Derandomized Non-Abelian Homomorphism Testing in Low Soundness Regime

We give a randomness-efficient homomorphism test in the low soundness regime for functions, $f: G\to \mathbb{U}_t$, from an arbitrary finite group $G$ to $t\times t$ unitary matrices. We show that if such a function passes a derandomized Blum--Luby--Rubinfeld (BLR) test (using small-bias sets), then (i) it correlates with a function arising from a genuine homomorphism, and (ii) it has a non-trivial Fourier mass on a low-dimensional irreducible representation. In the full randomness regime, such a test for matrix-valued functions on finite groups implicitly appears in the works of Gowers and Hatami [Sbornik: Mathematics '17], and Moore and Russell [SIAM Journal on Discrete Mathematics '15]. Thus, our work can be seen as a near-optimal derandomization of their results. Our key technical contribution is a "degree-2 expander mixing lemma'' that shows that Gowers' $\mathrm{U}^2$ norm can be efficiently estimated by restricting it to a small-bias subset. Another corollary is a "derandomized'' version of a useful lemma due to Babai, Nikolov, and Pyber [SODA'08] and Gowers [Comb. Probab. Comput.'08].