Deterministic constructions achieving strong convergence without randomness

Construct explicit deterministic sequences of unitary matrices U_1^N, …, U_r^N—e.g., via number-theoretic constructions such as Lubotzky–Phillips–Sarnak expanders—that strongly converge to free Haar unitaries, meaning that for every *-polynomial P, lim_{N→∞} ||P(U_1^N, …, U_r^N)|| = ||P(u_1, …, u_r)||.

Background

Strong convergence has thus far been established primarily for random models. The authors ask whether purely deterministic, number-theoretic constructions could yield strong convergence to free Haar unitaries, drawing analogy with optimal spectral constructions for regular graphs.

A positive resolution would bridge deep number-theoretic methods with free probability and random matrix theory, offering explicit models with strong convergence properties.

References

Could one hope to achieve strong convergence with no randomness at all, using number-theoretic constructions such as those that have been used to obtain regular graphs with optimal spectral properties ? These tantalizing questions remain very much open.

— Strong convergence: a short survey  (2510.12520 - Handel, 14 Oct 2025) in Section 2.2 (Strong asymptotic freeness)

The authors are also grateful to Stanis{aw J.~Szarek who pointed out that related classical derandomization problems have been approached through constructions based on expander graphs and it remains open how to construct a quantum version of it.

— Superadditivity of classical communication over quantum channels via random and deterministic permutations  (2608.25961 - Lovitz et al., 26 Aug 2026) in Acknowledgments and Disclosure of AI usage

Besides, in free probability theory, it remains a major challenge whether an explicit, deterministic strong convergence phenomenon can be constructed, see Section 6.4.

— Superadditivity of classical communication over quantum channels via random and deterministic permutations  (2608.25961 - Lovitz et al., 26 Aug 2026) in Section 1, subsection “Sketch proof of the main result”