In the same paper that was linked in the question, the authors mentioned the bounds of the TPEs, in Theorem 1.6: Let $v_C$ be a classical $(N, D, 1- l_C)$ TPE, and for $0<p<1$, define $v_Q = pv_C+ (1-p)\delta_F$. Suppose that $e_A$= $1-2(2k)^4k/\sqrt{N}>0$. Then $v_Q$ is a quantum $(N,D+1, 1-e_Q, k)$ TPE where $e_Q$ is greater than or equal to $e_A/12*\min (pe_C, 1-p)>0$. This bound is optimized when $p= 1/(1+l_C)$ in which case we have $e_Q$ is greater than or equal to $e_A e_C/24$. This means that any constant degree and gap $2k$ classical TPE gives a $k$ quantum TPE, with constant gap. If the classical TPE is efficient then the quantum one is too.
From these results, I believe that there is a similar geometric interpretation, except that it is limited when $2k$>N, for the quantum expanders.

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