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I am reading Harrow, Hassidim, and Lloyd's paper Quantum algorithms for linear systems of equations. On the third page of that paper, they write

Next we apply the conditional Hamiltonian evolution $\sum_{\tau=0}^{T-1} \left|\tau\right>\left<\tau\right|^{C}\otimes e^{iA\tau t_{o}/T}$ on $\left|\Psi_{0}\right>^{C}\otimes\left|b\right>\dots$

For the life of me, I cannot figure out the meaning of the $C$. What is it doing there? How would $\sum_{\tau=0}^{T-1} \left|\tau\right>\left<\tau\right|^{C}\otimes e^{iA\tau t_{o}/T}$ act on (say) $\left|0\right>\otimes \left|0\right>$ or $\left|1\right>\otimes \left|0\right>$?

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I remember struggling with this very same question! Ultimately I concluded that the $C$ is just a notational device (it doesn't represent any mathematical operation), just to indicate that, for a specific $\tau$, the Hamiltonian evolution operator $e^{iA\tau t_0/T}$ is performed on $\left|b\right>$, but only when conditioned on $\left|b\right>$ being tensored with $\left|\tau\right>$.

If you ignore the $C$'s, I think a calculation will show that you just apply the operator $\sum_{\tau=0}^{T-1} \left|\tau\right>\left<\tau\right|\otimes e^{iA\tau t_{o}/T}$ "as-is". I hope this helps.

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    $\begingroup$ Ah thank you. By the way, I found a free source for the published version of the paper, in which the $C$ disappears. It lives here: dspace.mit.edu/handle/1721.1/51753 $\endgroup$
    – user14717
    Apr 12 '12 at 10:03
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There's a Version 2 with appendix and there's write...

  1. Apply the Fourier transform to the register C. Denote the resulting basis states with |ki, for k = 0, . . . T − 1. Define λ˜ k := 2πk/t0.

if This does not answer properly, maybe take a look to this new version of this paper.

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