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.

References

On the mathematical side, an obvious open question concerns the optimality of our unitary design construction. While we have proven in Proposition~\ref{prop: lower bound design} that our $n$-dependence is optimal, and we inherit the optimal $k$-dependence of Ref. up to poly-logarithmic factors, the relation between these two dependencies is not known. More precisely, we cannot yet rule out the possibility that $\varepsilon$-approximate unitary $k$-designs over $n$ qubits can be created in depth $\mathcal{O}(k) + \mathcal{O}(\log(n / \varepsilon))$, whereas our construction requires a depth of $\tilde{\mathcal{O}(k) \times \mathcal{O}(\log(n / \varepsilon))$. Achieving a matching upper and lower bound on the circuit depth with respect to all three parameters $n, k, \varepsilon$ remains an outstanding challenge.

— Random unitaries in extremely low depth  (2407.07754 - Schuster et al., 2024) in Discussion

The construction leaves open whether all of these additional ingredients are necessary. In particular, our proof does not determine whether the perfect-matching ensemble alone forms a strong approximate $2$-design in logarithmic depth.

— Strong unitary designs in optimal depth and space  (2608.13491 - Parella-Dilmé et al., 13 Aug 2026) in Section Discussion

Despite these advances, the fundamental question of constructing depth-optimal unitary designs in 1D remains open.

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

Decoupling the design order $k$ from the approximation error $\varepsilon$ remains an open question in the high-design-order regime.

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