  For every $d\geq 1$, we consider the $d$-dimensional Hermitian fractional
Brownian motion (HfBm), that is the process with values in the space of
$(d\times d)$-Hermitian matrices and with upper-diagonal entries given by
complex fractional Brownian motions of Hurst index $H\in (0,1)$. We follow the
approach of [A. Deya and R. Schott: On the rough paths approach to
non-commutative stochastic calculus, JFA (2013)] to define a natural integral
with respect to the HfBm when $H>\frac13$, and identify this interpretation
with the rough integral with respect to the $d^2$ entries of the matrix. Using
this correspondence, we establish a convenient It{\^o}--Stratonovich formula
for the Hermitian Brownian motion. Finally, we show that at least when $H\geq
\frac12$, and as the size $d$ of the matrix tends to infinity, the integral
with respect to the HfBm converges (in the tracial sense) to the integral with
respect to the so-called non-commutative fractional Brownian motion.
