Optimal average-case fractional query complexity

Determine the optimal average-case query complexity for implementing U^t as a function of the dimension d and approximation error epsilon for Haar-random unitaries, without imposing a spectral-gap promise.

Background

The paper’s main theorem concerns worst-case oracle families satisfying a spectral gap from the branch cut. The authors contrast this with Haar-random unitaries, whose eigenphases typically approach the branch cut within a distance of order 1/d, and with an existing average-case algorithm that requires no spectral-gap promise.

The unresolved issue is to characterize the optimal query complexity in this average-case setting as a function of dimension and target error. The two-eigenvalue hard instance used for the worst-case lower bound does not determine the behavior for typical Haar-random oracles.