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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?

Mark Spinelli
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Hmecher
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  • What's wrong with having $\widetilde{A_k},\widetilde{U_k}$ up to a desired level of approximation? They say they use $\alpha^{O(1)}$ accuracy. – Mark Spinelli Oct 16 '21 at 14:00
  • I understand the approximation, but I don't understand what the state $|\tilde{A_k}\rangle$, for example, is supposed to mean. As in, what would this state be in terms of the computational basis? – Hmecher Oct 16 '21 at 21:52
  • I might have a major misunderstanding but I read these as being row-wise or column-wise unrollings/flattenings of the matrix elements, as in claim 5 of the paper. – Mark Spinelli Oct 17 '21 at 14:01
  • I see. So given a matrix $\begin{pmatrix}A_{00} & A_{01} \ A_{10} & A_{11} \end{pmatrix}$ the corresponding state would be $A_{00}|00\rangle + A_{01}|01\rangle + A_{10}|10\rangle + A_{11}|11\rangle$? What about normalization? This point is very poorly written about in the paper... – Hmecher Oct 17 '21 at 22:04

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I think some of the qubits that are initially part of the $0$ register in $\vert k,0\rangle$ get mapped to (approximations of) the coefficients $A_{min_k,min_k},A_{min_k,max_k},A_{max_k,min_k},A_{max_k,max_k}$, and some others will be mapped to the same for $U_k$.

I think this these coefficients are what the oracles are supposed to return.

I agree this feels like it's written at a very high level of abstraction, but if I understand it correctly, it's not dissimilar to the "code as data" duality in (classical) computer science.

Mark Spinelli
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  • This is still not clear to me. So in terms of the computational basis in the $2\times2$ subspace, what is this state: $|A_{\mathrm{min}_k,\mathrm{max}_k}\rangle$? Thanks for the "code as data" link, I will check it out. – Hmecher Oct 19 '21 at 16:58