Measure in grover's alg and shift phase

In summary, Grover's algorithm is a quantum algorithm that uses superposition and interference to search unsorted databases faster than classical algorithms. Measurement is crucial in collapsing the quantum state to reveal the solution, and the shift phase operator amplifies the correct solution's amplitude. It can only be used for unstructured search problems and the success rate is directly related to the number of iterations, with the optimal number being the square root of the number of possible solutions.
  • #1
liechmaster
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hi , i am new here so at first i want to say hello to all of You.
as title says i have two questions about qrover's algorithm, first one is more general i guess:
1. in the most of articles i have read about this algorithm the measure wasn't well explained,problem is that authors wrote "now we measure first n qubits"
when we are using n+1 ( one is as control qubit in oracles transformation)
my question is.. how we can measure only n qubits? what about that control qubit? i know it is not important one but i guess we can't just throw it out...
2. i am not sure if i well understand the shift phase , i mean the step when our solution qubit |x> changes its state to -|x>
is all about that in the oracle transformation? when we got gubit register |z> ( made from state |00..0> by H x H ... x H gates) and single qubit in state |1> and then we apply the H gate on it? and on that n+1 qubits we apply simple black box and in result : (-1)^f(z)|z>*[(|0>-|1>)/sqrt(2)] ( where f(x)=1 for solution and 0 for others)
is is all about changing the phase on solution qubit?
i hope my text is readable enough ; ) ( maybe little chaotic)
looking forward for replays.
 
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  • #2


Hello and welcome to the forum! It's great to have you here and I'm happy to answer your questions about qrover's algorithm.

To answer your first question, when we talk about measuring only n qubits, we are referring to the qubits that are involved in the actual computation and not the control qubit. The control qubit is used to control the operation of the oracle and is not part of the computation itself. Therefore, when we measure the first n qubits, we are not including the control qubit in the measurement.

As for your second question, you are correct that the shift phase is all about changing the phase on the solution qubit. This is done through the oracle transformation, where the state of the solution qubit is changed from |0> to -|1> (or vice versa) depending on the result of the function f(x). This is what allows us to identify the solution to the problem.

I hope this helps to clarify things for you. If you have any further questions, please don't hesitate to ask. Happy researching!
 
  • #3


Hello and welcome to the world of quantum computing!
To answer your first question, when we talk about measuring "n" qubits in Grover's algorithm, we are referring to the target qubits that we are trying to find the solution for. The control qubit is not included in this measurement because it is not part of the solution space that we are searching for. It is used as a control for the oracle transformation, but it does not affect the measurement of the solution qubits. So, in essence, we are measuring only the qubits that are relevant to our problem.

As for your second question, yes, the shift phase is a crucial step in Grover's algorithm and it is indeed part of the oracle transformation. The idea behind the shift phase is to amplify the amplitude of the solution qubit while suppressing the amplitudes of the other qubits. This is achieved by applying the Hadamard gate to the solution qubit and then applying the oracle transformation, which flips the phase of the solution qubit. This results in the solution qubit having a higher amplitude compared to the other qubits, making it more likely to be measured as the solution.

I hope this helps clarify your understanding of Grover's algorithm. Happy learning!
 

FAQ: Measure in grover's alg and shift phase

1. What is Grover's algorithm and how does it work?

Grover's algorithm is a quantum algorithm designed to search an unsorted database in a faster time than classical algorithms. It works by using quantum superposition and interference to amplify the correct solution, while canceling out the incorrect solutions.

2. What is the role of measurement in Grover's algorithm?

Measurement is a crucial step in Grover's algorithm as it allows the algorithm to collapse the quantum state into a classical state, revealing the solution to the search problem. Without measurement, the algorithm would not be able to return a definite answer.

3. How does the shift phase operator work in Grover's algorithm?

The shift phase operator is a key step in Grover's algorithm that is responsible for amplifying the amplitude of the correct solution and reducing the amplitudes of the incorrect solutions. It does this by applying a phase shift to the amplitude of the correct solution, making it more likely to be measured.

4. Can Grover's algorithm be used for any type of search problem?

No, Grover's algorithm is specifically designed for unstructured search problems, where the database is not sorted in a specific way. It cannot be used for structured search problems, like searching through a list of numbers in numerical order.

5. How does the number of iterations impact the success of Grover's algorithm?

The number of iterations in Grover's algorithm is directly related to its success rate. The optimal number of iterations for a search problem with N possible solutions is √N. This means that the more iterations the algorithm goes through, the higher the probability of measuring the correct solution becomes.

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