In the equations in section 3.4.2 of Aharonov and Ta-Shma's paper (pdf, arxiv abstract), they define the operator: $$T_1:|k,0\rangle\rightarrow|b_k,m_k,M_k,\tilde{A_k},\tilde{U_k},k\rangle,$$ where $b_k,m_k,M_k,$ and $k$ are all integers and $\tilde{A_k}$ and $\tilde{U_k}$ are $2\times2$ matrices. The state $|b_k\rangle$ can then be taken to be a computational basis state by writing the bit-string representation of $b_k$, and similarly for $m_k,M_k,$ and $k$.
My question is what is the meaning of $|\tilde{A_k}\rangle$ and $|\tilde{U_k}\rangle$? These are matrices with entries which are not integers, so how are they represented in terms of the computational basis?