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Plotting Superposition of Qubits on a Bloch Sphere in Qiskit

I've been trying to plot the state $\frac{|0>-|1>}{\sqrt{2}}$ using Qiskit. As a result, I need to apply the NOT gate on $|0>$ to get $|1>$ and then apply the Hadamard gate on $|1>$ to get the required state. Here's my code -

from qiskit import QuantumCircuit, Aer, execute
from qiskit.visualization import plot_bloch_vector
import numpy as np

# Create a quantum circuit with one qubit
qc = QuantumCircuit(1)

# Apply a Hadamard gate to the |1> state
qc.x(0)  # Apply an X gate to get |1>
qc.h(0)  # Apply a Hadamard gate

# Simulate the circuit on the local qasm simulator
backend = Aer.get_backend('statevector_simulator')
job = execute(qc, backend)
result = job.result()
statevector = result.get_statevector()

# Extract the Bloch vector components from the state vector
bloch_vector = [2 * np.real(statevector[0]), 2 * np.imag(statevector[1]), 0]

# Plot the Bloch vector on the Bloch sphere
plot_bloch_vector(bloch_vector).show()

But this gives me an output- enter image description here

This image corresponds to the state $\frac{|0>+|1>}{\sqrt{2}}$. Can someone point out the not-so-obvious mistake in my code that I've failed to debug so far?

Solution

  • The issue lies with extracting the Bloch vector from the state vector. Your equation

    bloch_vector = [2 * np.real(statevector[0]), 2 * np.imag(statevector[1]), 0]

    is not quite correct. To see why not, if you think about it, the statevectors of the two states are [1\sqrt(2), 1\sqrt(2)] and [1\sqrt(2), -1\sqrt(2)]. Clearly in both the cases, there are no imaginary parts.

    Given a single qubit state a|0> + b|1> , the correct equation for extracting the Bloch vectors is

    bloch_vector = [2*np.real(a*np.conjugate(b)), 2*np.imag(b*np.conjugate(a)), a*np.conjugate(a)-b*np.conjugate(b)].