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In the past, I thought I have seen quantum circuits/algorithm techniques to change the amplitude of state based on the state?

$\lvert \psi \rangle = \sum_x \ C_x \lvert x \rangle$, here $C_x$ is just constant (i.e. as in simple superpositions)

and after some $U$, $\lvert \psi \rangle = \sum_x \ C_x (x) \lvert x \rangle$

The amplitude changes based on the $\lvert x \rangle$ state.

Any pointer or reference would be appreciated.

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    $\begingroup$ Is there a difference between $C_x$ and $C_x(x)$ here? And between $|X\rangle$ and $|x\rangle$? In general, we can decompose a unitary matrix as $U=\sum_{x,y}u_{x,y}|x\rangle\langle y|$ with orthonormal $\{|x\rangle\}$ such that $U|\psi\rangle=\sum_x \left(\sum_y u_{x,y}C_y\right)|x\rangle$; that's the same as a transformation $C_x\to \left(\sum_y u_{x,y}C_y\right)$. $\endgroup$ Jun 14 at 18:25
  • $\begingroup$ @QuantumMechanic, sorry about the confusion about Cx, .. which is just meant to be a constant like the amplitude from the uniform superpositions. $\endgroup$ Jun 14 at 20:15
  • $\begingroup$ Maybe, you are looking for this article: arxiv.org/abs/quant-ph/0407010 $\endgroup$ Jun 14 at 20:42
  • $\begingroup$ I'm not sure exactly what you're looking for, so let me suggest a couple of possibilities: could it be amplitude amplification you're thinking of? Or, there's a step in the HHL algorithm that achieves something similar to this. $\endgroup$
    – DaftWullie
    Jun 15 at 6:59
  • $\begingroup$ I still don't understand what $C_x(x)$ represents. It should be the value of the coefficient of the $|x\rangle$ state after the evolution, but should it depend on the full $|\psi\rangle$ or only the index $x$? $\endgroup$
    – glS
    Jun 15 at 9:41
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The matrix contains information about the vectors(states). To see this, the matrix form can be written as $U_{ij}=\langle i\mid U\mid j\rangle$ or the total form of $U$ mentioned by @Quantum Mechanic, i.e., $U=\sum_{ij}U_{ij}\mid i\rangle\langle j\mid$. To show it more vividly, the stabilizer code will be a good example.

Another easier example is that: when $U=\left(\begin{array}{ll} 1 & 0 \\ 0 & -1 \end{array}\right) \\\\$, $U\mid 0\rangle$ will become $\mid 0\rangle$ while $U\mid 1\rangle$ will become $-\mid 1\rangle$.

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