Notation for a Conditional Hamiltonian Evolution Operator 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 $\sum_{\tau=0}^{T-1} \left|\tau\right&gt;\left&lt;\tau\right|^{C}\otimes e^{iA\tau t_{o}/T}$ on $\left|\Psi_{0}\right&gt;^{C}\otimes\left|b\right&gt;\dots$


For the life of me, I cannot figure out the meaning of the $C$. What is it doing there? How would $\sum_{\tau=0}^{T-1} \left|\tau\right&gt;\left&lt;\tau\right|^{C}\otimes e^{iA\tau t_{o}/T}$ act on (say) $\left|0\right&gt;\otimes \left|0\right&gt;$ or $\left|1\right&gt;\otimes \left|0\right&gt;$?
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 $\left|b\right&gt;$, but only when conditioned on $\left|b\right&gt;$ being tensored with $\left|\tau\right&gt;$. 

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