: $\Omega(\sqrt{n})$ quantum separation (or membership) queries are needed for optimization if the algorithm knows an interior point of the convex set, and $\Omega(n)$ quantum separation queries are needed if it does not.