Closing the non-Clifford resource gap

Determine the optimal number of non-Clifford T gates required to construct one-dimensional approximate unitary k-designs, and close the remaining gap between the O(nk log k) upper bound of the construction and the existing lower bound.

Background

The construction uses O(nk log k) magic T gates. The cited lower bound is Ω(nk / (log n log k + log² k)), leaving a multiplicative gap of O(log n log² k + log³ k). The authors conjecture that the true lower bound should be Ω(nk), suggesting that both improving the upper bound and strengthening the lower bound are relevant to resolving the resource-optimality question.

References

Relative to the lower bound on the magic resources, our construction leaves open a gap of scaling $O(\log n\,\log2 k+\log3k)$. We believe that one $\log(k)$ factor is due to residual inefficiency in our construction (similar to the excess depth overhead), and that the actual lower bound should be $\Omega(nk)$, with the lower bound derived in Ref. being lossy by a factor of $O(\log n\,\log k+\log2k)$. Closing this gap in both directions thus remains to be done.

— (Almost) quadruply optimal unitary designs in 1D  (2608.18650 - Liu et al., 19 Aug 2026) in Section 1, Discussion and outlook