Phase kickback is an interesting and useful phenomenon that is used in the design principles of many quantum algorithms.
We apply a Controlled-NOT operator, but the controller qubit will be affected!
CNOT is an operator defined on two qubits:
\[ CNOT = \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{pmatrix}\]
Its effect is very simple: if the state of the first qubit is one, then the state of the second qubit is flipped.
If the state of the first qubit is zero, then the state of the second qubit remains the same.
In summary:CNOT refers to as Controlled-NOT: NOT operator is applied in a controlled way.
Let’s create a quantum circuit with two qubits, say q[1] and q[0] in the reading order of Qiskit.
We start in quantum state |01> : - set the state of q[1] to |0> and - set the state of q[0] to |1> .
Apply Hadamard to both qubits.
Apply CNOT operator, where the controller qubit is q[1] and the target qubit is q[0].
Apply Hadamard to both qubits.
Measure the outcomes.
from qiskit import QuantumRegister, ClassicalRegister, QuantumCircuit, execute, Aer
qc = QuantumCircuit(2)
qc.x(0)
qc.h(0)
qc.h(1)
qc.cx(1, 0)
qc.h(0)
qc.h(1)
qc.measure_all(2, 2)
display(qc.draw(output='mpl',reverse_bits=True))
job = execute(qc,Aer.get_backend('qasm_simulator'),shots=100)
counts = job.result().get_counts(qc)
print("the measurument result is",counts)
the measurument result is {'11': 100}from qiskit.visualization import plot_bloch_multivector, plot_state_qsphere
out = execute(qc,Aer.get_backend('statevector_simulator')).result().get_statevector()
plot_bloch_multivector(out)
The CNOT-gate does not have any directly measurable implications. However, the control qubit switches its phase. It takes on the phase of the target qubit.
For the phase of the target qubit is kicked up to the control qubit i.e. control qubit takes up the phase of the target qubit, we call this phenomenon phase kickback.