Optimality and additive depth scaling for approximate unitary designs

Determine whether ε-approximate unitary k-designs over n qubits with relative error can be generated in circuit depth O(k) + O(log(n/ε)), rather than requiring a multiplicative scaling in k and log(n/ε) as in current constructions, and establish matching upper and lower bounds on the minimal circuit depth as a function of all three parameters n, k, and ε.

Background

The paper proves that gluing small designs yields ε-approximate unitary k-designs in depth scaling like log(n) times a near-linear function of k, achieving optimal dependence on n and inheriting near-optimal dependence on k from recent constructions. However, the authors point out that the joint optimality in n, k, and ε is unresolved: it may be possible to achieve an additive depth scaling across k and log(n/ε), rather than multiplicative. A definitive characterization would require matching upper and lower bounds that capture the full three-parameter dependence.