Circular Convolution Matrix of $ {H}^{H} {H} $
If $ H $ is a matrix form of Circular Convolution then it is a Circulant Matrix.
Being a Circulant Matrix means it can be diagonalized by the Fourier Matrix $ {F} $:

$$ H = {F}^{H} D F $$

Where the matrix $ D $ id a Diagonal Matrix with the Fourier Coefficients of $ \mathcal{F} \left( h \right) $ on it main diagonal.
Also pay attention that we use the Conjugate Transpose $ {\left( \cdot \right)}^{H} $ operator as $ F $ and $ D $ are built by complex numbers (The space is over the complex numbers, hence the Adjoint Operator is the Complex Conjugate).

Now, it is easy to see that:

$$ {H}^{H} = {F}^{H} {D}^{H} F $$

Which is again an operator diagonlaized with the Fourier Matrix hence it is a Circulant Matrix -> Cicrcular Operation.
Basically it is a multiplication in Fourier Domain (DFT) by the conjugate of the DFT of the vector $ h $. By the DFT properties it means it is the flipped version of $ h $.

By the way:

$$ {H}^{H} H = {F}^{H} {D}^{H} F {F}^{H} D F = {F}^{H} {D}^{H} D F $$

Now, pay attention to $ {D}^{H} D $. It is a diagonal matrix multiplied by its conjugate. Namely it is, in the Fourier Domain, the Squared Magnitude.
In time domain it means it can be done by circular convolution with the correlation function of $ h $ (Or just do circular convolution in one direction and then in the other which is basically convolution with the flipped version).