Qiskit is an open-source framework for quantum computing, developed by IBM. It allows users to simulate quantum circuits and run them on real quantum hardware. However, some users may encounter issues when simulating certain quantum states, such as the W-state, which can become entangled during X-basis measurement.
In this article, we will explain what the W-state is, why it becomes entangled during X-basis measurement, and how to address this issue in Qiskit.
What is the W-state?
The W-state is a type of entangled quantum state that can be defined as follows:
|W> = (1/sqrt(3))(|001> + |010> + |100>)
This state consists of three qubits, where each qubit can be in the state |0> or |1>. The W-state is entangled, which means that the state of each qubit cannot be described independently of the others. Instead, the state of the system as a whole is described by the superposition of all possible combinations of |0> and |1> states for each qubit.
Why does the W-state become entangled during X-basis measurement?
The W-state becomes entangled during X-basis measurement because of the way that the measurement process is defined in quantum mechanics. When a qubit is measured in the X-basis, it is projected onto the eigenstates of the X operator, which are |+> and |->. These states are defined as follows:
|+> = (1/sqrt(2))(|0> + |1>)
|-> = (1/sqrt(2))(|0> - |1>)
When a qubit in the W-state is measured in the X-basis, it is projected onto either the |+> or |-> state. However, the other qubits in the system are also affected by this measurement. This is because the W-state is an entangled state, which means that the state of each qubit is correlated with the state of the others. Therefore, when one qubit is measured, it affects the state of the other qubits in the system.
For example, suppose that the W-state is measured in the X-basis, and the first qubit is projected onto the |+> state. This means that the state of the system collapses to either |+00> or |+01>. However, the state of the other two qubits is still entangled. This means that the state of the system as a whole is still described by the superposition of all possible combinations of |0> and |1> states for the other two qubits.
How to address the issue in Qiskit
In Qiskit, the W-state can be simulated using the WStateSimulator class. However, this class does not provide a way to measure the qubits in the X-basis directly. Instead, the qubits must be measured in the Z-basis, and then transformed into the X-basis using a XGate.
To illustrate this, let's consider the following example:
from qiskit import QuantumCircuit, transpile, assemble, Aer, execute
from qiskit.visualization import plot\_histogram, plot\_bloch\_multivector
from qiskit.circuit.library import WStateSimulator
# Create a quantum circuit with three qubits
qc = QuantumCircuit(3)
# Apply a Hadamard gate to the first qubit
qc.h(0)
# Apply an X gate to the first qubit
qc.x(0)
# Apply a W state to the other two qubits
qc.append(WStateSimulator(2), [1, 2])
# Measure the qubits in the Z-basis
qc.measure\_all()
# Transpile the circuit to the backend
qc\_transpiled = transpile(qc, backend)
# Assemble the circuit
qc\_assembled = assemble(qc\_transpiled, backend)
# Execute the circuit
job = execute(qc\_assembled, backend, shots=1000)
# Get the results
result = job.result()
# Plot the histogram
plot\_histogram(result.get\_counts(qc))
In this example, we first create a quantum circuit with three qubits. We then apply a Hadamard gate to the first qubit, which puts it in a superposition of |0> and |1>. We then apply an X gate to the first qubit, which flips the state from |0> to |1> or from |1> to |0>. We then apply the W-state to the other two qubits using the WStateSimulator class. Finally, we measure the qubits in the Z-basis, and plot the histogram of the results.
However, this approach does not allow us to measure the qubits in the X-basis directly. To do this, we need to apply an X gate to each qubit before measuring it in the Z-basis. This will transform the Z-basis eigenstates into X-basis eigenstates. We can then measure the qubits in the Z-basis, and interpret the results as if they were measured in the X-basis.
Here is an example of how to do this:
from qiskit import QuantumCircuit, transpile, assemble, Aer, execute
from qiskit.visualization import plot\_histogram, plot\_bloch\_multivector
from qiskit.circuit.library import WStateSimulator
# Create a quantum circuit with three qubits
qc = QuantumCircuit(3)
# Apply a Hadamard gate to the first qubit
qc.h(0)
# Apply an X gate to the first qubit
qc.x(0)
# Apply a W state to the other two qubits
qc.append(WStateSimulator(2), [1, 2])
# Apply an X gate to each qubit
qc.x([0, 1, 2])
# Measure the qubits in the Z-basis
qc.measure\_all()
# Transpile the circuit to the backend
qc\_transpiled = transpile(qc, backend)
# Assemble the circuit
qc\_assembled = assemble(qc\_transpiled, backend)
# Execute the circuit
job = execute(qc\_assembled, backend, shots=1000)
# Get the results
result = job.result()
# Plot the histogram
plot\_histogram(result.get\_counts(qc))
In this example, we first create a quantum circuit with three qubits. We then apply a Hadamard gate to the first qubit, which puts it in a superposition of |0> and |1>. We then apply an X gate to the first qubit, which flips the state from |0> to |1> or from |1> to |0>. We then apply the W-state to the other two qubits using the WStateSimulator class. We then apply an X gate to each qubit, which transforms the Z-basis eigenstates into X-basis eigenstates. Finally, we measure the qubits in the Z-basis, and plot the histogram of the results. We can then interpret the results as if they were measured in the X-basis.
In this article, we have explained what the W-state is, why it becomes entangled during X-basis measurement, and how to address this issue in Qiskit. We have shown that the W-state can be simulated using the WStateSimulator class in Qiskit, and that it can be measured in the X-basis by applying an X gate to each qubit before measuring it in the Z-basis. This approach allows us to simulate the W-state in Qiskit, and to observe its entanglement properties during X-basis measurement.
References
| Title | Author | Year | Link |
|---|---|---|---|
| Quantum Computing for the Very Curious | Chris Bernhardt | 2019 | Link |
| Introduction to Quantum Computing | Andrew Steane | 2013 | Link |
| Quantum Computation and Quantum Information | Michael Nielsen and Isaac Chuang | 2010 | Link |