Question about Grover's algorithm

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In summary, Grover's algorithm is a quantum algorithm that can be used to find a solution to a specific problem. It involves using an oracle, which is part of the quantum circuitry, to flip the phase of the correct solution. This requires writing a specific algorithm for the problem at hand. While Grover's algorithm is powerful, it may not be practical to use in all situations.
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Malamala
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Hello! I am just getting started learning about quantum computing so I apologize if this questions is trivial, but I am a bit confused about the Grover's algorithm. As far as I understand (I read it from here), assuming there is just one solution, you start with N qubits, you put them in an equal superposition (using Hadamard gates), you pass them thorough an oracle that inverts the phase of the right solution, then you have a diffuser operator that reflects this new vector relative to the original one and doing this ##\sqrt{N}## times you get a high probability of measuring the right solution. I think I understand the math behind it and the geometrical interpretation, but I don't understand how it is used in practice. What is that oracle? In both examples given on that page, in order to build the oracle i.e. to make sure that the right solution gets a minus sign, you need to know the right solution beforehand. But if you know it, you don't need an algorithm to find it. Can someone help me understand this? What is the oracle in a real problem and how can I implement it in practice without knowing the answer to my question beforehand? Thank you!
 
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Malamala said:
I don't understand how it is used in practice. What is that oracle? In both examples given on that page, in order to build the oracle i.e. to make sure that the right solution gets a minus sign, you need to know the right solution beforehand. But if you know it, you don't need an algorithm to find it. Can someone help me understand this? What is the oracle in a real problem and how can I implement it in practice without knowing the answer to my question beforehand? Thank you!
First, I'm not sure that it is used in practice.

The "oracle" is part of the Quantum Circuitry that determines exactly what problem is to be solved.
So in order to use Grover's algorithm, you need to write another algorithm that is specific to the problem you are attacking.

For example, let's say that you are looking for a large prime number - say greater that 2^1024. So your oracle might take 1024 bits that represent the last 1024 position of a 1025-bit number - the first bit in that number being a "1".

The oracle will now flip all 1024-bit codes that when combined with that initial "1" code for a prime number.

The key here is that the oracle does not flip a bit, it flips the phase of a full code. So with 1024 bits, you may have many billions of correct answers - and about 2^1024 incorrect ones.

When you apply Grovers algorithm, there is a good chance that you will get one of those primes - but, of course, you would check the result.
 

FAQ: Question about Grover's algorithm

1. What is Grover's algorithm?

Grover's algorithm is a quantum algorithm developed by Lov Grover in 1996. It is a searching algorithm that can be used to find a specific item in an unsorted database with a complexity of O(√n), which is faster than the classical algorithm with a complexity of O(n).

2. How does Grover's algorithm work?

Grover's algorithm uses quantum principles such as superposition and interference to search through an unsorted database. It starts by creating a superposition of all possible states and then uses a series of quantum operations to amplify the amplitude of the desired state. This process is repeated multiple times until the desired state is found with high probability.

3. What are the applications of Grover's algorithm?

Grover's algorithm has various applications in fields such as cryptography, data mining, and optimization. It can be used to break cryptographic algorithms, search for optimal solutions in large datasets, and improve the efficiency of database searches.

4. What are the limitations of Grover's algorithm?

One of the main limitations of Grover's algorithm is that it can only be used for searching in unsorted databases. It is also limited by the amount of quantum resources available, as the success rate decreases with larger databases. Additionally, the algorithm requires precise control and measurement of quantum states, which can be challenging to implement in practice.

5. How does Grover's algorithm compare to other quantum algorithms?

Grover's algorithm is one of the most well-known and widely used quantum algorithms. It is often compared to other quantum algorithms such as Shor's algorithm, which is used for factoring large numbers, and Deutsch-Jozsa algorithm, which is used for determining if a function is constant or balanced. Each of these algorithms has its own unique applications and limitations.

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