With_Code

 

Copilot:

Certainly! Let’s break down the code step by step:

Statevector.from_label("0"): This line creates a statevector corresponding to the quantum state |0⟩. The from_label method initializes a statevector based on a label, where “0” represents the |0⟩ state1.

Statevector.from_label("1"): Similarly, this line creates a statevector corresponding to the quantum state |1⟩.

zero.tensor(one): The tensor method computes the tensor product of two statevectors. In this case, it combines the |0⟩ state with the |1⟩ state, resulting in a composite statevector representing the joint system.

draw("latex"): Finally, the draw method visualizes the composite statevector in LaTeX format. It generates a representation that you can display in Jupyter Notebooks or other LaTeX-compatible environments1.

Here’s the LaTeX representation of the combined state:

∣0⟩⊗∣1⟩=∣01⟩

This represents the joint state of a two-qubit system, where the first qubit is in the |0⟩ state and the second qubit is in the |1⟩ state. Feel free to explore this further in your quantum circuits! 😊

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Gemini:

Understanding the Qiskit Code

Import:

The code snippet seems to be using the Statevector class from the Qiskit library. This class represents a quantum state as a complex vector.

Code Breakdown:

1. plus = Statevector.from_label("+"):

   • Creates a quantum state plus representing the + state (superposition of |0> and |1>).
   • This state is often represented as |0> + |1> or (1/sqrt(2))|0> + (1/sqrt(2))|1>.

2. i_state = Statevector([1 / sqrt(2), 1j / sqrt(2)]):

   • Creates a quantum state i_state with complex amplitudes.
   • The values 1/sqrt(2) and 1j/sqrt(2) represent the amplitudes of the basis states |0> and |1> respectively.

3. psi = plus.tensor(i_state):

   • Creates a new quantum state psi by taking the tensor product of plus and i_state.
   • The tensor product combines two quantum states into a larger state representing the combined system.

4. psi.draw("latex"):

   • Renders the quantum state psi in LaTeX format for display.

Output:

The output shows the resulting quantum state psi in bra-ket notation:

1/2|00> + i/2|01> + 1/2|10> + i/2|11>

Interpretation:

The code creates two quantum states, plus and i_state, and then combines them into a new state psi using the tensor product. The final output shows the state psi as a superposition of four basis states: |00>, |01>, |10>, and |11>.

Additional Notes:

• The use of complex numbers in quantum states is essential for describing quantum phenomena.
• The tensor product is a fundamental operation in quantum mechanics for combining systems.
• The draw("latex") method provides a visual representation of the quantum state.

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This is the tensor product of the Identity gate and the X gate. Order matters; the tensor

product is not commutative...

                                                                        

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   This represents the CNOT matrix.

Meta:

                                                                            

                       

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This is a three qubit entangled circuit.

Copilot:

You

How can I create a W state in Qiskit?

To create a W state in Qiskit, follow these steps:

Initialize a quantum circuit with three qubits.

Apply the Hadamard gate (H) to the first qubit.

Apply the CNOT gate (CX) from the first qubit to the second and third qubits.

Apply the Toffoli gate (CCX) from the first qubit to the second and third qubits.

Measure the qubits.

Here’s the circuit in Qiskit:

Python ...

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Gemini:

The code you provided defines a quantum state vector in Qiskit. Let's break down how the input relates to the output:

Input:

• Statevector([0, 1, 1, 0, 1, 0, 0, 0] / sqrt(3)): This creates a quantum state vector using the Statevector function from Qiskit. The input is a list of complex numbers representing the amplitudes of the basis states.
  • [0, 1, 1, 0, 1, 0, 0, 0]: This list represents the amplitudes of the 8 possible basis states for a 3-qubit system (since 2^3 = 8). For example, the first element '0' corresponds to the basis state |000>, the second element '1' corresponds to |001>, and so on.
  • / sqrt(3): This normalizes the amplitudes so that the sum of their squared magnitudes is equal to 1, which is a requirement for a valid quantum state.

Output:

• W.draw("latex"): This part of the code renders the quantum state in a visually appealing LaTeX format, making it easier to read and understand.

The Output State:

The output you've shown is the LaTeX representation of the quantum state:

text

√3/3 |001> + √3/3 |010> + √3/3 |100>

Interpretation:

This state represents a superposition of three basis states:

• |001>: The first qubit is in state |1>, the second in state |0>, and the third in state |1>.
• |010>: The first qubit is in state |0>, the second in state |1>, and the third in state |0>.
• |100>: The first qubit is in state |1>, the second in state |0>, and the third in state |0>.

Each of these basis states has an amplitude of √3/3. This means that if you were to measure this state, you would have a 1/3 chance of collapsing to each of these states.

In summary:

The input to the Statevector function is a list of amplitudes, and the output is a representation of the quantum state in a visually clear LaTeX format. The LaTeX output shows the superposition of basis states and their corresponding amplitudes.

1/1

Asked for more info on the input numbers:

The input numbers are: [0, 1, 1, 0, 1, 0, 0, 0]. These are the amplitudes of a 3-qubit quantum state. Here's how they correspond to the basis states:

• 0: Represents the amplitude for the basis state |000>
• 1: Represents the amplitude for the basis state |001>
• 1: Represents the amplitude for the basis state |010>
• 0: Represents the amplitude for the basis state |011>
• 1: Represents the amplitude for the basis state |100>
• 0: Represents the amplitude for the basis state |101>
• 0: Represents the amplitude for the basis state |110>
• 0: Represents the amplitude for the basis state |111>

The sqrt(3) in the code means that each of these amplitudes is divided by the square root of 3, making the total probability of the state equal to 1.

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From Gemini:

You're asking a great question! Here's why you see only two numbers after measuring qubit 0:

• Original State: Your original state W is a superposition of three basis states:

  • √3/3 |001>
  • √3/3 |010>
  • √3/3 |100>

• Measuring Qubit 0: When you measure qubit 0, you're collapsing the wavefunction of the entire system. This means that the outcome of the measurement will determine which of the possible states the system will be in.

• Possible Outcomes:

  • Measuring 0: If you measure qubit 0 as being in state |0>, the possible outcomes are:
    • √3/3 |001> (since the first qubit is in state |0>)
    • √3/3 |010> (since the first qubit is in state |0>)
  • Measuring 1: If you measure qubit 0 as being in state |1>, the only possible outcome is:
    • √3/3 |100> (since the first qubit is in state |1>)

• New State: The measurement process eliminates the states that are incompatible with the outcome. This leaves you with a new state that is a superposition of only those states that are consistent with the measurement.

• Two Numbers: Since you measured qubit 0 as 0, the new state is a superposition of |001> and |010>, which is why you see only two numbers in the output. The amplitude of each of these states is √2/2, since they are both equally likely to be measured.

In Summary: When you measure a qubit, you collapse the wavefunction of the entire system. This means that the system is no longer in a superposition of all possible states, but rather in a superposition of only those states that are compatible with the measurement outcome.