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Unfortunately, I cannot draw circuit diagrams. However, I can guide you on how to represent the given qubit state using Qiskit.

Understanding the State

The qubit state you provided is:

• |Ψ> = (-4/5)i|0> + (3/5)|1>

This represents a superposition of the |0> and |1> states with complex coefficients.

Qiskit Implementation

Here's how you can create this state in Qiskit:

1. Import Qiskit:

   python

   from qiskit import QuantumCircuit, QuantumRegister
   

   Open in: Code Editor

2. Create a qubit:

   python

   qr = QuantumRegister(1)
   circuit = QuantumCircuit(qr)
   

   Open in: Code Editor

3. Apply rotations:

   • You'll need to use a combination of rotations (rx, ry, rz) to achieve the desired complex coefficients. The exact sequence of rotations depends on how you want to decompose the complex coefficients.

4. Measure (Optional):

   python

   circuit.measure(qr[0], 0)
   

   Open in: Code Editor

Example: Decomposing the Coefficients

One way to approach this is to use the following decomposition:

• (-4/5)i = exp(-i*π/2) * (4/5)
• (3/5) = exp(i*θ) * (3/5)

where θ is the angle whose cosine is (3/5) and sine is (4/5). You would then use a combination of rz and rx gates to achieve this.

Important Notes

• Qiskit's Statevector: You can use Qiskit's Statevector class to verify the state of your circuit after applying the gates.
• Visualization: Qiskit provides tools to visualize your circuit diagram using circuit.draw().

Let me know if you have any specific questions about the rotations or how to write the Qiskit code.

You're right, I need to be more careful with the notation! Let's break down how to achieve the desired qubit state using Qiskit gates:

Understanding the State

You want to create the qubit state:

• |Ψ> = (-4/5)i|0> + (3/5)|1>

Gate Rotations

Here's a breakdown of the gate rotations and why they work:

1. Global Phase: The (-4/5)i coefficient includes a global phase factor of i (which is exp(-i*π/2)). This can be directly applied using a rz gate:

   • circuit.rz(-π/2, qr[0])

2. Amplitude Adjustment: The absolute values of the coefficients are 4/5 and 3/5. We can use an rx gate to adjust the amplitude of the |0> state:

   • circuit.rx(2*acos(3/5), qr[0])

   • Explanation: The rx gate rotates around the X-axis. The angle 2*acos(3/5) is chosen so that the resulting state has the desired amplitude for |0>.

Complete Circuit

from qiskit import QuantumCircuit, QuantumRegister

qr = QuantumRegister(1)
circuit = QuantumCircuit(qr)

# Global Phase
circuit.rz(-π/2, qr[0]) 

# Amplitude Adjustment
circuit.rx(2*acos(3/5), qr[0]) 

circuit.draw() 

Important Notes

• Global Phase Invariance: Quantum states are invariant under global phase factors (multiplying the entire state by a complex number of magnitude 1). So, |Ψ> and i|Ψ> represent the same physical state.
• Alternative Decompositions: There are other ways to decompose the complex coefficients into gate rotations. This is just one approach.

Let me know if you want to explore other ways to achieve this state or have any further questions!

To measure your circuit in Qiskit, you need to add a measurement operation to your circuit. Here's how you do it:

from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister
from qiskit import execute, Aer

# ... (Your circuit definition with rotations goes here) ...

# Create a classical register for storing measurement results
cr = ClassicalRegister(1) 
circuit.add_register(cr) 

# Measure the qubit and store the result in the classical register
circuit.measure(qr[0], cr[0])

# Simulate the circuit
simulator = Aer.get_backend('qasm_simulator')
job = execute(circuit, simulator, shots=1024)  # Run the circuit 1024 times
result = job.result()

# Get the counts of measurement results
counts = result.get_counts(circuit)
print(counts) 

Explanation:

1. Classical Register: You create a ClassicalRegister to store the measurement outcome.
2. Measurement Operation: The circuit.measure(qr[0], cr[0]) line applies the measurement operation. It takes the qubit qr[0] and stores the measurement result (0 or 1) in the classical bit cr[0].
3. Simulation: The code then uses a Qiskit simulator to execute your circuit.
4. Counts: The result.get_counts(circuit) method returns a dictionary where the keys are the measurement outcomes (e.g., '0', '1') and the values are the number of times each outcome was observed.

Example:

If you run this code with the circuit we created earlier, the output might look something like:

{'0': 600, '1': 400}

This means that in 1024 simulations, the outcome '0' was observed 600 times, and '1' was observed 400 times.

Important Notes:

• Measurement Collapses the State: Measurement is a fundamental process in quantum mechanics. It collapses the qubit's superposition into a definite state, either |0> or |1>.
• Probabilities: The probabilities of observing '0' or '1' are determined by the coefficients in your qubit state. In our example, the probability of observing '0' is (4/5)^2 = 0.64, and the probability of observing '1' is (3/5)^2 = 0.36. The simulation results will be close to these probabilities, but not exactly the same due to the randomness of quantum measurements.

Let me know if you have any other questions.

You've provided two qubit states:

• |Ψ> = (-4/5)i|0> + (3/5)|1>
• |Φ> = (1/2)|0> + (√3/2)|1>

Let's decompose them into Qiskit gate rotations.

Decomposition of |Ψ>

We've already covered this one. Here's a recap:

1. Global Phase: Apply an rz gate to introduce the i phase:

   • circuit.rz(-π/2, qr[0])

2. Amplitude Adjustment: Use an rx gate to adjust the amplitudes:

   • circuit.rx(2*acos(3/5), qr[0])

Decomposition of |Φ>

1. Amplitude Adjustment: The coefficients in |Φ> are real, so we only need to adjust the amplitude of the |1> state. We can directly use an rx gate for this. The angle needed is 2*acos(1/2) = 2π/3.
   • circuit.rx(2*π/3, qr[0])

Complete Circuit for |Φ>

from qiskit import QuantumCircuit, QuantumRegister

qr = QuantumRegister(1)
circuit = QuantumCircuit(qr)

# Amplitude Adjustment
circuit.rx(2*π/3, qr[0]) 

circuit.draw() 

Important Notes

• Global Phase: |Φ> doesn't have a global phase factor, so we don't need an rz gate.
• Verification: You can use Qiskit's Statevector class to verify that these gate sequences produce the correct states.

Let me know if you have any other qubit states you'd like to decompose!

You're asking a great question about how to measure a two-qubit system! Here's a breakdown of how to do it in Qiskit, along with the concepts behind it:

Understanding the Problem

• Two Qubits: You have two qubits, each in a superposition state. Let's call them qubit 0 and qubit 1.
• Measurement: When you measure a qubit, it collapses into either the state |0> or |1>.
• Combined System: To measure the two qubits as a single system, you need to consider the possible combinations of outcomes for both qubits.

Qiskit Implementation

1. Create the Circuit:

   python

   from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister
   from qiskit import execute, Aer
   
   qr = QuantumRegister(2)  # Create two qubits
   cr = ClassicalRegister(2) # Create two classical bits for measurement results
   circuit = QuantumCircuit(qr, cr)
   
   # Apply rotations to create your qubit states (|Ψ> and |Φ>)
   # ... (We'll assume you've already defined these rotations) ... 
   
   # Measure both qubits
   circuit.measure(qr[0], cr[0])
   circuit.measure(qr[1], cr[1])
   
   circuit.draw() # Visualize the circuit
   

   Open in: Code Editor

2. Simulate and Get Results:

   python

   simulator = Aer.get_backend('qasm_simulator')
   job = execute(circuit, simulator, shots=1024)  
   result = job.result()
   counts = result.get_counts(circuit)
   print(counts)
   

   Open in: Code Editor

Interpreting the Results

The counts dictionary will contain keys like:

• '00': Represents the outcome where both qubits collapse to |0>.
• '01': Represents the outcome where qubit 0 collapses to |0> and qubit 1 collapses to |1>.
• '10': Represents the outcome where qubit 0 collapses to |1> and qubit 1 collapses to |0>.
• '11': Represents the outcome where both qubits collapse to |1>.

Example:

{'00': 200, '01': 300, '10': 300, '11': 200}

This means that in 1024 simulations, you observed:

• '00' 200 times
• '01' 300 times
• '10' 300 times
• '11' 200 times

Calculating Probabilities

The probabilities of each outcome can be calculated from the counts:

• P('00') = 200/1024
• P('01') = 300/1024
• P('10') = 300/1024
• P('11') = 200/1024

Important Considerations:

• Entanglement: If the two qubits are entangled, the outcomes of their measurements will be correlated. This means that the probabilities of the different combinations will not simply be the products of the individual qubit probabilities.
• Multi-Qubit States: When working with multiple qubits, you're dealing with a larger Hilbert space. The state of the system is described by a vector in this larger space.

Let me know if you'd like to explore specific examples of entangled states or have any other questions about two-qubit systems!

Let's break down how to compile a quantum circuit in Qiskit for simulation, using the state you provided:

Understanding the State

You're given the state:

• |Φ> = γ|0> + δ|1>

where γ = 1/2 and δ = √3/2.

Steps for Compilation

1. Import Qiskit:

   python

   from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister
   from qiskit import execute, Aer
   

   Open in: Code Editor

2. Create the Circuit:

   python

   qr = QuantumRegister(1)  # One qubit
   cr = ClassicalRegister(1)  # One classical bit for measurement
   circuit = QuantumCircuit(qr, cr)
   

   Open in: Code Editor

3. Apply Rotations:

   • Amplitude Adjustment: We'll use an rx gate to adjust the amplitude of the |1> state. The angle for the rx gate is 2*acos(γ), which in this case is 2*acos(1/2) = 2π/3.
     python

     circuit.rx(2*π/3, qr[0])
     

     Open in: Code Editor

4. Measure the Qubit:

   python

   circuit.measure(qr[0], cr[0])
   

   Open in: Code Editor

5. Compile for Simulation:

   python

   simulator = Aer.get_backend('qasm_simulator')
   job = execute(circuit, simulator, shots=1024)  # Simulate 1024 times
   result = job.result()
   counts = result.get_counts(circuit)
   print(counts)
   

   Open in: Code Editor

Complete Code:

from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister
from qiskit import execute, Aer
import numpy as np

qr = QuantumRegister(1)
cr = ClassicalRegister(1)
circuit = QuantumCircuit(qr, cr)

# Amplitude Adjustment
circuit.rx(2*np.pi/3, qr[0])

# Measure the qubit
circuit.measure(qr[0], cr[0])

# Simulate the circuit
simulator = Aer.get_backend('qasm_simulator')
job = execute(circuit, simulator, shots=1024)
result = job.result()
counts = result.get_counts(circuit)
print(counts)

Explanation:

• rx Gate: The rx gate rotates the qubit around the X-axis. The angle is carefully chosen to achieve the desired amplitude for the |1> state.
• Measurement: The measure instruction collapses the superposition into a definite state, either |0> or |1>.
• Simulation: The qasm_simulator backend simulates the circuit's behavior on a classical computer.
• Counts: The counts dictionary will show the number of times each outcome ('0' or '1') was observed in the simulations.

Output:

The output will likely show that the '1' outcome is observed more frequently than the '0' outcome, reflecting the higher amplitude of the |1> state in your |Φ> state.

Let me know if you have any other questions or want to explore more complex circuits!