How to implement a doubly controlled phase gate?

[quantumadmin]

A doubly controlled phase gate is a multi-qubit gate that introduces a phase factor to the quantum state of the target qubit if all of the control qubits are in the |1⟩ state. It's a generalization of the controlled phase gate (CZ gate) and can be implemented using basic quantum gates in Qiskit.

Here's how you can implement a doubly controlled phase gate (also known as a Toffoli-like gate) using Qiskit:

Code: Select all

from qiskit import QuantumCircuit, Aer, execute

# Create a 3-qubit quantum circuit
qc = QuantumCircuit(3)

# Apply the doubly controlled phase gate
qc.cu1(theta, 0, 2)  # Controlled-Z gate between qubit 0 and qubit 2
qc.cx(1, 2)           # Controlled-X (CNOT) gate between qubit 1 and qubit 2
qc.cu1(-theta, 0, 2)  # Controlled-Z gate with a negative phase between qubit 0 and qubit 2
qc.cx(1, 2)           # Controlled-X (CNOT) gate between qubit 1 and qubit 2
qc.cu1(theta, 0, 2)   # Controlled-Z gate between qubit 0 and qubit 2

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

print("Final state vector:")
print(statevector)

In this example, the doubly controlled phase gate is constructed using a sequence of controlled-Z and controlled-X (CNOT) gates. The cu1 gate is used for the controlled-Z operations with the appropriate phase factor (theta). The target qubit is qubit 2, and the control qubits are qubit 0 and qubit 1.

The theta parameter determines the phase factor applied by the gate. You can adjust this value according to your requirements.

Please note that this is a simulation of the doubly controlled phase gate using a state vector simulator. On actual quantum hardware, the gate might be implemented using a different set of gates due to hardware constraints and gate decompositions.