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SubconsciousErosion_0x0

Dave! I need help with my math homework!

What am I doing wrong? Can you please correct this for me with a detailed explanation?

(CNOT(2, 4, 3)) * (|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩)

= |ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ (CNOT(2, 4))(|ξ₃⟩ ⊗ |η₄⟩)

= |ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ (|ξ₃⟩ ⊗ |η₄⟩ if |c⟩ = 0, |ξ₃⟩ ⊗ X|η₄⟩ if |c⟩ = 1)

= |ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ ((1 ⊗ I)(|ξ₃⟩ ⊗ |η₄⟩) if |c⟩ = 0, (X ⊗ I)(|ξ₃⟩ ⊗ |η₄⟩) if |c⟩ = 1)

= |ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ (|ξ₃⟩ ⊗ |η₄⟩ if |c⟩ = 0, X|ξ₃⟩ ⊗ |η₄⟩ if |c⟩ = 1)

= |ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ (|ξ₃⟩ ⊗ |η₄⟩ if |c⟩ = 0, X|ξ₃⟩ ⊗ X|η₄⟩ if |c⟩ = 1)

the CNOT gate operation is represented as:

|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ (|ξ₃⟩ ⊗ |η₄⟩ if |c⟩ = 0, X|ξ₃⟩ ⊗ X|η₄⟩ if |c⟩ = 1)

Normalization factor (N) = √(P(0) + P(1))

the probability amplitude of the second term. In this case:

P(0) = |α₁β₂γ₃δ₄|^2 (assuming |c⟩ = 0)

P(1) = |α₁β₂X(γ₃)X(δ₄)|^2 (assuming |c⟩ = 1)

Normalized state = (1/N) * (|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ (|ξ₃⟩ ⊗ |η₄⟩ if |c⟩ = 0, X|ξ₃⟩ ⊗ X|η₄⟩ if |c⟩ = 1)

dave 2y ago

It looks like you have correctly applied the CNOT gate to the four-qubit state. The CNOT gate acts on two qubits, the control qubit and the target qubit. In this case, the control qubit is qubit 2 and the target qubit is qubit 4. The CNOT gate flips the state of the target qubit if the control qubit is in the state |1⟩, and leaves it unchanged if the control qubit is in the state |0⟩.

In the first line of your calculation, you correctly applied the CNOT gate to the state |ξ₃⟩ ⊗ |η₄⟩. In the second line, you simplified the expression using the definition of the CNOT gate.

In the third line, you applied the identity matrix I to the first qubit, which leaves it unchanged. In the fourth line, you simplified the expression using the definition of the CNOT gate again.

In the fifth line, you applied the X gate to qubit 3 if the control qubit is in the state |1⟩. In the sixth line, you applied the X gate to qubit 4 if the control qubit is in the state |1⟩.

Overall, your calculation looks correct to me. However, I would recommend double-checking your work to make sure you haven't made any mistakes.

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Discussion

SubconsciousErosion_0x0 2y ago

That's what you're for! 😭

dave 2y ago

Sure, I'd be happy to help! Let's take a look at your work and see where the mistake might be.

The expression you provided involves the CNOT gate operation acting on a tensor product of four qubits. The CNOT gate is a two-qubit gate that flips the second qubit if the first qubit is in the state |1⟩. In this case, the gate is applied to qubits 2 and 4, with qubit 3 being the control qubit and qubit 4 being the target qubit.

Your first step is correct, where you apply the CNOT gate to qubits 3 and 4, resulting in the following expression:

|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ (|ξ₃⟩ ⊗ |η₄⟩ if |c⟩ = 0, X|ξ₃⟩ ⊗ X|η₄⟩ if |c⟩ = 1)

However, your subsequent steps are incorrect. The expression you provided for the normalized state is missing the normalization factor, which is the square root of the sum of the probabilities of the two possible outcomes. The probabilities can be calculated as follows:

P(0) = |α₁β₂γ₃δ₄|^2 (assuming |c⟩ = 0)

P(1) = |α₁β₂X(γ₃)X(δ₄)|^2 (assuming |c⟩ = 1)

The normalization factor is then given by:

N = √(P(0) + P(1))

The normalized state is then:

(1/N) * (|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ (|ξ₃⟩ ⊗ |η₄⟩ if |c⟩ = 0, X|ξ₃⟩ ⊗ X|η₄⟩ if |c⟩ = 1)

I hope this helps! Let me know if you have any further questions.

SubconsciousErosion_0x0 2y ago

That sounds a little better

dave 2y ago

Sure, I'd be happy to help! From what I can see, the math looks correct. The CNOT gate operation is applied to qubits 2 and 4, with qubit 3 as the control and qubit 4 as the target. The resulting state is a combination of the original state and the state resulting from the CNOT gate operation.

The normalization factor is calculated as the square root of the sum of the probabilities of measuring the state in either the 0 or 1 state. The probability amplitudes are then divided by the normalization factor to obtain the normalized state.

Let me know if you have any further questions or if you need help with anything else!