Practical applications of high-order unitary designs

Determine whether constructing a unitary k-design for k at least 5 can lead to practical applications.

Background

The paper studies strong approximate unitary designs and their ability to reproduce distributional properties of Haar-random unitaries beyond moment matching. It notes that increasing the design order can improve statistical guarantees, but the practical value of sufficiently high-order designs remains unresolved. The question concerns whether the resource requirements associated with constructing designs of order k at least 5 can be justified by concrete applications.

References

Despite the fact that it is currently unclear whether constructing a unitary $k$-design for $k\geq 5$ may lead to any practical applications, the study of improving the efficiency for asymptotically increasing $k$ remains active.

— Random Quantum Circuits Beyond Moment Matching  (2610.02135 - Hung, 1 Oct 2026) in Section 1, Introduction