DeepAI
Log In Sign Up

Convex optimization using quantum oracles

09/03/2018
by   Joran van Apeldoorn, et al.
0

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 Õ(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 Õ(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.

READ FULL TEXT
09/04/2018

Quantum algorithms and lower bounds for convex optimization

While recent work suggests that quantum computers can speed up the solut...
06/22/2017

Efficient Convex Optimization with Membership Oracles

We consider the problem of minimizing a convex function over a convex se...
07/29/2020

Online Convex Optimization with Classical and Quantum Evaluation Oracles

As a fundamental tool in AI, convex optimization has been a significant ...
12/30/2021

Set membership with two classical and quantum bit probes

We consider the following problem: Given a set S of at most n elements f...
11/04/2021

Quantum search-to-decision reductions and the state synthesis problem

It is a useful fact in classical computer science that many search probl...
12/15/2021

Learning Graph Partitions

Given a partition of a graph into connected components, the membership o...
03/08/2019

Active Learning a Convex Body in Low Dimensions

Consider a set P ⊆R^d of n points, and a convex body C provided via a se...