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In particular I'm hoping to understand what is written in this paper better: https://arxiv.org/abs/quant-ph/9802037 (On the Power of One Bit of Quantum Information, Knill and Laflamme 1998)

In the section titled "Deterministic quantum computation with one qubit (DQC1)" and around equations (6) and (7), they say that within DQC1 we can evaluate $\alpha _b$ with two computations. What are these computations exactly? I see the traces provided but what circuits/ sequence of operations does this correspond to?

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If you perform the circuit shown below, the density matrix of the first qubit before measurement reads:

$$ \rho=\frac{1}{2}(\sigma_0+\frac{Re(Tr(U_n))}{2^n}\sigma_1+\frac{Im(Tr(U_n))}{2^n}\sigma_2)$$

enter image description here

As you can see this circuit belongs to DQC1 as only the first qubit is pure.

Hence, if you decide to choose $U_n=\sigma_b U$, by measuring the Pauli $\sigma_1$ and $\sigma_2$ of the first qubit in a repeated manner, you can access to the real and imaginary part of $\alpha_b \equiv Tr(\sigma_b U)/2^n$.

You only need a polynomial number of repetition as a function of $n$ to get the good result with a polynomial accuracy (ref)

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    $\begingroup$ Thanks for your answer. I wonder if you could point to me how this was implied by the paper? In other words, apart from independently coming up with the circuit you have shown, how could I have read this from the paper? $\endgroup$
    – shashvat
    May 15, 2022 at 17:00
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    $\begingroup$ Hi. I agree with you that it is not so clear from the paper you refer to. The reason why I thought about this comes from the fact that normalized trace estimation is a routine often used in the context of DQC1. $\endgroup$ May 16, 2022 at 9:16

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