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Let's break down how this circuit solves Simon's algorithm for s = 11.

Understanding Simon's Algorithm

Simon's algorithm is a quantum algorithm used to solve a specific problem: finding a hidden string 's' in a function f(x) that has the property f(x) = f(x ⊕ s). In simpler terms, if you XOR any input 'x' with 's', you'll get the same output from the function.

The Circuit

1. Initialization:

   • The circuit starts with four qubits in the state |0⟩.

2. Hadamard Gates (H):

   • Hadamard gates are applied to the first two qubits (|0⟩_1 and |0⟩_1). This puts these qubits into a superposition of all possible states.

3. Oracle Function (f(x)):

   • The circuit represents the oracle function f(x) as a series of controlled-NOT (CNOT) gates. Here's how it works:
   • The first two qubits (|0⟩_1 and |0⟩_1) are the control qubits.
   • The last two qubits (|0⟩_2 and |0⟩_2) are the target qubits.
   • The control qubits are connected to the target qubits with CNOT gates. The placement of these gates defines the function f(x).

4. More Hadamard Gates (H):

   • Hadamard gates are applied again to the first two qubits.

5. Measurement:

   • The first two qubits are measured.

How it Solves for s = 11

Let's focus on the specific case where s = 11 (which in binary is 1011).

1. Oracle Function (f(x)):

   • The placement of the CNOT gates in the circuit encodes the function f(x) with the hidden string s = 11. In this case, the CNOT gates are arranged so that:
     • If the first two qubits are |10⟩, the third qubit flips.
     • If the first two qubits are |11⟩, the fourth qubit flips.

2. Measurement:

   • When you measure the first two qubits, you'll get a random outcome. However, the outcome is related to the hidden string 's'. The crucial point is that every measurement result will be a linear combination of the hidden string 's' (1011) and another random string.

   • For example, you might measure:

     • |00⟩ (which is 0000) ⊕ s (1011) = 1011
     • |01⟩ (which is 0001) ⊕ s (1011) = 1010
     • |10⟩ (which is 0010) ⊕ s (1011) = 1001
     • |11⟩ (which is 0011) ⊕ s (1011) = 1000

3. Finding s:

   • By repeating the measurement process a few times, you'll get several different linear combinations of 's'. You can then use these results to solve a system of equations and find the hidden string 's'.

Key Points

• The placement of the CNOT gates in the oracle function is crucial. It defines the function f(x) and encodes the hidden string 's'.
• The Hadamard gates create superposition, which allows the algorithm to explore multiple input states simultaneously.
• Measurement results provide information about the hidden string 's'.
• By repeating the process and solving a system of equations, you can determine the hidden string.

Let me know if you'd like a more detailed explanation of any specific aspect of the circuit or Simon's algorithm!

You are absolutely right to ask this! The circuit doesn't directly "identify" the hidden string 's'. Instead, it provides us with information that allows us to calculate 's'. Here's how:

1. The Oracle Function: The CNOT gates in the circuit implement the oracle function f(x). As we discussed earlier, the specific placement of these gates encodes the hidden string 's'.

2. Superposition and Measurement: The Hadamard gates create a superposition of states, allowing the algorithm to effectively query the oracle function with multiple inputs simultaneously. When we measure the first two qubits (after the second set of Hadamard gates), we get a random output.

3. The Key is Linear Combinations: The crucial point is that each measurement result is a linear combination of the hidden string 's' and another random string. This is because the oracle function flips the target qubits based on the input, and the Hadamards introduce a superposition of these flips.

4. Finding 's' Through Equations: To find 's', we need to repeat the measurement process several times. Each measurement will give us a different linear combination of 's'. We then have a system of equations where the unknowns are the bits of 's'. By solving this system, we can determine the value of 's'.

Example:

Let's say we get these measurement results:

• 1011 (which is 's' itself)
• 1010
• 1001
• 1000

We can see that:

• 1010 = 1011 ⊕ 0001
• 1001 = 1011 ⊕ 0010
• 1000 = 1011 ⊕ 0011

Therefore, we can deduce that the hidden string 's' is 1011.

In Summary:

The circuit doesn't directly reveal 's'. It provides us with a set of linear combinations of 's' that we can use to solve a system of equations and ultimately determine the hidden string.

Let me know if you have any further questions or would like to explore this with a specific example!