Minimax or average-case optimality of orthogonal 3-designs

Determine whether orthogonal 3-designs are optimal, in either the minimax worst-case-observable sense or an average-observable sense, among admissible orthogonal measurement ensembles.

Background

The paper proves that all orthogonal 3-designs produce identical shadow channels, estimators, variances, and seminorms. It also constructs an orthogonal 2-design that is not a 3-design and that performs better for a particular observable, showing that 3-designs are not universally variance-optimal among 2-designs.

This leaves open whether the exact orthogonal 3-design predictions nevertheless optimize a worst-case observable criterion or an average-case criterion over observables.

References

We leave it as an open problem to decide whether the orthogonal 3-design is optimal in a minimax (worst-case-observable) or average sense.

— Real Classical Shadows with Noise  (2608.18935 - Hingane et al., 19 Aug 2026) in Section 6, Discussion, second item (Optimality of the ensemble)

A circuit family of finite depth realizes at best an approximate one, and whether logarithmic-depth approximate orthogonal designs exist is open. We leave it as an open problem to establish or rule this out, and to settle the resulting trade-off between gate cost and the sample-complexity advantage.

— Real Classical Shadows with Noise  (2608.18935 - Hingane et al., 19 Aug 2026) in Section 6, Discussion, fifth item (Circuit depth)