Conditional phase shift for Grover's algorithm

In summary, the conversation discusses the understanding of a deduction involving the conditional phase shift and the unitary operator. The author is confused about the representation of the operator and the treatment of certain cases. It is noted that there are some typos and that the discussion may be part of a student project.
  • #1
Francis
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TL;DR Summary
Dear fellas, I am reading a quantum physics tutorial to understand Grover's algorithm and I am stuck with a (very simple) deduction.
I have the pdf attached page 23 at the top.
I am trying to understand the following deduction:
"The conditional phase shift can be represented by the unitary operator 2|0> <0| - I:"

1609892386212.png
for eq. 4a) I was expecting to be:

[2 |0><0| - I] |0> =
2 |0> <0|0> - I|0> = 2|0> - |0> =
|0>

as for eq. 4b I can't understand it at all. Why does the author considers I|0> = I and I|x> = I? What am I missing?
 

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  • #2
Just a lot of typos. As to 4b), I think the author just treats the case where ##x=0## separately since it isn't picking up the minus sign.
 
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  • #3
It seems that this a student project.

It looks like ##\left| x \right>## has has been split into two cases: ##\left| x= 0 \right>## in (4a); ##\left| x \neq 0\right> ## with ##\left| 0 \right>## and ##\left| x \right>## orthogonal to each other in (4b),

@Haborix relied with similar comments while I was typing.
 
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1. What is a conditional phase shift in Grover's algorithm?

A conditional phase shift is a step in Grover's algorithm where the amplitudes of the target states are inverted based on the outcome of a specific condition. This helps to amplify the amplitude of the desired target state, making it more likely to be measured as the final solution.

2. How does a conditional phase shift work in Grover's algorithm?

In Grover's algorithm, a conditional phase shift is achieved by applying a controlled-phase gate to the target qubit and an ancillary qubit, where the ancillary qubit is in the state of 0. This gate will only affect the target qubit if the ancillary qubit is in the state of 1, which is determined by the outcome of the condition.

3. What is the purpose of a conditional phase shift in Grover's algorithm?

The main purpose of a conditional phase shift in Grover's algorithm is to increase the amplitude of the target state, making it more likely to be measured as the final solution. This is achieved by inverting the amplitudes of the target states based on the outcome of a specific condition.

4. How does a conditional phase shift contribute to the efficiency of Grover's algorithm?

A conditional phase shift contributes to the efficiency of Grover's algorithm by reducing the number of iterations required to find the solution. By amplifying the amplitude of the target state, it makes it more likely to be measured as the final solution, reducing the overall number of iterations needed.

5. Are there any limitations to using a conditional phase shift in Grover's algorithm?

Yes, there are limitations to using a conditional phase shift in Grover's algorithm. One limitation is that the condition used to determine the inversion of the amplitudes must be carefully chosen to ensure that the target state is amplified and not any other states. Additionally, the use of a conditional phase shift may also introduce errors in the algorithm, which can affect the accuracy of the final solution.

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