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What would be the simplest way to construct a block encoding circuit $U_A$ for a $2^n\times 2^n$ matrix $A$ proportional to $\operatorname{diag}(0,1,2,\ldots,2^n-1)$?

A couple of options I can imagine:

  • Use quantum adder $|x, y\rangle\mapsto|x, y+x\rangle$. Requires a number of ancillas equal to $n$.
  • Somehow first prepare a diagonal matrix $\operatorname{e}^{iA}$ (probably won't require ancillas) and then calculate its logarithm using QSP.

Due to a very particular form of $A$ I'm hoping that some efficient block encoding may exist, with a number of ancillas logarithmic in $n$ or even constant.

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    $\begingroup$ I'm not sure about block encoding unitaries, but you should be able to recognize $\mathrm{diag}(1,2)^{\otimes{n}}=\mathrm{diag}(1,2,2^2,\cdots,2^n)$ and do something useful $\endgroup$ Commented Mar 8, 2023 at 20:20

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Assume that, $U_n = \operatorname{diag}(0,1,2,\ldots,2^n-1)$

Then,

$U_1 = \frac{1}{2}(I-Z)$

$U_2 = \frac{1}{2}(3II-2ZI-IZ)$

$U_3 = \frac{1}{2}(7III-4ZII-2IZI - IIZ)$

And in general,

$U_n = \frac{1}{2}((2^n - 1)I^{\otimes n}-2^{n-1}ZI^{\otimes (n-1)}-2^{n-2}IZI^{\otimes (n-2)} - ... -I^{\otimes (n-1)}Z$

That is, we can write $U_n$ as a linear combination of $n+1$ unitaries. Which means we can use LCU technique to build the circuit using $\lceil \log(n+1) \rceil$ ancillas.

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  • $\begingroup$ I'm confused: none of these are unitary $\endgroup$ Commented Mar 9, 2023 at 13:51
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    $\begingroup$ What do you mean? Each Pauli operator is unitary. The final matrix is not unitary, and not supposed to be. $\endgroup$
    – mavzolej
    Commented Mar 9, 2023 at 19:13
  • $\begingroup$ @mavzolej I did not realize the final matrix need not be unitary, that helps! $\endgroup$ Commented Mar 9, 2023 at 21:01

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