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.