  We prove a quantitative Fourth Moment Theorem for Wigner integrals of any
order with symmetric kernels, generalizing an earlier result from Kemp et al.
(2012). The proof relies on free stochastic analysis and uses a new biproduct
formula for bi-integrals. A consequence of our main result is a
Nualart-Ortiz-Latorre type characterization of convergence in law to the
semicircular distribution for Wigner integrals. As an application, we provide
Berry-Esseen type bounds in the context of the free Breuer-Major theorem for
the free fractional Brownian motion.
