Linear-in-dimension T-count synthesis or stronger lower bound

Determine whether arbitrary effectively specified unitaries on a d-dimensional Hilbert space can be synthesized over Clifford+T with a T-count of \widetilde{O}(d), or establish a stronger worst-case lower bound than the known \widetilde{\Omega}(d) bound at constant accuracy.

Background

The paper develops a Clifford+T synthesis method for arbitrary effectively specified d-dimensional unitaries, achieving a T-count of O(d^{5/4}L^{5/8}\log d) in the regime L\le d, where L=n+\log(1/\epsilon). This improves the preceding exponent but leaves a multiplicative gap between the upper bound and the known \widetilde{\Omega}(d) lower bound at constant accuracy.

The authors identify closing this gap in either direction as the central unresolved issue: one possibility is an essentially linear T-count construction, while the other is a stronger lower bound demonstrating that such a construction is impossible.