Using Grover's algorithm, find the following states: '101', '001', and '010'.

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Grover's algorithm is a quantum search algorithm that aims to find marked items in an unsorted database more efficiently than classical algorithms. It provides a quadratic speedup over classical search algorithms. To apply Grover's algorithm to find the marked states '101', '001', and '010', you need to follow these steps:

Initialization: Start with an equal superposition of all possible states.
Oracle: Apply a phase inversion oracle to mark the desired states.
Amplification: Apply Grover diffusion operator to amplify the amplitude of the marked states.
Let's go through the steps for each of the states:

1. Finding '101':
The target state '101' can be represented in binary as '00000 00101'. To implement Grover's algorithm, you need to mark this state using an oracle.

Oracle: Apply a Z gate (also known as a phase-flip gate) to qubits where the state '101' has a '1' bit:

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|0⟩ -> |101⟩

Amplification: Apply the Grover diffusion operator (Hadamard - Z - Hadamard) to amplify the marked state:

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H^n Z H^n |101⟩

2. Finding '001':
The target state '001' can be represented in binary as '00000 00001'. Again, you need to mark this state using an oracle.

Oracle: Apply a Z gate to qubits where the state '001' has a '1' bit:

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|0⟩ -> |001⟩

Amplification: Apply the Grover diffusion operator:

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H^n Z H^n |001⟩

3. Finding '010':
The target state '010' can be represented in binary as '00000 00010'. Mark the state using an oracle.

Oracle: Apply a Z gate to qubits where the state '010' has a '1' bit:

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|0⟩ -> |010⟩

Amplification: Apply the Grover diffusion operator:

Code: Select all

H^n Z H^n |010⟩

Please note that the above descriptions outline the general steps for implementing Grover's algorithm for each state. In practice, you would need to carefully implement these steps using quantum gates in a quantum programming framework or on actual quantum hardware. Additionally, the number of Grover iterations needed depends on factors like the number of marked states and the size of the search space. You might need to adjust the number of iterations for optimal performance.