  For a fixed quadratic polynomial $\mathfrak{p}$ in $n$ non-commuting
variables, and $n$ independent $N\times N$ complex Ginibre matrices
$X_1^N,\dots, X_n^N$, we establish the convergence of the empirical spectral
distribution of $P^N =\mathfrak{p}(X_1^N,\dots, X_n^N)$ to the Brown measure of
$\mathfrak{p}$ evaluated at $n$ freely independent circular elements
$c_1,\dots, c_n$ in a non-commutative probability space. The main step of the
proof is to obtain quantitative control on the pseudospectrum of $P^N$. Via the
well-known linearization trick this hinges on anti-concentration properties for
certain matrix-valued random walks, which we find can fail for structural
reasons of a different nature from the arithmetic obstructions that were
illuminated in works on the Littlewood--Offord problem for discrete scalar
random walks.
