Near-linear quantum lower bound for membership-based convex optimization

Establish an a9(n) worst-case quantum membership-query lower bound for implementing an optimization oracle over a family of n-dimensional convex bodies, even when the algorithm is given an interior point of the body.

Background

The paper studies the quantum query complexity of convex optimization when the feasible convex body is accessed through a membership oracle and the objective is explicitly given. Prior work established an \Omega(\sqrt n) lower bound and asked whether the dependence on dimension could be strengthened to linear.

The conjecture concerns whether some family of n-dimensional convex bodies is intrinsically difficult enough that every quantum algorithm implementing an optimization oracle must make linearly many membership queries, even with access to an interior point. The paper's main optimization lower bound proves this conjecture up to logarithmic factors for an explicit family of centered ellipsoids.