  In this work we focus on computing the conformal dimensions $D(j_L,j_R)$ of
local fields that transform in an irreducible representation of $SU(2) \times
SU(2)$ labeled with $(j_L,j_R)$ at the $O(4)$ Wilson-Fisher fixed point using
the Monte Carlo method. In the large charge expansion, among the sectors with a
fixed large value of $j = {\rm max}(j_L,j_R)$, the leading sector has
$|j_L-j_R| = 0$ and the sub-leading one has $|j_L-j_R| = 1$. Since Monte Carlo
calculations at large $j$ become challenging in the traditional lattice
formulation of the $O(4)$ model, a qubit regularized $O(4)$ lattice model was
used recently to compute $D(j,j)$. Here we extend those calculations to the
sub-leading sector. Our Monte Carlo results up to $j=20$ fit well to the form
$D(j,j-1)-D(j) \sim \lambda_{1/2}/\sqrt{j} + \lambda_1/j +
\lambda_{3/2}/j^{3/2}$, consistent with recent predictions of the large charge
expansion. Taking into account systematic effects in our fitting procedures we
estimate the two leading coefficients to be $\lambda_{1/2}=2.08(5)$,
$\lambda_1=2.2(3)$.
